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Belyaev +U.S. Bank, Minneapolis, MN, USA +February 1, 2023 +Abstract +A new approach to Local Volatility implementation in the interest +rate model is presented. The major tool of this approach is a small +volatility approximation. This approximation works very well and it +can be used to calibrate all ATM swaptions. It works fast and accu- +rate. In order to reproduce all available swaption prices we need to +take into account the dependence of forward volatility on the current +swap rate. Here we assume that forward volatility is a deterministic +function on strike, tenor, and expiration at every point on the grid. +We determine these functions and apply them in Monte-Carlo calcu- +lations. +It was demonstrated that this approach works well. However, in +the case of short term and low tenor swaptions we observed errors in +swaption pricing. To fix this problem we need to modify the scenario +generation process. +1 +Introduction +Local Volatility Model was presented by Dupire [1] in 1994. According to this +model forward volatility is a deterministic function of a current underlying +S(t) price and time. Model dynamics in this case has the following form: +dS(t) = µ(t)S(t)dt + σL(S(t), t)S(t)dW(t); +(1) +where µ(T) = r(t)−y(t) is a arbitrage free drift; r(t) and y(t) are risk free rate +and dividend yield; dW(t) is Brownian motion; σL(S(t), t) is a deterministic +Local Volatility. +1 +arXiv:2301.13595v1 [q-fin.PR] 31 Jan 2023 + +Representing the option price as a function of forward one C(F(T), X, T) +where F(T) = S(0)e +� T +0 (r(t)−y(t))dt we have +σ2 +L(X, t) = +∂C +∂T +1 +2X2 ∂2C +∂X2 +. +(2) +where C(F(T), X, T) is a call option price; X is a strike. +Having arbitrage-free interpolated/extrapolated volatility surface we can +calculate local volatility by eq.(2) and generate scenarios which are perfectly +calibrated to all available option prices. This procedure is deterministic and +fast. +Later Gatheral [2] derived the following formula for local volatility which +is expressed in terms of implied volatility itself: +σ2 +L(X, t) = +∂w +∂T +1 − y +w +∂w +∂y + 1 +4 +� +− 1 +4 − 1 +w + y2 +w2 +� � +∂w +∂y +�2 + 1 +2 +∂2w +∂y2 +; +(3) +where w(y, T) is an implied variance of the option with strike X and time +to expiration T; y = ln(X/F(T)). The formula makes it easier to calculate +Local Volatility. +In the case of normal volatilities models where +dS(t) = µ(t)dt + σ(S(t), t)dW(t); +(4) +µ(T) = ∂F(T) +∂T , formula for Local Volatility is also known [3]: +σ2 +L(X, t) = +dw +dT +1 − y +w +∂w +∂y + 1 +4 +� +− 1 +w + y2 +w2 +� � +∂w +∂y +�2 + 1 +2 +∂2w +∂y2 +. +(5) +Here we use the following notation: y = X − F(T). +To calibrate interest rate model we will use small volatility approximation +[4]. This approximation works very well and it means that bond price dy- +namic approximately is a normal one. So, Local Volatility can be calculated +according to eq.(5). +In Section 2 we describe a Small Volatility approximation and demon- +strate its accuracy. This approximation can be used to calibrate the model +for selected OTM strikes as well. In Section 3 we apply this approximation in +order to to calculate deterministic sensitivities to strikes at every point of the +2 + +grid. In Section 4 fixed tenor dynamics and forward volatility calculation are +discussed. In Section 5 we use these sensitivities to calculate current forward +volatilities according to eq.(5) to generate interest rate scenarios. +All calculations are completed for March 29, 2022 SOFR rate and swap- +tions market data. Even though the fact that the approach works well we +observe relatively big errors in case of short-term and low-tenor swaptions. +2 +Small Volatility Approximation +Here we consider HJM interest rate model. HJM model [5] has the following +dynamics: +df(t, T) = α(t, T)dt + σ(t, T)dW(t); +(6) +where f(t, T) is a forward rate: +B(t, T) = e−� T +t f(t,τ)dτ; +(7) +B(t, T) is a zero coupon risk-free bond; σ(t, T) is a normal volatility; dW(t, T) +is a Brownian motion; and +α(t, T) = σ(t, T) +� T +t +σ(t, τ)dτ +(8) +is a drift. +The drift is chosen to satisfy martingale condition on bond prices +B(0, T) = +� +e−� t +0 r(τ)dτB(t, T) +� +; ∀t ∈ [0, T]. +(9) +Distribution of discounted bond prices at time T can be presented in the +following form: +e−� T +0 r(τ)dτB(T, T1) = += B(0, T1)e−� T +0 dτ � T1 +τ +α(τ,t)dt−� T +0 dW(τ)� T1 +τ +σ(τ,t)dt = += B(0, T1) +� +1 − +� T +0 dW(τ) +� T1 +τ +σ(τ, t)dt + o(σ) +� +. +(10) +3 + +It means that the distribution of SOFR swap present value is: +PV (T) = e−� T +0 r(t)dt +N +� +n=1 +B(T, Tn) +� +rs + 1 − B(T, Tn−1) +B(T, Tn) +� += += e−� T +0 r(t)dt +� +rs +N +� +n=1 +B(T, Tn) − B(T, T) + B(T, TN) +� +≃ +≃ (rs − rATM) +N +� +n=1 +B(0, Tn) + Σ(T, N)ξ +√ +T; +(11) +where Tn are times of payments; rs and rATM = B(0,T)−B(0,TN) +� +n=1NB(0,Tn) are swap rate +and ATM rate; T0 = T. +Volatility Σ(T, N) is calculated according to the following formulas: +Σ2(T, N)T = +� T +0 v2(t, Ndt)dt; +v(t, N) = rs +N +� +n=1 +B(0, Tn) +� Tn +t +σ(t, τ)dτ − +−B(0.T) +� T +t +σ(t, τ)dτ + B(0, TN) +� TN +t +σ(t, τ)dτ. +(12) +In order to calibrate the interest rate model we can use interpolated +volatilities or to assume that all unknown volatilities are equal to each other. +Here we use the first approach. +In the case of the grid with 3 month time step the first swaption is a tenor +1 expired in 3 months. According to (12) we have: +v(dt, 1) = rsB(0, 5dt) +4 +� +k=0 +(k + 1)σ(0, k)dt − +−B(0, dt)σ(0, 0)dt + ++B(0, 5dt) +4 +� +k=0 +(k + 1)σ(0, k)dt; +(13) +where dt = 0.25. +Assuming that all unknown volatilities are equal in (13) +σ(0, k) = σ(0, 0); ∀k < 5; +(14) +we can calculate volatilities in (13). +4 + +We can apply this procedure to other expirations and tenors, taking into +account already defined volatility values. For every available next tenor and +time to expiration we obtain the following equation: +Σ2(i, j) = Aσ2 + Bσ + C; +(15) +where A, B, C are factors that can be determined by using bond prices and +already calculated volatilities; σ is an unknown forward volatility. +Schematically calibration process can be represented in the following way: +0 +0.25 +0.5 +0.75 +1.0 +0 +0.25 +0.5 +0.75 +- +Te + Tenor +6 +Te +� +� +� +� +� +� +� +� +� +� +� +� +� +�� +f +f +f +f +f +f +f +f +f +f +f +v +f +f +f +f +f +f +v +f +f +f +Σ(1, 5) +Σ(2, 6) +σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 5) σ(0, 5) +σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) +where Σ(1, 5) and Σ(2, 6) are input volatilities of 1-year swaptions with +3- and 6-months to expiration. +This procedure leads to a good calibrated prices for all ATM swaptions, +see Fig.1 where we compare input ATM volatilities with volatilities calculated +from Monte-Carlo prices. +Forward volatility surface is shown on Fig.2. +3 +Sensitivity Calculation. +In the previous section we describe how to calibrate forward volatilities to +reproduce all available ATM swaptions prices. This procedure can be applied +to find forward volatilities calibrated to all swaption prices with shifted strikes +from their ATM values. +5 + +Sswaption quotes are available within ±2% shifts. The calibration process +is similar to the ATM swaption calibration. The only difference is a change of +ATM volatility to the shifted one for selected strike. The calibration quality +is good (see Fig.3). +In the selected data set for SOFR swaption volatility smile, we have +strikes up to ±2% rate shift from ATM rate. This smile can be fitted by +quadratic polynomial as it is demonstrated in Fig.4. Splines can also be used +but obtained results are very similar. It means that we can use quadratic +interpolation. +However, we need to take care of extrapolation, due to the swap rates can +go out of [−2%, 2%] range rate shifts. This extrapolation need to be contin- +uously differentiable and it would be very helpful if we can use extrapolation +with limited volatility, in order to to use small volatility approximation. +We choose the following quadratic extrapolation: +v(x) = αu + βu(x − x0) + γu(x − x0)2; +x0 < x < xu; +(16) +v(x) = αd + βd(x − (−x0)) + γd(x − (−x0))2; +xd < x < −x0; +where x0 = 2% is the current maximal available rate shift in input data; final +points of extrapolation are chosen as xd = −10% and xu = 10%. +To obtain continuously differentiable function, we need to have +αu = v(x0); +αd = v(−x0); +βu = dv(x) +dx +|x=x0 +; +βd = dv(x) +dx +|x=−x0 +. +(17) +To get zero slope at the end points xu,d we have +γu(xu − x0) = −1 +2βu; +γd(xd − (−x0)) = −1 +2βd; +(18) +Above end points we assume that +v(x) = +� +v(xd); +if x < xd; +v(xu); +if x > xu; +(19) +It leads to the smooth volatility curve depicted in Fig.5. +6 + +4 +Fixed Tenor Dynamics +Before going to Scenario Generation Process let us consider fixed tenor dy- +namics. +In the case of selected tenor N we consider the following ATM +swaption values distribution which present value is: +PV (S(T, N)) = e−� T +0 r(τ)dτ +� +rATM(T, N) +N +� +n=1 +B(T, Tn) − 1 + B(T, TN) +� += += e−� T +0 r(τ)dτ +�B(0.T) − B(0, TN) +�N +n=1 B(0, Tn) +N +� +n=1 +B(T, Tn) − 1 + B(T, TN) +� +.(20) +In small volatility approximation Eq.(20) has the following form: +PV (S(T, N)) = Σ(T, N)ξ +√ +T. +(21) +In the case of non-zero constant difference between current swaption rate and +ATM rate +δ = r(T, N) − rATM(T, N); +(22) +swap present value distribution is +PV (S(T, N, δ)) = δ +N +� +n=1 +B(0, Tn) + Σ(T, N, δ)ξ +√ +T = += A(T, N) +� +δ + Σ(T, N, δ) +A(T, N) ξ +� +; +(23) +where Σ(T, N, δ) is implied volatility of fixed tenor underlying for selected +rate shift; +A(T, N) = +N +� +n=1 +B(0, tn). +(24) +As we can see from (23) we can use (5) to determine forward volatility +assuming deterministic dependence of local forward volatility for selected +point on the grid. +7 + +5 +Local Volatility Scenarios +In the scenario generation process we can use formula for Local Volatility +of normal volatility model. Here we assume that local forward volatility on +every point on the grid can be defined deterministically from eq.(5). Below +we describe this approach. +In addition to forward volatility, the formula depends on total variance +w. +Deterministic variance can be calculated in the following form: +w(−2%, n, k) = v2 +f(−2%, n − 1, k)dt + w(−2%, n − 1, k); +w(ATM, n, k) = v2 +f(ATM, n − 1, k)dt + w(ATM, n − 1, k); +w(2%, n, k) = v2 +f(2%, n − 1, k)dt + w(2%, n − 1, k); +(25) +where vf(shift, n, k) is a forward volatility for selected time step n and point +on the grid k. +Then we use interpolation-extrapolation formulas to get function w(x, n, k) +for every point on the grid. +This procedure gives us all deterministic functions needed to calculate +forward volatility according to equation (5). +To generate scenarios we need to calculate current swap rate which is a +difference between current and initial ATM swap rates. +Initial ATM swap rate is: +rS(0, ti, tenor) = B(0, ti) − B(0, ti + tenor) +�tenor +n=1 B(0, ti + n) +. +(26) +Observed swap rate at time ti is: +rS(ti, ti, tenor) = 1 − B(ti, ti + tenor) +�tenor +n=1 B(ti + n) +. +(27) +So, the current rate strike is +X(ti, ti, tenor) = rS(ti, ti, tenor) − rS(0, ti, tenor). +(28) +In the case of tenor = 1 we assume that this strike is the same for all +time steps between 0 and tenor = 1. For tenor = 2 we use strike for all +times between tenor = 1 and tenor = 2 only etc.. As was mentioned in the +8 + +previous section this procedure works. We use it because we already have +calculated volatility sensitivities on swap rates. +In calculations we apply +calculated variance for selected strike and point on the grid (25). +We apply this procedure and generated 100,000 scenarios. ATM swaption +prices for tenors 1, 5, 10 and 30 are shown in Fig.6. +1 Year Tenor 1 smile looks good (see Fig.7). All Tenor 1 expirations also +in a good agreement with maximal errors for 5-year expiration, see Fig.8. +All other expirations of Tenor 1 swaptions are in better agreement with the +input prices, Fig.9. +Errors in higher tenors swaptions are smaller. You can see it in case of 5 +Year Tenor 2 swaption, Fig.10. +Note, that using 1 month time step improve the quality of calibrated +scenario set, see Fig.11. +6 +Conclusion +Implementation of Local Volatility Model in interest rate model is presented. +Calibration is deterministic, it works fast and is accurate. Observed short +term and low tenor swaption errors can be improved by modifying scenario +generation process. +Approach and results were presented on QuantMinds International Con- +ference 2022, Barcelona, Spain [6]. +9 + +References +[1] Dupire, B. (1994). ”Pricing With a Smile.” Risk 7, pp. 18-20. +[2] Gatheral, J. (2006). ”The Volatility Surface: A Practitioner´ıs Guide.” +New York, NY: John Wiley & Sons. +[3] Costeanu, V. & Pirjol D. ”Asymptotic expansion for the normal im- +plied volatility in local volatility models” , arXiv:1105.3359v1, q-fin.CP, +(2011); +[4] V.M. Belyaev : “Swaption Prices in HJM Model. Nonparametric Fit”, +arXiv:1697.01619, [ q-fin.PR], (2016); QuantMinds International Con- +ferences (2017-2021); +[5] Heath, D., R. Jarrow, and A. Morton (1990): +”Bond Pricing and +the Term Structure of Interest Rates: A Discrete Time Approxima- +tion”.Journal of Financial and Quantitative Analysis, 25: 419−440. +[6] V.M. Belyaev : “Local Volatility in Interest Rate Models”, QuantMinds +International Conference 2022, Barcelona, Spain. +10 + +FIGURES +Figure 1: ATM Swaption Normal Volatilities. +11 + +Tenor 1 +1.6% +1.4% +1.2% +1.0% +0.8% +●Vol +0.6% +OMC +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Time to ExpirationTenor 5 +1.4% +1.2% +1.0% +0.8% +0.6% +●Vol +0.4% +OMC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Time to ExpirationTenor10 +1.2% +1.0% +0.8% +0.6% +.Vol +0.4% +OMC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Time to ExpirationTenor30 +1.2% +1.0% +0.8% +0.6% +.Vol +0.4% +OMC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Time to ExpirationFigure 2: ATM Forward Volatility Surface. +12 + +0.01 +0.008 +Forward Volatility +0.006 +0.004 +0.002 +30 +07 +25 +30 +20 +25 +20 +15 +15 +10 +10 +5 +5 +Timeto Exp +Tenor +0 +0Figure 3: OTM Swaption Normal Volatilities. +13 + +Tenor 10, strike shift-2% +1.6% +1.4% +1.2% +1.0% +0.8% +●Vol +0.6% +0.4% +●MC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Timeto ExpirationTenor 10, strike shift 2% +1.80% +1.60% +1.40% +1.20% +1.00% +0.80% +Vol +0.60% +OMC +0.40% +0.20% +0.00% +0 +5 +10 +15 +20 +25 +30 +35 +Time to ExpirationTenor 30, strike shift-2% +1.6% +1.4% +1.2% +1.0% +0.8% +●Vol +0.6% +0.4% +●MC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Timeto ExpirationTenor30,strike shift 2% +1.60% +1.40% +1.20% +1.00% +0.80% +Vol +0.60% +0.40% +OMC +0.20% +0.00% +0 +5 +10 +15 +20 +25 +30 +35 +Time to ExpirationTenor 1, strike shift-2% +1.8% +1.6% +1.4% +1.2% +1.0% +0.8% +.Vol +0.6% +●MC +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Timeto ExpirationTenor 1, strike shift 2% +2.00% +1.80% +1.60% +1.40% +1.20% +1.00% +0.80% +.Vol +0.60% +OMC +0.40% +0.20% +0.00% +0 +5 +10 +15 +20 +25 +30 +35 +Time to ExpirationTenor 5, strike shift -2% +1.6% +1.4% +1.2% +1.0% +0.8% +0.6% +·Vol +0.4% +●MC +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30 +Timeto ExpirationTenor 5, strike shift 2% +2.00% +1.80% +1.60% +1.40% +1.20% +1.00% +0.80% +.Vol +0.60% +OMC +0.40% +0.20% +0.00% +0 +5 +10 +15 +20 +25 +30 +35 +Time to ExpirationFigure 4: 1 Year Tenor 1. Swaption volatility smile. +14 + +2.0% +1.9% +y= 5.668x*+ 0.0831x + 0.0148 +R²=0.9971 +1.8% +1.7% +1.6% +Input +....... Inter polation. +1.5% +1.4% +1.3% +1.2% +-2.5% +-2.0% +-1.5% +1.0% +0.5% +0.0% +0.5% +1.0% +1.5% +2.0% +2.5%Figure 5: 1 Year Tenor 1 Volatility Extrapolation. +Figure 6: ATM Swaption Normal Volatilities with smile. +15 + +3.5% +3.0% +2.5% +2.0% +Input +Model +1.0% +0.5% +0.0% +-10% +-5% +%0 +5% +10%Tenor 1 +1.8% +1.6% +1.4% +1.2% +1.0% +oInput +0.8% +eMcVol +0.6% +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30Tenor 5 +1.4% +1.2% +1.0% +0.8% +oInput +0.6% +eMcVol +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30Tenor 10 +1.2% +1.0% +0.8% +0.6% +oInput +o McVol +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30Tenor 30 +1.2% +1.0% +0.8% +0.6% +oInput +o McVol +0.4% +0.2% +0.0% +0 +5 +10 +15 +20 +25 +30Figure 7: +Swaption volatility smile. +16 + +1 Year, Tenor 1 +2.0% +1.8% +1.6% +. +1.2% +1.0% +oInput +0.8% +.McVol +0.6% +0.4% +0.2% +0.0% +-2.5% +1.5% +-0.5% +0.5% +1.5% +2.5%Figure 8: +Swaption volatility smile. +17 + +5 Years, Tenor 1 +1.4% +. +1.2% +: +1.0% +0.8% +.Vol +0.6% +OMC +0.4% +0.2% +0.0% +-0.03 +-0.02 +-0.01 +0 +0.01 +0.02 +0.03Figure 9: +Swaption volatility smile. +18 + +10Years,Tenor1 +1.2% +1.0% +. 8.8% +0.6% +.Vol +OMC +0.4% +0.2% +0.0% +-0.03 +-0.02 +-0.01 +0 +0.01 +0.02 +0.03Figure 10: +Swaption volatility smile. +Figure 11: Swaption volatility smiles. 1 month time steps. +19 + +5 Years, Tenor 2 +1.4% +1.2% +0.8% +.Vol +0.6% +OMC +0.4% +0.2% +0.0% +-0.03 +0.02 +-0.01 +0 +0.01 +0.02 +0.033 Years, Tenor 1 +1.8% +1.6% +1.4% +. +1.2% +1.0% +eInput +0.8% +●MC +0.6% +0.4% +0.2% +0.0% +-2.5% +1.5% +%5'0- +0.5% +1.5% +2.5%5 Years, Tenor 1 +1.4% +1.2% +1.0% +0.8% +Input +0.6% +●MC +0.4% +0.2% +0.0% +-2.5% +-1.5% +%5'0- +0.5% +1.5% +2.5%10 Years, Tenor 1 +1.2% +1.0% +0.6% +oInput +●MC +0.4% +0.2% +0.0% +-2.5% +1.5% +-0.5% +0.5% +1.5% +2.5%30 Years, Tenor 1 +0.8% +0.7% +0.6% +0.5% +0.4% +oInput +0.3% +●MC +0.2% +0.1% +0.0% +-2.5% +1.5% +%5'0- +0.5% +1.5% +2.5% \ No newline at end of file diff --git a/0NFRT4oBgHgl3EQfkTex/content/tmp_files/load_file.txt b/0NFRT4oBgHgl3EQfkTex/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34fab28dfb534299c7f42a69040a5904f3ddc74f --- /dev/null +++ b/0NFRT4oBgHgl3EQfkTex/content/tmp_files/load_file.txt @@ -0,0 +1,519 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf,len=518 +page_content='Local Volatility in Interest Rate Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Belyaev U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Bank, Minneapolis, MN, USA February 1, 2023 Abstract A new approach to Local Volatility implementation in the interest rate model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The major tool of this approach is a small volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This approximation works very well and it can be used to calibrate all ATM swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' It works fast and accu- rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In order to reproduce all available swaption prices we need to take into account the dependence of forward volatility on the current swap rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Here we assume that forward volatility is a deterministic function on strike, tenor, and expiration at every point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' We determine these functions and apply them in Monte-Carlo calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' It was demonstrated that this approach works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' However, in the case of short term and low tenor swaptions we observed errors in swaption pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' To fix this problem we need to modify the scenario generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 1 Introduction Local Volatility Model was presented by Dupire [1] in 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' According to this model forward volatility is a deterministic function of a current underlying S(t) price and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Model dynamics in this case has the following form: dS(t) = µ(t)S(t)dt + σL(S(t), t)S(t)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (1) where µ(T) = r(t)−y(t) is a arbitrage free drift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' r(t) and y(t) are risk free rate and dividend yield;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' dW(t) is Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' σL(S(t), t) is a deterministic Local Volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='13595v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='PR] 31 Jan 2023 Representing the option price as a function of forward one C(F(T), X, T) where F(T) = S(0)e � T 0 (r(t)−y(t))dt we have σ2 L(X, t) = ∂C ∂T 1 2X2 ∂2C ∂X2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (2) where C(F(T), X, T) is a call option price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' X is a strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Having arbitrage-free interpolated/extrapolated volatility surface we can calculate local volatility by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (2) and generate scenarios which are perfectly calibrated to all available option prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This procedure is deterministic and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Later Gatheral [2] derived the following formula for local volatility which is expressed in terms of implied volatility itself: σ2 L(X, t) = ∂w ∂T 1 − y w ∂w ∂y + 1 4 � − 1 4 − 1 w + y2 w2 � � ∂w ∂y �2 + 1 2 ∂2w ∂y2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (3) where w(y, T) is an implied variance of the option with strike X and time to expiration T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' y = ln(X/F(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The formula makes it easier to calculate Local Volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In the case of normal volatilities models where dS(t) = µ(t)dt + σ(S(t), t)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (4) µ(T) = ∂F(T) ∂T , formula for Local Volatility is also known [3]: σ2 L(X, t) = dw dT 1 − y w ∂w ∂y + 1 4 � − 1 w + y2 w2 � � ∂w ∂y �2 + 1 2 ∂2w ∂y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (5) Here we use the following notation: y = X − F(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' To calibrate interest rate model we will use small volatility approximation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This approximation works very well and it means that bond price dy- namic approximately is a normal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' So, Local Volatility can be calculated according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In Section 2 we describe a Small Volatility approximation and demon- strate its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This approximation can be used to calibrate the model for selected OTM strikes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In Section 3 we apply this approximation in order to to calculate deterministic sensitivities to strikes at every point of the 2 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In Section 4 fixed tenor dynamics and forward volatility calculation are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In Section 5 we use these sensitivities to calculate current forward volatilities according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (5) to generate interest rate scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' All calculations are completed for March 29, 2022 SOFR rate and swap- tions market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Even though the fact that the approach works well we observe relatively big errors in case of short-term and low-tenor swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 2 Small Volatility Approximation Here we consider HJM interest rate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' HJM model [5] has the following dynamics: df(t, T) = α(t, T)dt + σ(t, T)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (6) where f(t, T) is a forward rate: B(t, T) = e−� T t f(t,τ)dτ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (7) B(t, T) is a zero coupon risk-free bond;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' σ(t, T) is a normal volatility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' dW(t, T) is a Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' and α(t, T) = σ(t, T) � T t σ(t, τ)dτ (8) is a drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The drift is chosen to satisfy martingale condition on bond prices B(0, T) = � e−� t 0 r(τ)dτB(t, T) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (9) Distribution of discounted bond prices at time T can be presented in the following form: e−� T 0 r(τ)dτB(T, T1) = = B(0, T1)e−� T 0 dτ � T1 τ α(τ,t)dt−� T 0 dW(τ)� T1 τ σ(τ,t)dt = = B(0, T1) � 1 − � T 0 dW(τ) � T1 τ σ(τ, t)dt + o(σ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (10) 3 It means that the distribution of SOFR swap present value is: PV (T) = e−� T 0 r(t)dt N � n=1 B(T, Tn) � rs + 1 − B(T, Tn−1) B(T, Tn) � = = e−� T 0 r(t)dt � rs N � n=1 B(T, Tn) − B(T, T) + B(T, TN) � ≃ ≃ (rs − rATM) N � n=1 B(0, Tn) + Σ(T, N)ξ √ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (11) where Tn are times of payments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' rs and rATM = B(0,T)−B(0,TN) � n=1NB(0,Tn) are swap rate and ATM rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' T0 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Volatility Σ(T, N) is calculated according to the following formulas: Σ2(T, N)T = � T 0 v2(t, Ndt)dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' v(t, N) = rs N � n=1 B(0, Tn) � Tn t σ(t, τ)dτ − −B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='T) � T t σ(t, τ)dτ + B(0, TN) � TN t σ(t, τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (12) In order to calibrate the interest rate model we can use interpolated volatilities or to assume that all unknown volatilities are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Here we use the first approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In the case of the grid with 3 month time step the first swaption is a tenor 1 expired in 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' According to (12) we have: v(dt, 1) = rsB(0, 5dt) 4 � k=0 (k + 1)σ(0, k)dt − −B(0, dt)σ(0, 0)dt + +B(0, 5dt) 4 � k=0 (k + 1)σ(0, k)dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (13) where dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Assuming that all unknown volatilities are equal in (13) σ(0, k) = σ(0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ∀k < 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (14) we can calculate volatilities in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 4 We can apply this procedure to other expirations and tenors, taking into account already defined volatility values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' For every available next tenor and time to expiration we obtain the following equation: Σ2(i, j) = Aσ2 + Bσ + C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (15) where A, B, C are factors that can be determined by using bond prices and already calculated volatilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' σ is an unknown forward volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Schematically calibration process can be represented in the following way: 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='75 Te + Tenor 6 Te � � � � � � � � � � � � � �� f f f f f f f f f f f v f f f f f f v f f f Σ(1, 5) Σ(2, 6) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 5) σ(0, 5) σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) where Σ(1, 5) and Σ(2, 6) are input volatilities of 1-year swaptions with 3- and 6-months to expiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This procedure leads to a good calibrated prices for all ATM swaptions, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='1 where we compare input ATM volatilities with volatilities calculated from Monte-Carlo prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Forward volatility surface is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 3 Sensitivity Calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In the previous section we describe how to calibrate forward volatilities to reproduce all available ATM swaptions prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This procedure can be applied to find forward volatilities calibrated to all swaption prices with shifted strikes from their ATM values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 5 Sswaption quotes are available within ±2% shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The calibration process is similar to the ATM swaption calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The only difference is a change of ATM volatility to the shifted one for selected strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' The calibration quality is good (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In the selected data set for SOFR swaption volatility smile, we have strikes up to ±2% rate shift from ATM rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This smile can be fitted by quadratic polynomial as it is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Splines can also be used but obtained results are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' It means that we can use quadratic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' However, we need to take care of extrapolation, due to the swap rates can go out of [−2%, 2%] range rate shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This extrapolation need to be contin- uously differentiable and it would be very helpful if we can use extrapolation with limited volatility, in order to to use small volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' We choose the following quadratic extrapolation: v(x) = αu + βu(x − x0) + γu(x − x0)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' x0 < x < xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (16) v(x) = αd + βd(x − (−x0)) + γd(x − (−x0))2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' xd < x < −x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' where x0 = 2% is the current maximal available rate shift in input data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' final points of extrapolation are chosen as xd = −10% and xu = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' To obtain continuously differentiable function, we need to have αu = v(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' αd = v(−x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' βu = dv(x) dx |x=x0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' βd = dv(x) dx |x=−x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (17) To get zero slope at the end points xu,d we have γu(xu − x0) = −1 2βu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' γd(xd − (−x0)) = −1 2βd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (18) Above end points we assume that v(x) = � v(xd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' if x < xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' v(xu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' if x > xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (19) It leads to the smooth volatility curve depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 6 4 Fixed Tenor Dynamics Before going to Scenario Generation Process let us consider fixed tenor dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In the case of selected tenor N we consider the following ATM swaption values distribution which present value is: PV (S(T, N)) = e−� T 0 r(τ)dτ � rATM(T, N) N � n=1 B(T, Tn) − 1 + B(T, TN) � = = e−� T 0 r(τ)dτ �B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='T) − B(0, TN) �N n=1 B(0, Tn) N � n=1 B(T, Tn) − 1 + B(T, TN) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (20) In small volatility approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (20) has the following form: PV (S(T, N)) = Σ(T, N)ξ √ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (21) In the case of non-zero constant difference between current swaption rate and ATM rate δ = r(T, N) − rATM(T, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (22) swap present value distribution is PV (S(T, N, δ)) = δ N � n=1 B(0, Tn) + Σ(T, N, δ)ξ √ T = = A(T, N) � δ + Σ(T, N, δ) A(T, N) ξ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (23) where Σ(T, N, δ) is implied volatility of fixed tenor underlying for selected rate shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' A(T, N) = N � n=1 B(0, tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (24) As we can see from (23) we can use (5) to determine forward volatility assuming deterministic dependence of local forward volatility for selected point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 7 5 Local Volatility Scenarios In the scenario generation process we can use formula for Local Volatility of normal volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Here we assume that local forward volatility on every point on the grid can be defined deterministically from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Below we describe this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In addition to forward volatility, the formula depends on total variance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Deterministic variance can be calculated in the following form: w(−2%, n, k) = v2 f(−2%, n − 1, k)dt + w(−2%, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' w(ATM, n, k) = v2 f(ATM, n − 1, k)dt + w(ATM, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' w(2%, n, k) = v2 f(2%, n − 1, k)dt + w(2%, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (25) where vf(shift, n, k) is a forward volatility for selected time step n and point on the grid k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Then we use interpolation-extrapolation formulas to get function w(x, n, k) for every point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' This procedure gives us all deterministic functions needed to calculate forward volatility according to equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' To generate scenarios we need to calculate current swap rate which is a difference between current and initial ATM swap rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Initial ATM swap rate is: rS(0, ti, tenor) = B(0, ti) − B(0, ti + tenor) �tenor n=1 B(0, ti + n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (26) Observed swap rate at time ti is: rS(ti, ti, tenor) = 1 − B(ti, ti + tenor) �tenor n=1 B(ti + n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (27) So, the current rate strike is X(ti, ti, tenor) = rS(ti, ti, tenor) − rS(0, ti, tenor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (28) In the case of tenor = 1 we assume that this strike is the same for all time steps between 0 and tenor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' For tenor = 2 we use strike for all times between tenor = 1 and tenor = 2 only etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='. As was mentioned in the 8 previous section this procedure works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' We use it because we already have calculated volatility sensitivities on swap rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' In calculations we apply calculated variance for selected strike and point on the grid (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' We apply this procedure and generated 100,000 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ATM swaption prices for tenors 1, 5, 10 and 30 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 1 Year Tenor 1 smile looks good (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' All Tenor 1 expirations also in a good agreement with maximal errors for 5-year expiration, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' All other expirations of Tenor 1 swaptions are in better agreement with the input prices, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Errors in higher tenors swaptions are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' You can see it in case of 5 Year Tenor 2 swaption, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Note, that using 1 month time step improve the quality of calibrated scenario set, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 6 Conclusion Implementation of Local Volatility Model in interest rate model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Calibration is deterministic, it works fast and is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Observed short term and low tenor swaption errors can be improved by modifying scenario generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Approach and results were presented on QuantMinds International Con- ference 2022, Barcelona, Spain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 9 References [1] Dupire, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ”Pricing With a Smile.” Risk 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 18-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' [2] Gatheral, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ”The Volatility Surface: A Practitioner´ıs Guide.” New York, NY: John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' [3] Costeanu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' & Pirjol D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' ”Asymptotic expansion for the normal im- plied volatility in local volatility models” , arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='3359v1, q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='CP, (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Belyaev : “Swaption Prices in HJM Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Nonparametric Fit”, arXiv:1697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='01619, [ q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='PR], (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' QuantMinds International Con- ferences (2017-2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' [5] Heath, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Jarrow, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Morton (1990): ”Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approxima- tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='Journal of Financial and Quantitative Analysis, 25: 419−440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' Belyaev : “Local Volatility in Interest Rate Models”, QuantMinds International Conference 2022, Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 10 FIGURES Figure 1: ATM Swaption Normal Volatilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 11 Tenor 1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0 5 10 15 20 25 30 Time to ExpirationTenor 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='4% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='6% Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='4% OMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0 5 10 15 20 25 30 Time to ExpirationTenor10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='6% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='4% OMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0 5 10 15 20 25 30 Time to ExpirationTenor30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='6% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='4% OMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='0% 0 5 10 15 20 25 30 Time to ExpirationFigure 2: ATM Forward Volatility Surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='008 Forward Volatility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} +page_content='002 30 07 25 30 20 25 20 15 15 10 10 5 5 Timeto Exp Tenor 0 0Figure 3: OTM Swaption Normal Volatilities.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'} diff --git a/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf b/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aecf68f0e544d8d21ca9b2b06c8d481220f119a7 --- /dev/null +++ b/29FQT4oBgHgl3EQfGTVE/content/2301.13244v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f3fb24ca2d61541b1d945f030fb9f15a6ab2b045ddb122d59915df4e9beed81 +size 5140466 diff --git a/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/2301.01688v1.pdf.txt b/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/2301.01688v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a433ceb24e0ea3c40bc98fb764b3903652f1f13c --- /dev/null +++ b/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/2301.01688v1.pdf.txt @@ -0,0 +1,1641 @@ +arXiv:2301.01688v1 [math.OC] 4 Jan 2023 +Feedback Stabilization of Tank-Liquid +System with Robustness to Surface Tension +Iasson Karafyllis, Filippos Vokos +Department of Mathematics, +National Technical University of Athens, +Zografou Campus, 15780, Athens, Greece, +emails: iasonkar@central.ntua.gr, fivojean@mail.ntua.gr +Miroslav Krstic +Department of Mechanical and Aerospace Eng., +University of California, La Jolla, San Diego, +CA 92093-0411, USA email: krstic@ucsd.edu +January 5, 2023 +Abstract +We construct a robust stabilizing feedback law for the viscous Saint- +Venant system of Partial Differential Equations (PDEs) with surface tension +and without wall friction. The Saint-Venant system describes the move- +ment of a tank which contains a viscous liquid. We assume constant con- +tact angles between the liquid and the walls of the tank and we achieve +a spill-free exponential stabilization with robustness to surface tension by +using a Control Lyapunov Functional (CLF). The proposed CLF provides a +parameterized family of sets which approximate the state space from the +interior. Based on the CLF, we construct a nonlinear stabilizing feedback +law which ensures that the closed-loop system converges exponentially to +the desired equilibrium point in the sense of an appropriate norm. +1 +Introduction +The Saint-Venant model, which was derived in [2], constitutes a significant and +very influential mathematical model in fluid mechanics. It is also referred in +literature as the shallow water model. Recent extensions of the Saint-Venant +model take into account various types of forces such as viscous stresses, surface +tension and friction forces (see [10,19,29,33,41,43]). +The feedback stabilization problem of the Saint-Venant model is a challeng- +ing problem. The dominant cases studied in the litterature include the inviscid +model - which ignores forces such as viscous stresses and surface tension - and +1 + +the linearized model (see [1,3–5,11–14,16–18,30,35,36]). In [11,12,14,35,36] +the problem of the movement of an 1-D tank which contains a fluid is stud- +ied. More specifically, [11,12,14] provide controllability results for the Saint- +Venant model without viscosity, without friction and without surface tension, +while [35] suggests a new variational formulation of Saint-Venant equations +and proves the steady-state controllability of the linear approximations of sev- +eral control configurations. In [36] the inviscid Saint-Venant model is studied +and appropriate stabilizing full-state feedback and output feedback control +laws are constructed. In [3–5, 12, 13, 16–18, 30] the movement of a fluid in an +open channel is studied. Stabilization results are provided in [1,3,4,13,17,18]. +In [3, 4, 17, 18, 30] the linearized Saint-Venant model is being used while [1] +deals with a general linear hyperbolic system which appears in Saint-Venant +equations among other linear hyperbolic laws. The works [5, 12, 13, 16] study +the nonlinear Saint-Venant model. In [5] the feedforward control problem of +general nonlinear hyperbolic systems is studied and an application using the +Saint-Venant model with friction is provided. In [12, 13] local convergence of +the state of hyperbolic systems of conservation laws is guaranteed using a strict +Lypaunov function which exploits Riemann invariants. An application to the +inviscid, frictonless Saint-Venant model is provided as well. The paper [16] +achieves regulation of the water flow and level in water-ways using the invis- +cid Saint-Venant model without friction and without surface tension. +Very few studies in the literature deal with the nonlinear viscous Saint- +Venant model that is used for the description of the movement of a tank which +contains an incompressible, Newtonian fluid. The first work that studied the +nonlinear viscous Saint-Venant model without wall friction and without sur- +face tension was [23]. In [23] an appropriate nonlinear feedback law is con- +structed which provides semiglobal stabilization results by following a CLF +methodology. The work [25] extends the results obtained in [23] in the case +where wall friction forces are taken into account. In [25] both the case of a +velocity independent friction coefficient and the general case of friction co- +efficient are studied. A robust with respect to wall friction stabilizing feed- +back law is constructed. Another study which deals with the nonlinear viscous +Saint-Venant model is [24]. In [24] a stabilizing output-feedback control law for +the viscous Saint-Venant PDE system without wall friction and without surface +tension is constructed. The output-feedback control law is utilized through a +functional-observer methodology and a CLF methodology. +The study of the movement of a fluid which interacts with a gas bound- +ary and a solid boundary is inevitably intertwined with the notion of the sur- +face tension and the notions of contact angle and wettability (see [27, 34]). +Surface tension is crucial as it acts in the interface between liquid and gas. +From a mathematical point of view surface tension is very important because +it changes the order of the PDEs (it is expressed by a third order term). Con- +tact angle is the angle at which the fluid surface intersects with a solid bound- +ary as stated in [34], and it is a measure of wettability of the solid surface. +There is a wide literature concerning the topic of contact angles (see for in- +stance [20,21,27,28,37,38,42,43]). The concept of contact angle is significant +2 + +in our study because it provides an additional boundary condition. +In this paper we solve the feedback stabilization problem for a tank con- +taining a liquid modeled by the viscous Saint-Venant system of PDEs with sur- +face tension and without wall friction. We consider the case of constant contact +angles between the liquid and the walls of the tank, as in [37,38]. We utilize a +specific form of the feedback law initially presented in [25], which constitutes +a more general form of the feedback law in [23] with robustness to surface +tension. Indeed, we saw that the proposed feedback law guarantees stabiliza- +tion no matter what the value of the surface tension coefficient is. Therefore, the +knowledge of the surface tension coefficient is not necessary and the feedback +law is independent of the surface tension coefficient. We achieve a spill-free +exponential stabilization, with robustness to surface tension. As in [23–25] we +follow a CLF methodology and we design the feedback law based on an appro- +priate functional, which is the CLF. The CLF determines a specific parameter- +ized set which approximates the state space of the control problem from the +interior. +Although this work presents enough technical similarities with [25], there +are some crucial differences. Firstly, in contrast with [25], the system of PDEs +contains an extra term due to surface tension and does not contain a friction +term. Moreover, in order for the model to be complete and for the problem +to be well-posed, an additional boundary condition is used. The additional +boundary condition is provided by the assumption of a constant contact angle. +Here we use only one CLF while in [25] two different functionals are proposed. +As a consequence this work does not provide a bound for the sup-norm of +the fluid velocity, as in [25], due to the absence of an appropriate functional. +Here the CLF is different from the corresponding one in [25], as it contains an +additional potential energy term due to the effect of the surface tension. +This paper is organized as follows. In Section 2 the control problem is +described as well as its main objective. In Section 3 we provide the intuitive +ideas and the statements of the results of this work along with some auxiliary +lemmas. Section 4 includes all the proofs of the results presented in Section 3. +Finally, Section 5 points out the conclusions of this work and suggests topics +for future research. +Notation +∗ R+ = [0,+∞) is the set of non-negative real numbers. +∗ Let S ⊆ Rn be an open set and let A ⊆ Rn be a set such that S ⊆ A ⊆ cl(S). +By C0(A;Ω), we denote the class of continuous functions on A, which +take values in Ω ⊆ Rm. By Ck(A;Ω), where k ≥ 1 is an integer, we denote +the class of functions on A ⊆ Rn, which takes values in Ω ⊆ Rm and has +continuous derivatives of order k. In other words, the functions of class +Ck(A;Ω) are the functions which have continuous derivatives of order k +in S = int(A) that can be continued continuously to all points in ∂S ∩ A. +When Ω = R then we write C0(A) or Ck(A). When I ⊆ R is an interval and +3 + +G ∈ C1(I) is a function of a single variable, G′(h) denotes the derivative +with respect to h ∈ I. +∗ Let I ⊆ R be an interval, let a < b be given constants and let u : I ×[a,b] → +R be a given function. We utilize the notation u[t] to denote the profile +at certain t ∈ I, i.e., (u[t])(x) = u(t,x) for all x ∈ [a,b]. When u(t,x) is three +times differentiable with respect to x ∈ [a,b], we use the notation ux(t,x), +uxx(t,x) and uxxx(t,x) for the first, second and third derivative of u with +respect to x ∈ [a,b] respectively, i.e., +ux(t,x) = ∂u +∂x (t,x),uxx(t,x) = ∂2 u +∂x2 (t,x) and uxxx(t,x) = ∂3 u +∂x3 (t,x) +When u(t,x) is differentiable with respect to t, we use the notation ut(t,x) +for the derivative of u with respect to t, i.e., +ut(t,x)= ∂u +∂t (t,x) +∗ Given a set U ⊆ Rn, χU denotes the characteristic function of U defined +by +χU(x) := +� +1 +for all x ∈ U +0 +for all x � U +The sign function sgn : R → R is the function defined by +sgn(x) := + +1 +for x > 0 +0 +for x = 0 +−1 +for x < 0 +∗ Consider given constants a,b such that a < b . For p ∈ [1,+∞), Lp(a,b) +denotes the set of equivalence classes of Lebesgue measurable functions +u : (a,b) → R with +∥u∥p := +�� b +a +|u(x)|p dx +�1/p +< +∞. +L∞(a,b) denotes the set of equivalence classes of Lebesgue measurable +functions u : (a,b) → R with +∥u∥∞ := esssup +x∈(a,b) +(|u(x)|) < +∞. +For an integer k ≥ 1, Hk(a,b) denotes the Sobolev space of functions in +L2(a,b) with all its weak derivatives up to order k ≥ 1 in L2(a,b). +4 + +2 +The Control Problem +We want to manipulate the motion of a tank which contains a viscous, Newto- +nian, incompressible liquid. Viscosity is utilized as a gain in the controller on +the difference between the boundary liquid levels and to settle a region of at- +traction. The tank is subject to an acceleration which we consider as the control +input and obeys Newton’s second law. The problem is described by the viscous +Saint-Venant equations. We restrict our study to the one-dimensional (1-D) +case of the model. Moreover, contrary to prior works, in this work we do not +neglect the surface tension that acts on the free surface (liquid-gas interface) +but we neglect friction with the tank walls. +We intend to drive asymptotically the tank to a specified position. The +aforementioned goal must be achieved without liquid spillage and by having +both the tank and the liquid within the tank at rest. The equations describ- +ing the motion of the liquid in the tank can be derived by performing mass +and momentum balances (from first principles assuming that the liquid pres- +sure is the combination of hydrostatic pressure and capillary pressure given +by the Young-Laplace equation (see [15]) and by ignoring friction with the +tank walls). The equations can also be derived by using approximations of the +Navier-Stokes equations for the incompressible fluid (see [6–8, 28, 32, 37, 38]; +but see also [21,29] for fluid equations involving capillary phenomena). +We denote by a(t) the position of the left side of the tank at time t ≥ 0 and +we consider the length of the tank to be L > 0 (a constant). The evolution of the +liquid level and of the liquid velocity is described by the following equations +Ht + (Hu)z = 0, for t > 0, z ∈ [a(t),a(t) + L] +(1) +(Hu)t + +� +Hu2 + 1 +2gH2� +z +− σH + + +Hzz +� +1 + H2z +�3/2 + + +z += µ(Huz)z +for t > 0, z ∈ (a(t),a(t) + L) +(2) +where H(t,z) > 0, u(t,z) ∈ R are the liquid level and the liquid velocity, respec- +tively, at time t ≥ 0 and position z ∈ [a(t),a(t) + L], while g,µ,σ > 0 (constants) +are the acceleration of gravity, the kinematic viscosity of the liquid and the ra- +tio of the surface tension and liquid density, respectively. In some papers the +term + + +Hzz +� +1 + H2z +�3/2 + + +z +is replaced by Hzzz (see [6–8,32], but here we prefer a more +accurate description of the surface tension. +The liquid velocities at the walls of the tank are equal with the tank velocity. +Consequently: +u(t,a(t)) = u(t,a(t) + L) = w(t), for t ≥ 0 +(3) +where w(t) = ˙a(t) is the velocity of the tank at time t ≥ 0. Moreover, we get for +the tank +¨a(t) = −f (t), for t > 0 +(4) +5 + +where −f (t), the control input to the problem, is the tank acceleration. Defin- +ing the quantities +v(t,x) := u(t,a(t) + x) − w(t) +(5) +h(t,x) := H(t,a(t) + x) +(6) +ξ(t) := a(t) − a∗ +(7) +where a∗ ∈ R is the position (a constant) which we want the left side of the tank +to reach, we get the model: +˙ξ = w, for t ≥ 0 +(8) +˙w = −f , for t ≥ 0 +(9) +ht + (hv)x = 0, for t > 0, x ∈ [0,L] +(10) +(hv)t + +� +hv2 + 1 +2gh2� +x +− σh + + +hxx +� +1 + h2x +�3/2 + + +x += µ(hvx)x + hf , +for t > 0, x ∈ (0,L) +(11) +v(t,0) = v(t,L) = 0, for t ≥ 0 +(12) +Equations (10) and (12) imply that every classical solution of (8)-(12) satisfies +the following +d +d t +�� L +0 +h(t,x)dx +� += 0 for all t > 0 +(13) +Consequently, the total mass of the liquid m > 0 is constant, and without loss of +generality we can assume that every solution of (8)-(12) satisfies the equation +� L +0 +h(t,x)dx ≡ m +(14) +Due to the nature of our problem it is important to mention that the liquid +level h(t,x) must be positive for all times, i.e., we must have: +min +x∈[0,L](h(t,x)) > 0, for t ≥ 0 +(15) +Contrary to prior works, model (8)-(12), (14) is not a complete mathematical +description of the system. This can be seen directly by studying the lineariza- +tion of model (8)-(12), (14) but also can be seen by studying the literature +(see [27, 37, 38, 42, 43] and references therein). For a complete mathematical +model of the system we need two additional boundary conditions that describe +the interaction between the liquid and the solid walls of the tank. There are +many ways to describe the evolution of the angle of contact of a liquid with a +solid boundary (see the detailed presentation in [27]). In [37, 38], Schweizer +suggested (based on energy arguments and the fact that there might be a dis- +crepancy between the actual microscopic and the apparent macroscopic con- +tact angle) the use of a constant contact angle. Moreover, the assumption of +6 + +a constant contact angle allows the well-posedness of the overall problem (at +least for small data; see [37,38,42]). The constant contact angle approach has +been used extensively in the literature (see for instance [20,42,43]). +In this work, we adopt the constant contact angle approach by imposing a +contact angle equal to π/2. Therefore, the model (8)-(12), (14) is accompanied +by the following boundary conditions: +hx(t,0) = hx(t,L) = 0, for t ≥ 0 +(16) +In order to avoid liquid spillage the following condition must be satisfied: +max +x∈[0,L](h(t,x)) < Hmax, for t ≥ 0 +(17) +where Hmax > 0 is the height of the tank walls. We consider classical solutions +for the system (8)-(12), (14), (16), i.e., we consider +ξ ∈ C2 (R+), w ∈ C1 (R+), h ∈ C1 ([0,+∞) × [0,L]; (0,+∞)) ∩C3 ((0,+∞) ×(0,L)), +v ∈ C0([0,+∞) × [0,L]) ∩C1 ((0,+∞) ×[0,L]) with v[t] ∈ C2 ((0,L)) for each t > 0 +that satisfy equations (8)-(12), (14), (16) for a given input f ∈ C0 (R+). +For the system (8)-(12), (14), (16) with f (t) ≡ 0 (which is the open loop +system), there exists a continuum of equilibrium points, i.e., the points +h(x) ≡ h∗,v(x) ≡ 0, for x ∈ [0,L] +(18) +ξ ∈ R,w = 0 +(19) +where h∗ = m/L. We assume that the equilibrium points satisfy the condition +(17), i.e., h∗ < Hmax. +We intend to construct a robust with respect to surface tension control law +of the form +f (t) = F (h[t],v[t],ξ(t),w(t)), for t > 0, +(20) +which stabilizes the equilibrium point with ξ = 0. In addition to that we im- +pose the condition (17). +It follows from (18), (19) that the desired equilibrium point is not asymp- +totically stable for the open-loop system. Consequently the described control +problem is not at all trivial. +3 +The feedback law +3.1 +The Control Lyapunov Functional (CLF) +We define the set S ⊂ R2 × +� +C0 ([0,L]) +�2 as follows: +(ξ,w,h,v) ∈ S ⇔ + +h ∈ C0 ([0,L];(0,+∞)) ∩ H1(0,L) +v ∈ C0 ([0,L]) +� L +0 +h(x)dx = m +(ξ,w) ∈ R2,v(0) = v(L) = 0 +(21) +7 + +The above definition guarantees that every (ξ,w,h,v) ∈ S satisfies (12) and (14). +In addition to that, we define the following functionals for all (ξ,w,h,v) ∈ S: +V (ξ,w,h,v) := δE(h,v) + W(h,v) + qk2 +2 ξ2 + q +2 (w + kξ)2 +(22) +E(h,v) := 1 +2 +� L +0 +h(x)v2(x)dx + g +2 +���h − h∗χ[0,L] +���2 +2 ++σ +� L +0 +� � +1 + (h′(x))2 − 1 +� +dx +(23) +W(h,v) := 1 +2 +� L +0 +h−1(x)(h(x)v(x) + µh′(x))2 dx + g +2 +���h − h∗χ[0,L] +���2 +2 ++σ +� L +0 + + +� +1 + (h′(x))2 − 1 + +dx +(24) +where k,q > 0 are position error and velocity gains (to be selected) respectively, +δ > 0 and h∗ = m/L. In particular: +• the functional E is the mechanical energy of the liquid within the tank as +it is the sum of the potential energy +g +2 +���h − h∗χ[0,L] +���2 +2 + σ +� L +0 +� � +1 + (h′(x))2 − 1 +� +dx +and the kinetic energy +1 +2 +� L +0 +h(x)v2(x)dx +of the liquid. It should be noticed that there is no contribution to the +mechanical energy of the tank-liquid interface which allows to give the +interpretation that the boundary condition (16) (a constant contact angle) +is a result of the absence of interaction between liquid and solid. +• the functional W is a kind of mechanical energy of the liquid within +the tank and has been used extensively in the literature of isentropic, +compressible liquid flow (see [22,31,39,40]) as well as in [23–25]. +The functional V (ξ,w,h,v) defined by (22) will be utilized as a CLF for the +system, and for the derivation of useful bounds for the function h as guaran- +teed by the following lemma. +8 + +Lemma 1. Let constants q,k,δ > 0 be given, and define the increasing function +G ∈ C0(R) ∩ C1((−∞,0) ∪ (0,+∞)) as follows +G(h) := + +sgn(h − h∗) +�2 +3h +√ +h − 2h∗ √ +h + 4 +3h∗ √ +h∗ +� +for h > 0 +−4 +3h∗ √ +h∗ + h +for h ≤ 0 +(25) +Denote by G−1 : R → R the inverse function of G and define the constant +c := +1 +µ +� +δg +(26) +Then for every (ξ,w,h,v) ∈ S, the following inequality holds: +Q1 (V (ξ,w,h,v)) ≤ h(x) ≤ Q2 (V (ξ,w,h,v)), for all x ∈ [0,L], +(27) +where the functions Qi : R+ → R (i = 1,2) are defined as follows for all s ≥ 0: +Q1(s) := max +� +G−1 (−cs),N1(s),N2(s) +� +(28) +Q2(s) := min +� +G−1 (cs),P1(s),P2(s) +� +(29) +with the functions Ni : R+ → R (i = 1,2) and Pi : R+ → R (i = 1,2) defined by the +following expressions for all s ≥ 0: +N1(s) := h∗ − +� +2m(1 + δ)s +δµ2 +, +(30) +N2(s) := h∗ − +�� +s +σ(δ + 1) + L +�2 +− L2, +(31) +P1(s) := h∗ + +� +2m(1 + δ)s +δµ2 +, +(32) +P2(s) := h∗ + +�� +s +σ(δ + 1) + L +�2 +− L2 +(33) +Remark 1. It follows from (25), (26), (28) and the fact that h∗ = m/L that +Q1 (V (ξ,w,h,v)) > 0 when +V (ξ,w,h,v) < max(θ1,θ2,θ3) +(34) +with +θ1 := 4 +3µh∗ � +δgh∗, θ2 := +µ2h∗δ +2L(1 + δ) +and +θ3 := σ (δ + 1) +� � +(h∗)2 + L2 − L +� +9 + +Definitions (28) and (29) imply that Q2 : R+ → R is an increasing function +while Q1 : R+ → R is a decreasing function. +It is important to mention that Lemma 1 is more general than Lemma 1 +in [23] and Lemma 1 in [25]. Lemma 1 in [23] can be applied only for the case +δ = 1 and σ = 0, while Lemma 1 in [25] can be applied only for the case σ = 0. +Here Lemma 1 can be applied for all δ > 0 and σ ≥ 0. +3.2 +The state space +As in [23–25] the state space will be appropriately defined in order to exclude +states of the set S defined by (21) that violate the condition (17), i.e, the states +that cause liquid spillage. We define the following +X := +� +(ξ,w,h,v) ∈ S : max +x∈[0,L](h(x)) < Hmax +� +(35) +R := 2µ +� +δgh∗ +3 +(Hmax − h∗)min(ζ1,ζ2) +(36) +where +ζ1 := max(Γ1,Γ2,Γ3) and +(37) +ζ2 := +h∗ +Hmax − h∗ max(2,∆1,∆2) +(38) +with Γ1,Γ2,Γ3,∆1 and ∆2 defined as follows: +Γ1 := +� +Hmax +h∗ +− +2 +√ +h∗ +√Hmax + +√ +h∗ , +(39) +Γ2 := 3µ +√ +δ(Hmax − h∗) +4m(1 + δ) +� +gh∗ , +(40) +Γ3 := +3σ(δ + 1) +� � +L2 + (Hmax − h∗)2 − L +� +2µ +� +δgh∗ (Hmax − h∗) +, +(41) +∆1 := +3µ +√ +δ +4L +� +gh∗ (1 + δ) +, +(42) +∆2 := +3σ(δ + 1) +√ +h∗ +2µ +� +δg +� � +(h∗)2 + L2 + L +� +(43) +The aforementioned definition (36), the fact that h∗ < Hmax and Lemma 1 +imply for all (ξ,w,h,v) ∈ S with V (ξ,w,h,v) < R the following +0 < Q1 (V (ξ,w,h,v)) ≤ h(x) ≤ Q2 (V (ξ,w,h,v)) < Hmax, for all x ∈ [0,L] +(44) +10 + +Consequently, the conditions (17) for avoiding liquid spillage are satisfied when +(ξ,w,h,v) ∈ S with V (ξ,w,h,v) < R. +The set X defined by (35) is the state space of system (8)-(12), (14), (16). +In particular, we consider as state space the metric space X ⊂ R2 × H1 (0,L) × +L2 (0,L) with metric induced by the norm of the underlying normed linear +space R2 × H1 (0,L) × L2 (0,L), i.e., +∥(ξ,w,h,v)∥X = +� +ξ2 + w2 + ∥h∥2 +2 + +���h′���2 +2 + ∥v∥2 +2 +�1/2 +(45) +However, we need to approximate the state space from its interior by using +certain parameterized sets that allow us to obtain useful estimates. We define +XV (r) := {(ξ,w,h,v) ∈ S : V (ξ,w,h,v) ≤ r }, for r ≥ 0 +(46) +Inequalities (44) imply that +XV (r) ⊆ X, for all r ∈ [0,R) +(47) +As indicated by the following proposition the set XV (r) for r > 0 contains a +neighborhood of +� +0,0,h∗χ[0,L],0 +� +(in the topology of X with metric induced by +the norm ∥ ∥X defined by (45)). +Proposition 1. Let constants q,k,δ > 0 be given. Then for every (ξ,w,h,v) ∈ S +satisfying the inequality +���(0,w,h − h∗χ[0,L],v) +���X ≤ ε +(48) +for some ε > 0 with +ε < min(h∗,Hmax − h∗)/ +√ +L, +(49) +the following inequality holds: +V (ξ,w,h,v) ≤ C1 +���(ξ,w,h − h∗χ[0,L],v) +���2 +X + C2 +���(ξ,w,h − h∗χ[0,L],v) +���X +(50) +where +C1 := max + + +µ2 +h∗ − ε +√ +L +, δ + 1 +2 +g, (δ + 2)Hmax +2 +,q, 3qk2 +2 + +, +(51) +C2 := σ(δ + 1) +√ +L +(52) +and ∥·∥X is defined by (45). +3.3 +Stabilization results +The following theorem guarantees exponential stabilization of the state of the +system (8)-(12), (14), (16) by means of the nonlinear feedback law (55). +11 + +Theorem 1 (Stabilization of the Tank-Liquid System). +Let arbitrary constants ω,k,q,δ > 0 be given and define R by means of (36). Let +arbitrary r ∈ [0,R) be given and assume that +k < qθ(r) +(53) +where +θ(r) := +ωgµδπ2Q1(r) +gµδπ2Q1(r) + 2ωL(mgLHmax(δ + 1)2 + 2µ2δπ2Q1(r)) +(54) +where Q1 is defined by (28). Then there exist constants M,λ > 0 with the following +property: +(P) Every classical solution of the system (8)-(12), (14), (16) and +f (t) = −ω +� +(δ + 1) +� L +0 +h(t,x)v(t,x)dx + µ(h(t,L) − h(t,0)) − q(w(t) + kξ(t)) +� +, +for t > 0 +(55) +with (ξ(0),w(0),h[0],v[0]) ∈ XV (r), satisfies (ξ(t),w(t),h[t], v[t]) ∈ XV (r) and the +following estimate for t ≥ 0: +���� +� +ξ(t),w(t),h[t] − h∗χ[0,L],v[t] +�����X +≤ M exp(−λt) +���� +� +ξ(0),w(0),h[0] − h∗χ[0,1],v[0] +�����X +(56) +Remarks on Theorem 1. +1) The arbitrary quantities ω,k,q,δ > 0 are the control parameters. We should +point out that the ratio k/q must be sufficiently small due to (53), and this is +the only restriction for the control parameters. +2) The set XV (r) is the set for which exponential stabilization is achieved. As +indicated by Proposition 1, the set XV (r) for r > 0 contains a neighborhood +of +� +0,0,h∗χ[0,L],0 +� +(in the topology of X with metric induced by the norm ∥ ∥X +defined by (45)). The size of the set XV (r) depends on r ∈ [0,R) and on δ,q,k +(recall (36) and (22)). It is straightforward to see that the larger the parameter +q (or k) the smaller the set XV (r). However, the dependence of XV (r) on δ +(through the dependence of R on δ) is not clear. On the contrary it is a very +complicated, non-monotonic dependence. +3) The feedback law (55) only requires the measurement of the four following +quantities: +• the position of the tank denoted by ξ(t), and the velocity of the tank +denoted by w(t), +• the total momentum of the liquid, i.e., the quantity +� L +0 +h(t,x)v(t,x)dx, +and +12 + +• the difference the liquid level at the tank walls, i.e., the quantity h(t,L) − +h(t,0). +It should be emphasized that the feedback law (55) does not require the mea- +surement of the whole liquid level and liquid velocity profile whereas it is +completely independent of the surface tension coefficient. +4) The feedback law (55) is the same feedback law that was used in [23, 25]. +When the results in [23,25] and Theorem 1 are taken into account then it fol- +lows that the feedback law (55) guarantees robustness with respect to surface +tension as well as robustness with respect to wall friction forces. From a con- +trol point of view, this is an ideal situation: the feedback law (55) is robust +with respect to all possible perturbations of the basic model, its measurement +requirements are minimal and guarantees exponential stabilization of the cor- +responding closed-loop (nonlinear; not the linearized) system. +5) In contrast with [25], Theorem 1 does not provide an estimate for the norm +∥vx[t]∥2, and consequently an estimate for the sup-norm of the fluid velocity. +A topic for future research is the contruction of an appropriate CLF based on +which an estimate for the norm ∥vx[t]∥2 can be obtained. +4 +Proofs +Proof of Lemma 1. The proof is exactly the same with the proof of Lemma 1 +in [25]. The only difference is that here we can obtain an additional estimate for +���h − h∗χ[0,L] +���∞. Indeed, due to the fact that the function ϕ : R+ → R+defined +by +ϕ(s) = +√ +s2 + 1 − 1, for s ≥ 0 +(57) +is increasing and convex, we can use Jensen’s inequality (see page 120 in [9]) +and get for all h ∈ C0 ([0,L];(0,+∞)) ∩ H1(0,L) with +� L +0 +h(x)dx = m: +ϕ +�1 +L +���h′���1 +� += ϕ +�1 +L +� L +0 +���h′(x) +���dx +� +≤ 1 +L +� L +0 +ϕ +����h′(x) +��� +� +dx = 1 +L +� L +0 +� � +(h′(x))2 + 1 − 1 +� +dx +(58) +Using (58), the inequality +���h − h∗χ[0,L] +���∞ ≤ ∥h′∥1 (which is a direct consequence +of the fact that there exists x∗ ∈ [0,L] such that h(x∗) = h∗; a consequence of +continuity of h, the mean value theorem and the facts that +� L +0 +h(x)dx = m, h∗ = +m/L), the fact that the function ϕ−1 : R+ → R+ (the inverse function of ϕ) is +increasing with ϕ−1(s) = +� +(s + 1)2 − 1 for s ≥ 0 and the inequality +� L +0 +� � +(h′(x))2 + 1 − 1 +� +dx ≤ V (ξ,w,h,v) +σ(δ + 1) +(59) +13 + +which is a direct consequence of definitions (22), (23), (24), we get for all +(ξ,w,h,v) ∈ S: +���h − h∗χ[0,L] +���∞ ≤ +�� +L + V (ξ,w,h,v) +σ(δ + 1) +�2 +− L2 +(60) +Using the additional estimate (60) in conjunction with the estimates shown in +the proof of Lemma 1 in [25] and definitions (26), (28) and (29) we get (27) . +The proof is complete. +□ +Proof of Proposition 1. Consider arbitrary (ξ, w,h,v) ∈ S satisfying (48) and (49). +Definitions (22), (23), (24) and the inequalities +(h(x)v(x) + µh′(x))2 ≤ 2h2(x)v2(x) + 2µ2 (h′(x))2 , +(61) +(w + kξ)2 ≤ 2w2 + 2k2ξ2, +(62) +� +1 + (h′(x))2 − 1 ≤ +���h′(x) +��� +(63) +imply: +V (ξ,w,h,v) ≤ δ + 2 +2 +� L +0 +h(x)v2(x)dx + µ2 +� L +0 +h−1(x)(h′(x))2 dx ++δ + 1 +2 +g +���h − h∗χ[0,L] +���2 +2 + 3qk2 +2 +ξ2 + qw2 + σ(δ + 1) +���h′���1 +(64) +Following the arguments of the proof of Proposition 2.5 in [25] we obtain from +(64) the following: +V (ξ,w,h,v) ≤ δ + 2 +2 +Hmax ∥v∥2 +2 + qw2 + 3qk2 +2 +ξ2 ++µ2 � +h∗ − ε +√ +L +�−1 ���h′���2 +2 + δ + 1 +2 +g +���h − h∗χ[0,L] +���2 +2 + σ(δ + 1) +√ +L +���h′���2 +(65) +Inequality (50) is a direct consequence of (65) and definition (45). The proof is +complete. +□ +In order to give the proof of the main result of this study, we need to provide +some preliminary lemmas along with their proofs. +Lemma 2. Every classical solution of the system (8)-(12), (14), (16) satisfies the +following equations for all t > 0: +d +dt E(h[t],v[t]) = −µ +� L +0 +h(t,x)v2 +x(t,x)dx + f (t) +� L +0 +h(t,x)v(t,x)dx +(66) +14 + +d +dt W(h[t],v[t]) = −µg ∥hx[t]∥2 +2 − µσ +� L +0 +h2xx(t,x)dx +� +1 + h2x(t,x) +�3/2 ++f (t) +� L +0 +(h(t,x)v(t,x) + µhx(t,x))dx +(67) +where E,W are defined by (23), (24), respectively. +Proof. Due to (10) and (11) we get for t > 0, x ∈ (0,L): +vt(t,x) + v(t,x)vx(t,x) + ghx(t,x) += σh−1(t,x) + + +1 + h2x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x ++µh−1(t,x)(h(t,x)vx(t,x))x + f (t) +(68) +Combining definition (23), (10) and (68) we get for all t > 0 the following ex- +pression for the time derivative of the functional (23) : +d +dt E(h[t],v[t]) = −1 +2 +� L +0 +(h(t,x)v(t,x))xv2(t,x)dx +− +� L +0 +h(t,x)v2(t,x)vx(t,x)dx − g +� L +0 +h(t,x)v(t,x)hx(t,x)dx ++σ +� L +0 +v(t,x) + + +1 + h2 +x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +dx ++µ +� L +0 +v(t,x)(h(t,x)vx(t,x))x dx + f (t) +� L +0 +h(t,x)v(t,x)dx +−g +� L +0 +(h(t,x)v(t,x))x(h(t,x) − h∗)dx +−σ +� L +0 +hx(t,x) +� +1 + h2x(t,x) +(h(t,x)v(t,x))xxdx +(69) +Using (69), integration by parts as in the proof of Lemma 2.11 in [25], (12), +15 + +(16) and the fact that for all t > 0 +σ +� L +0 +v(t,x) + + +1 + h2x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +dx += −σ +� L +0 +vx(t,x)1 + h2 +x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 +dx +(70) +− σ +� L +0 +hx(t,x) +� +1 + h2x(t,x) +(h(t,x)v(t,x))xxdx += σ +� L +0 +vx(t,x)1 + h2x(t,x) + hxx(t,x)h(t,x) +� +1 + h2x(t,x) +�3/2 +dx +(71) +as a consequence of integration by parts as well, we obtain equation (66). +Next we define for all t ≥ 0 and x ∈ [0,L]: +ϕ(t,x) := h(t,x)v(t,x) + µhx(t,x) +(72) +Definition (72), (10) and (11) imply for all t > 0 and x ∈ (0,L): +ϕt(t,x) = − + +v(t,x)ϕ(t,x) + 1 +2gh2(t,x) − σ 1 + h2 +x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x ++h(t,x)f (t) +(73) +Using definition (24) along with (73) and (10), we get for all t > 0 : +d +dt W(h[t],v[t]) = 1 +2 +� L +0 +h−2(t,x)ϕ2(t,x)(h(t,x)v(t,x))xdx +− +� L +0 +h−1(t,x)ϕ(t,x) +� +ϕ(t,x)v(t,x) + 1 +2gh2(t,x) +� +x +dx ++σ +� L +0 +h−1(t,x)ϕ(t,x) + + +1 + h2x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +dx ++f (t) +� L +0 +ϕ(t,x)dx − g +� L +0 +(h(t,x) − h∗)(h(t,x)v(t,x))xdx +−σ +� L +0 +hx(t,x)(h(t,x)v(t,x))xx +� +1 + h2x(t,x) +dx +(74) +Using (12) and integration by parts as in proof of Lemma 2.11 in [25], we obtain +16 + +from (74) and definition (72) for all t > 0: +d +dt W(h[t],v[t]) = −µg ∥hx[t]∥2 +2 + f (t) +� L +0 +(h(t,x)v(t,x) + µhx(t,x))dx ++σ +� L +0 +v(t,x) + + +1 + h2x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +dx ++µσ +� L +0 +h−1(t,x)hx(t,x) + + +1 + h2x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +dx +−σ +� L +0 +hx(t,x)(h(t,x)v(t,x))xx +� +1 + h2x(t,x) +dx +(75) +Using (16), (70), (71) and the fact that +h(t,x) + + +hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x += + + +1 + h2 +x(t,x) + h(t,x)hxx(t,x) +� +1 + h2x(t,x) +�3/2 + + +x +(76) +we obtain from (75) equation (67) for all t > 0. The proof is complete. +□ +Lemma 3. Let constants q,k,δ > 0 be given. Then there exists a non-decreasing +function Λ : [0,R) → (0,+∞), where R > 0 is defined by (36) such that for every +(ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) < R, the following +inequality holds: +V (ξ,w,h,v) +Λ(V (ξ,w,h,v)) ≤ +���h′���2 +2 + +� L +0 +(h′′(x))2 +� +1 + (h′(x))2�3/2 dx ++ +� L +0 +h(x)(v′(x))2 dx + ξ2 + (w + kξ)2 +(77) +Proof. Let arbitrary (ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) < +R be given. Using the same arguments as in the proof of Lemma 2.12 in [25] +and the fact that +� L +0 +� � +1 + (h′(x))2 − 1 +� +dx ≤ +���h′���2 +2 +(78) +17 + +we obtain the following estimate: +V (ξ,w,h,v) ≤ L2 (δ + 2)Q2(V (ξ,w,h,v)) +2π2Q1(V (ξ,w,h,v)) +� L +0 +h(x)(v′(x))2 dx ++ + + +(δ + 1) +� +gL2 + 2σ +� +2 ++ +µ2 +Q1 (V (ξ,w,h,v)) + + +���h′���2 +2 + qk2 +2 ξ2 + q +2 (w + kξ)2 +≤ Λ(V (ξ,w,h,v)) +× +����h′���2 +2 + +� L +0 +(h′′(x))2 +� +1 + (h′(x))2�3/2 dx + +� L +0 +h(x)(v′(x))2 dx + ξ2 + (w + kξ)2 +� +(79) +where +Λ(s) := 1 +2 max +� +κ1 + 2µ2 +Q1 (s), κ2Q2(s) +Q1(s) ,κ3 +� +, for s ∈ [0,R) +(80) +with κ1 := (δ + 1) +� +gL2 + 2σ +� +, κ2 := L2 (δ + 2)/π2 and κ3 := qmax(1,k2). Defini- +tion (80) and the fact that Q2 : R+ → R is an increasing function and Q1 : R+ → +R is a decreasing function imply that Λ : [0,R) → (0,+∞) is a non-decreasing +function. Inequality (77) holds as a direct consequence of (79). The proof is +complete. +□ +Lemma 4. Let constants q,k,δ > 0 be given. Then there exist non-decreasing func- +tions Gi : [0,R) → (0,+∞), i = 1,2, where R > 0 is defined by (36), such that for +every (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R, the following inequalities hold: +���(ξ,w,h − h∗χ[0,L],v) +���2 +X ≤ V (ξ,w,h,v)G1 (V (ξ,w,h,v)) +(81) +V (ξ,w,h,v) +G2 (V (ξ,w,h,v)) ≤ +���(ξ,w,h − h∗χ[0,L],v) +���2 +X +(82) +where ∥·∥X is defined by (45). +Proof. Let arbitrary (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R be given. Using defini- +tions (22), (23), (24), inequalities (61), (62), the inequality +� +1 + (h′(x))2 ≤ 1 + (h′(x))2 +(83) +and (44) we obtain +V (ξ,w,h,v) ≤ δ + 2 +2 +Hmax ∥v∥2 +2 + δ + 1 +2 +g +���h − h∗χ[0,L] +���2 +2 ++ +� +µ2 +Q1 (V (ξ,w,h,v)) + σ (δ + 1) +����h′���2 +2 + 3qk2 +2 +ξ2 + qw2 +(84) +18 + +Inequality (84) implies inequality (82) with +G2 (s) := max +�δ + 2 +2 +Hmax, δ + 1 +2 +g, +µ2 +Q1 (s) + σ (δ + 1), 3qk2 +2 +,q +� +, +for s ∈ [0,R) +(85) +The fact that Q1 : R+ → R is a decreasing function and the above definition +imply that G2 : [0,R) → (0,+∞) is a non-decreasing function. +The proof of inequality (81) is exactly the same with the proof of Lemma 4 +in [25]. The proof is complete. +□ +Lemma 5. Let constants ω,k,q,δ > 0 and r ∈ [0,R) be given, where R > 0 is defined +by (36). Then every classical solution of the system (8)-(12), (14), (16) and (55) +satisfies the following inequality for all t > 0 for which V (ξ(t),w(t),h[t],v[t]) < R: +d +dt V (ξ(t),w(t),h[t],v[t]) ≤ −3µg +4 ∥hx[t]∥2 +2 − qk3ξ2(t) +− +µδ +2Hmax +� +2Hmax − Q1(r)Q2 (V (t)) +Q1 (V (t)) +�� L +0 +h(t,x)v2 +x(t,x)dx +−µσ +� L +0 +h2xx(t,x) +� +1 + h2x(t,x) +�3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2 +(86) +where V (t) = V (ξ(t),w(t),h[t],v[t]), θ(r) is defined by (54) and Qi : R+ → R (i = +1,2) are the functions defined by (28) and (29). +Proof. Let ω,k,q,δ > 0 be given constants and let r ∈ [0,R) be a constant, where +R > 0 is defined by (36). In addition to that we consider a classical solution of +the system (8)-(12), (14), (16) and (55) at a time t > 0 for which V (ξ(t),w(t),h[t],v[t]) < +R. Using Lemma 2, (66), (67) and definition (22) and by following the same +procedure as in the proof of Lemma 2.14 in [25] by assuming zero friction +coefficient, we establish the following inequality: +d +dt V (ξ(t),w(t),h[t],v[t]) ≤ −3µg +4 +∥hx[t]∥2 +2 − µδ +� L +0 +h(t,x)v2 +x(t,x)dx +−µσ +� L +0 +h2xx(t,x) +� +1 + h2x(t,x) +�3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2 − qk3ξ2(t) ++µδπ2Q1(r) +2L2Hmax +� L +0 +h(t,x)v2(t,x)dx +(87) +Since v(t,0) = v(t,L) = 0 (recall (12)), by virtue of Wirtinger’s inequality and +(44), we get: +∥v[t]∥2 +2 ≤ L2 +π2 ∥vx[t]∥2 +2 ≤ +L2 +π2Q1 (V (t)) +� L +0 +h(t,x)v2 +x(t,x)dx +(88) +Combining (44), (87) and (88), we obtain (86). The proof is complete. +□ +19 + +We can now present the proof of Theorem 1. +Proof of Theorem 1. Let constants ω,q,k,δ > 0 satisfying (53). Let constant r ∈ +[0,R) be given. Consider a classical solution of the system (8)-(12), (14), (16) +with (55) that satisfies V (ξ(0),w(0),h[0],v[0]) ≤ r. Let r ∈ (r,R) be a constant +that satisfies: +Q2 (r) +Q1 (r) < 2Hmax +Q1(r) +(89) +The existence of ¯r ∈ (r,R) is a direct consequence of the continuity of the func- +tions involved in (89). +Due to (53), Lemma 5, (86) and (89) the following implication is true: +If t > 0 and V (ξ(t),w(t),h[t],v[t]) ≤ r then d +d t V (ξ(t),w(t),h[t],v[t]) ≤ 0 +(90) +A contradiction argument as in the proof of Theorem 2.6 in [25] implies that +V (ξ(t),w(t), h[t],v[t]) ≤ r for all t ≥ 0. +Implication (90) and the fact V (ξ(t),w(t),h[t],v[t]) ≤ r for all t ≥ 0 imply +that +d +d t V (ξ(t),w(t),h[t],v[t]) ≤ 0 for all t > 0 +(91) +Due to the above and the continuity of the mapping t → V (ξ(t),w(t),h[t], v[t]), +we get that +V (ξ(t),w(t),h[t],v[t]) ≤ V (ξ(0),w(0),h[0],v[0]) ≤ r < R,for all t ≥ 0 +(92) +Consequently, (ξ(t),w(t),h[t],v[t]) ∈ XV (r) for all t ≥ 0 (recall (46)). Using (92) +and Lemma 5, we conclude that (86) holds for all t > 0. Using (92), (86) and +the fact that Q2 : R+ → R is an increasing function while Q1 : R+ → R is a +decreasing function, we obtain the following estimate for t > 0 +d +dt V (ξ(t),w(t),h[t],v[t]) +≤ −β(r) +� +∥hx[t]∥2 +2 + +� L +0 +h(t,x)v2 +x(t,x)dx + +� L +0 +h2xx(t,x) +� +1 + h2x(t,x) +�3/2 dx ++ξ2(t) + (w(t) + kξ(t))2 +� +(93) +where +β(r) := min +�3µg +4 , µδ(2Hmax − Q2 (r)) +2Hmax +,qk3,q(qθ(r) − k),µσ +� +(94) +Notice that (53) and the fact that r ∈ [0,R) in conjunction with definitions (29), +(36), (93) imply that β(r) > 0. It follows from Lemma 3, (77), the continuity +of the mapping t → V (ξ(t),w(t),h[t],v[t]), (recall that v ∈ C0 (R+ ;H1 (0,L) +� +, +20 + +h ∈ C1 (R+ × [0,L];(0,+∞)) and v ∈ C0 (R+ × [0,L])), estimates (92), (93), Lemma +4, (81) and (82) that the following estimate holds for all t ≥ 0: +���(ξ(t),w(t),h[t] − h∗χ[0,L],v[t]) +���2 +X +≤ Ω(r)exp +� +−β(r)t +Λ(r) +����(ξ(0),w(0),h[0] − h∗χ[0,L],v[0]) +���2 +X +(95) +with +Ω(r) := G1 (r)G2 (r) +(96) +where Λ is the non-decreasing function involved in (77) and Gi : [0,R) → +(0,+∞) (i = 1,2) are the non-decreasing functions involved in (81), (82). Es- +timate (56) with M = +� +Ω(r) and λ = β(r) +2Λ(r) is a consequence of estimate (95). +The proof is complete. +□ +5 +Concluding Remarks +In this work we managed to show that the robust with respect to wall friction +nonlinear feedback law proposed in [25] provides also robust stabilization re- +sults with respect to surface tension. This shows even more the significance of +the CLFs as stabilizing tools for the infinite-dimensional case of systems de- +scribed by PDEs and illustrates the fact that robustness is inherent in the CLF +methodology. +The present study deals with the case of viscous Saint-Venant system with +surface tension and without wall friction. It is of interest to study the more +challenging problem of the viscous Saint-Venant system with surface tension +and with wall friction as well as the construction of an additional functional +which provides a bound for the sup-norm of the fluid velocity. In addition to +that, other topics for future research are the study of existence and unique- +ness of the solutions for the closed-loop system, the study of the problem with +non constant (dynamic) contact angles, the study of the output feedback stabi- +lization problem, the construction of appropriate numerical schemes and the +derivation of stability estimates in stronger spatial norms. 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Stern, A Sharp Interface Method for Two-Phase Flows Inter- +acting with Moving Bodies, Proceedings of the 18th AIAA Computational +Fluid Dynamics Conference, Miami, FL, 2007. +24 + diff --git a/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/load_file.txt b/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c4540182ddafb06e838191ba42bc02e30971299 --- /dev/null +++ b/2dAzT4oBgHgl3EQfuP2J/content/tmp_files/load_file.txt @@ -0,0 +1,808 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf,len=807 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='01688v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='OC] 4 Jan 2023 Feedback Stabilization of Tank-Liquid System with Robustness to Surface Tension Iasson Karafyllis, Filippos Vokos Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece, emails: iasonkar@central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='ntua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='gr, fivojean@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='ntua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='gr Miroslav Krstic Department of Mechanical and Aerospace Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', University of California, La Jolla, San Diego, CA 92093-0411, USA email: krstic@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='edu January 5, 2023 Abstract We construct a robust stabilizing feedback law for the viscous Saint- Venant system of Partial Differential Equations (PDEs) with surface tension and without wall friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The Saint-Venant system describes the move- ment of a tank which contains a viscous liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We assume constant con- tact angles between the liquid and the walls of the tank and we achieve a spill-free exponential stabilization with robustness to surface tension by using a Control Lyapunov Functional (CLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proposed CLF provides a parameterized family of sets which approximate the state space from the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Based on the CLF, we construct a nonlinear stabilizing feedback law which ensures that the closed-loop system converges exponentially to the desired equilibrium point in the sense of an appropriate norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 1 Introduction The Saint-Venant model, which was derived in [2], constitutes a significant and very influential mathematical model in fluid mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It is also referred in literature as the shallow water model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Recent extensions of the Saint-Venant model take into account various types of forces such as viscous stresses, surface tension and friction forces (see [10,19,29,33,41,43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The feedback stabilization problem of the Saint-Venant model is a challeng- ing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The dominant cases studied in the litterature include the inviscid model - which ignores forces such as viscous stresses and surface tension - and 1 the linearized model (see [1,3–5,11–14,16–18,30,35,36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [11,12,14,35,36] the problem of the movement of an 1-D tank which contains a fluid is stud- ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' More specifically, [11,12,14] provide controllability results for the Saint- Venant model without viscosity, without friction and without surface tension, while [35] suggests a new variational formulation of Saint-Venant equations and proves the steady-state controllability of the linear approximations of sev- eral control configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [36] the inviscid Saint-Venant model is studied and appropriate stabilizing full-state feedback and output feedback control laws are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [3–5, 12, 13, 16–18, 30] the movement of a fluid in an open channel is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Stabilization results are provided in [1,3,4,13,17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [3, 4, 17, 18, 30] the linearized Saint-Venant model is being used while [1] deals with a general linear hyperbolic system which appears in Saint-Venant equations among other linear hyperbolic laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The works [5, 12, 13, 16] study the nonlinear Saint-Venant model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [5] the feedforward control problem of general nonlinear hyperbolic systems is studied and an application using the Saint-Venant model with friction is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [12, 13] local convergence of the state of hyperbolic systems of conservation laws is guaranteed using a strict Lypaunov function which exploits Riemann invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' An application to the inviscid, frictonless Saint-Venant model is provided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The paper [16] achieves regulation of the water flow and level in water-ways using the invis- cid Saint-Venant model without friction and without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Very few studies in the literature deal with the nonlinear viscous Saint- Venant model that is used for the description of the movement of a tank which contains an incompressible, Newtonian fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The first work that studied the nonlinear viscous Saint-Venant model without wall friction and without sur- face tension was [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [23] an appropriate nonlinear feedback law is con- structed which provides semiglobal stabilization results by following a CLF methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The work [25] extends the results obtained in [23] in the case where wall friction forces are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [25] both the case of a velocity independent friction coefficient and the general case of friction co- efficient are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' A robust with respect to wall friction stabilizing feed- back law is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Another study which deals with the nonlinear viscous Saint-Venant model is [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [24] a stabilizing output-feedback control law for the viscous Saint-Venant PDE system without wall friction and without surface tension is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The output-feedback control law is utilized through a functional-observer methodology and a CLF methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The study of the movement of a fluid which interacts with a gas bound- ary and a solid boundary is inevitably intertwined with the notion of the sur- face tension and the notions of contact angle and wettability (see [27, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Surface tension is crucial as it acts in the interface between liquid and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' From a mathematical point of view surface tension is very important because it changes the order of the PDEs (it is expressed by a third order term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Con- tact angle is the angle at which the fluid surface intersects with a solid bound- ary as stated in [34], and it is a measure of wettability of the solid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' There is a wide literature concerning the topic of contact angles (see for in- stance [20,21,27,28,37,38,42,43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The concept of contact angle is significant 2 in our study because it provides an additional boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In this paper we solve the feedback stabilization problem for a tank con- taining a liquid modeled by the viscous Saint-Venant system of PDEs with sur- face tension and without wall friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We consider the case of constant contact angles between the liquid and the walls of the tank, as in [37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We utilize a specific form of the feedback law initially presented in [25], which constitutes a more general form of the feedback law in [23] with robustness to surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Indeed, we saw that the proposed feedback law guarantees stabiliza- tion no matter what the value of the surface tension coefficient is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Therefore, the knowledge of the surface tension coefficient is not necessary and the feedback law is independent of the surface tension coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We achieve a spill-free exponential stabilization, with robustness to surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' As in [23–25] we follow a CLF methodology and we design the feedback law based on an appro- priate functional, which is the CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The CLF determines a specific parameter- ized set which approximates the state space of the control problem from the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Although this work presents enough technical similarities with [25], there are some crucial differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Firstly, in contrast with [25], the system of PDEs contains an extra term due to surface tension and does not contain a friction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Moreover, in order for the model to be complete and for the problem to be well-posed, an additional boundary condition is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The additional boundary condition is provided by the assumption of a constant contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Here we use only one CLF while in [25] two different functionals are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' As a consequence this work does not provide a bound for the sup-norm of the fluid velocity, as in [25], due to the absence of an appropriate functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Here the CLF is different from the corresponding one in [25], as it contains an additional potential energy term due to the effect of the surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In Section 2 the control problem is described as well as its main objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In Section 3 we provide the intuitive ideas and the statements of the results of this work along with some auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Section 4 includes all the proofs of the results presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Finally, Section 5 points out the conclusions of this work and suggests topics for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Notation ∗ R+ = [0,+∞) is the set of non-negative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' ∗ Let S ⊆ Rn be an open set and let A ⊆ Rn be a set such that S ⊆ A ⊆ cl(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' By C0(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Ω), we denote the class of continuous functions on A, which take values in Ω ⊆ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' By Ck(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Ω), where k ≥ 1 is an integer, we denote the class of functions on A ⊆ Rn, which takes values in Ω ⊆ Rm and has continuous derivatives of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In other words, the functions of class Ck(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Ω) are the functions which have continuous derivatives of order k in S = int(A) that can be continued continuously to all points in ∂S ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' When Ω = R then we write C0(A) or Ck(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' When I ⊆ R is an interval and 3 G ∈ C1(I) is a function of a single variable, G′(h) denotes the derivative with respect to h ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' ∗ Let I ⊆ R be an interval, let a < b be given constants and let u : I ×[a,b] → R be a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We utilize the notation u[t] to denote the profile at certain t ∈ I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', (u[t])(x) = u(t,x) for all x ∈ [a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' When u(t,x) is three times differentiable with respect to x ∈ [a,b], we use the notation ux(t,x), uxx(t,x) and uxxx(t,x) for the first, second and third derivative of u with respect to x ∈ [a,b] respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', ux(t,x) = ∂u ∂x (t,x),uxx(t,x) = ∂2 u ∂x2 (t,x) and uxxx(t,x) = ∂3 u ∂x3 (t,x) When u(t,x) is differentiable with respect to t, we use the notation ut(t,x) for the derivative of u with respect to t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', ut(t,x)= ∂u ∂t (t,x) ∗ Given a set U ⊆ Rn, χU denotes the characteristic function of U defined by χU(x) := � 1 for all x ∈ U 0 for all x � U The sign function sgn : R → R is the function defined by sgn(x) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 1 for x > 0 0 for x = 0 −1 for x < 0 ∗ Consider given constants a,b such that a < b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' For p ∈ [1,+∞), Lp(a,b) denotes the set of equivalence classes of Lebesgue measurable functions u : (a,b) → R with ∥u∥p := �� b a |u(x)|p dx �1/p < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' L∞(a,b) denotes the set of equivalence classes of Lebesgue measurable functions u : (a,b) → R with ∥u∥∞ := esssup x∈(a,b) (|u(x)|) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' For an integer k ≥ 1, Hk(a,b) denotes the Sobolev space of functions in L2(a,b) with all its weak derivatives up to order k ≥ 1 in L2(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 4 2 The Control Problem We want to manipulate the motion of a tank which contains a viscous, Newto- nian, incompressible liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Viscosity is utilized as a gain in the controller on the difference between the boundary liquid levels and to settle a region of at- traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The tank is subject to an acceleration which we consider as the control input and obeys Newton’s second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The problem is described by the viscous Saint-Venant equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We restrict our study to the one-dimensional (1-D) case of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Moreover, contrary to prior works, in this work we do not neglect the surface tension that acts on the free surface (liquid-gas interface) but we neglect friction with the tank walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We intend to drive asymptotically the tank to a specified position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The aforementioned goal must be achieved without liquid spillage and by having both the tank and the liquid within the tank at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The equations describ- ing the motion of the liquid in the tank can be derived by performing mass and momentum balances (from first principles assuming that the liquid pres- sure is the combination of hydrostatic pressure and capillary pressure given by the Young-Laplace equation (see [15]) and by ignoring friction with the tank walls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The equations can also be derived by using approximations of the Navier-Stokes equations for the incompressible fluid (see [6–8, 28, 32, 37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' but see also [21,29] for fluid equations involving capillary phenomena).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We denote by a(t) the position of the left side of the tank at time t ≥ 0 and we consider the length of the tank to be L > 0 (a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The evolution of the liquid level and of the liquid velocity is described by the following equations Ht + (Hu)z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' z ∈ [a(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='a(t) + L] (1) (Hu)t + � Hu2 + 1 2gH2� z − σH \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Hzz � 1 + H2z �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 z = µ(Huz)z for t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' z ∈ (a(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='a(t) + L) (2) where H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='z) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='z) ∈ R are the liquid level and the liquid velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' respec- tively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' at time t ≥ 0 and position z ∈ [a(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='a(t) + L],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' while g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='σ > 0 (constants) are the acceleration of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the kinematic viscosity of the liquid and the ra- tio of the surface tension and liquid density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In some papers the term \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Hzz � 1 + H2z �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 z is replaced by Hzzz (see [6–8,32], but here we prefer a more accurate description of the surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The liquid velocities at the walls of the tank are equal with the tank velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Consequently: u(t,a(t)) = u(t,a(t) + L) = w(t), for t ≥ 0 (3) where w(t) = ˙a(t) is the velocity of the tank at time t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Moreover, we get for the tank ¨a(t) = −f (t), for t > 0 (4) 5 where −f (t), the control input to the problem, is the tank acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Defin- ing the quantities v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) := u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='a(t) + x) − w(t) (5) h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) := H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='a(t) + x) (6) ξ(t) := a(t) − a∗ (7) where a∗ ∈ R is the position (a constant) which we want the left side of the tank to reach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we get the model: ˙ξ = w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t ≥ 0 (8) ˙w = −f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t ≥ 0 (9) ht + (hv)x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' x ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] (10) (hv)t + � hv2 + 1 2gh2� x − σh \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed hxx � 1 + h2x �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x = µ(hvx)x + hf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' x ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L) (11) v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='0) = v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t ≥ 0 (12) Equations (10) and (12) imply that every classical solution of (8)-(12) satisfies the following d d t �� L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx � = 0 for all t > 0 (13) Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the total mass of the liquid m > 0 is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' and without loss of generality we can assume that every solution of (8)-(12) satisfies the equation � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx ≡ m (14) Due to the nature of our problem it is important to mention that the liquid level h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) must be positive for all times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', we must have: min x∈[0,L](h(t,x)) > 0, for t ≥ 0 (15) Contrary to prior works, model (8)-(12), (14) is not a complete mathematical description of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' This can be seen directly by studying the lineariza- tion of model (8)-(12), (14) but also can be seen by studying the literature (see [27, 37, 38, 42, 43] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' For a complete mathematical model of the system we need two additional boundary conditions that describe the interaction between the liquid and the solid walls of the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' There are many ways to describe the evolution of the angle of contact of a liquid with a solid boundary (see the detailed presentation in [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In [37, 38], Schweizer suggested (based on energy arguments and the fact that there might be a dis- crepancy between the actual microscopic and the apparent macroscopic con- tact angle) the use of a constant contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Moreover, the assumption of 6 a constant contact angle allows the well-posedness of the overall problem (at least for small data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' see [37,38,42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The constant contact angle approach has been used extensively in the literature (see for instance [20,42,43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In this work, we adopt the constant contact angle approach by imposing a contact angle equal to π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Therefore, the model (8)-(12), (14) is accompanied by the following boundary conditions: hx(t,0) = hx(t,L) = 0, for t ≥ 0 (16) In order to avoid liquid spillage the following condition must be satisfied: max x∈[0,L](h(t,x)) < Hmax, for t ≥ 0 (17) where Hmax > 0 is the height of the tank walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We consider classical solutions for the system (8)-(12), (14), (16), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', we consider ξ ∈ C2 (R+), w ∈ C1 (R+), h ∈ C1 ([0,+∞) × [0,L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (0,+∞)) ∩C3 ((0,+∞) ×(0,L)), v ∈ C0([0,+∞) × [0,L]) ∩C1 ((0,+∞) ×[0,L]) with v[t] ∈ C2 ((0,L)) for each t > 0 that satisfy equations (8)-(12), (14), (16) for a given input f ∈ C0 (R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' For the system (8)-(12), (14), (16) with f (t) ≡ 0 (which is the open loop system), there exists a continuum of equilibrium points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', the points h(x) ≡ h∗,v(x) ≡ 0, for x ∈ [0,L] (18) ξ ∈ R,w = 0 (19) where h∗ = m/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We assume that the equilibrium points satisfy the condition (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', h∗ < Hmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We intend to construct a robust with respect to surface tension control law of the form f (t) = F (h[t],v[t],ξ(t),w(t)), for t > 0, (20) which stabilizes the equilibrium point with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In addition to that we im- pose the condition (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It follows from (18), (19) that the desired equilibrium point is not asymp- totically stable for the open-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Consequently the described control problem is not at all trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 3 The feedback law 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='1 The Control Lyapunov Functional (CLF) We define the set S ⊂ R2 × � C0 ([0,L]) �2 as follows: (ξ,w,h,v) ∈ S ⇔ \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 h ∈ C0 ([0,L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='(0,+∞)) ∩ H1(0,L) v ∈ C0 ([0,L]) � L 0 h(x)dx = m (ξ,w) ∈ R2,v(0) = v(L) = 0 (21) 7 The above definition guarantees that every (ξ,w,h,v) ∈ S satisfies (12) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In addition to that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we define the following functionals for all (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S: V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) := δE(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) + W(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) + qk2 2 ξ2 + q 2 (w + kξ)2 (22) E(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) := 1 2 � L 0 h(x)v2(x)dx + g 2 ���h − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] ���2 2 +σ � L 0 � � 1 + (h′(x))2 − 1 � dx (23) W(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) := 1 2 � L 0 h−1(x)(h(x)v(x) + µh′(x))2 dx + g 2 ���h − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] ���2 2 +σ � L 0 \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � 1 + (h′(x))2 − 1 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8dx (24) where k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='q > 0 are position error and velocity gains (to be selected) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' δ > 0 and h∗ = m/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In particular: the functional E is the mechanical energy of the liquid within the tank as it is the sum of the potential energy g 2 ���h − h∗χ[0,L] ���2 2 + σ � L 0 � � 1 + (h′(x))2 − 1 � dx and the kinetic energy 1 2 � L 0 h(x)v2(x)dx of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It should be noticed that there is no contribution to the mechanical energy of the tank-liquid interface which allows to give the interpretation that the boundary condition (16) (a constant contact angle) is a result of the absence of interaction between liquid and solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the functional W is a kind of mechanical energy of the liquid within the tank and has been used extensively in the literature of isentropic, compressible liquid flow (see [22,31,39,40]) as well as in [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The functional V (ξ,w,h,v) defined by (22) will be utilized as a CLF for the system, and for the derivation of useful bounds for the function h as guaran- teed by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 8 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='δ > 0 be given,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' and define the increasing function G ∈ C0(R) ∩ C1((−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='0) ∪ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='+∞)) as follows G(h) := \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 sgn(h − h∗) �2 3h √ h − 2h∗ √ h + 4 3h∗ √ h∗ � for h > 0 −4 3h∗ √ h∗ + h for h ≤ 0 (25) Denote by G−1 : R → R the inverse function of G and define the constant c := 1 µ � δg (26) Then for every (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the following inequality holds: Q1 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) ≤ h(x) ≤ Q2 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for all x ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (27) where the functions Qi : R+ → R (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='2) are defined as follows for all s ≥ 0: Q1(s) := max � G−1 (−cs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='N1(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='N2(s) � (28) Q2(s) := min � G−1 (cs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='P1(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='P2(s) � (29) with the functions Ni : R+ → R (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='2) and Pi : R+ → R (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='2) defined by the following expressions for all s ≥ 0: N1(s) := h∗ − � 2m(1 + δ)s δµ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (30) N2(s) := h∗ − �� s σ(δ + 1) + L �2 − L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (31) P1(s) := h∗ + � 2m(1 + δ)s δµ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (32) P2(s) := h∗ + �� s σ(δ + 1) + L �2 − L2 (33) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It follows from (25), (26), (28) and the fact that h∗ = m/L that Q1 (V (ξ,w,h,v)) > 0 when V (ξ,w,h,v) < max(θ1,θ2,θ3) (34) with θ1 := 4 3µh∗ � δgh∗, θ2 := µ2h∗δ 2L(1 + δ) and θ3 := σ (δ + 1) � � (h∗)2 + L2 − L � 9 Definitions (28) and (29) imply that Q2 : R+ → R is an increasing function while Q1 : R+ → R is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It is important to mention that Lemma 1 is more general than Lemma 1 in [23] and Lemma 1 in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Lemma 1 in [23] can be applied only for the case δ = 1 and σ = 0, while Lemma 1 in [25] can be applied only for the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Here Lemma 1 can be applied for all δ > 0 and σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='2 The state space As in [23–25] the state space will be appropriately defined in order to exclude states of the set S defined by (21) that violate the condition (17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e, the states that cause liquid spillage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We define the following X := � (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S : max x∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L](h(x)) < Hmax � (35) R := 2µ � δgh∗ 3 (Hmax − h∗)min(ζ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='ζ2) (36) where ζ1 := max(Γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Γ3) and (37) ζ2 := h∗ Hmax − h∗ max(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='∆2) (38) with Γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='Γ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='∆1 and ∆2 defined as follows: Γ1 := � Hmax h∗ − 2 √ h∗ √Hmax + √ h∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (39) Γ2 := 3µ √ δ(Hmax − h∗) 4m(1 + δ) � gh∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (40) Γ3 := 3σ(δ + 1) � � L2 + (Hmax − h∗)2 − L � 2µ � δgh∗ (Hmax − h∗) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (41) ∆1 := 3µ √ δ 4L � gh∗ (1 + δ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (42) ∆2 := 3σ(δ + 1) √ h∗ 2µ � δg � � (h∗)2 + L2 + L � (43) The aforementioned definition (36),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the fact that h∗ < Hmax and Lemma 1 imply for all (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S with V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) < R the following 0 < Q1 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) ≤ h(x) ≤ Q2 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) < Hmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for all x ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] (44) 10 Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the conditions (17) for avoiding liquid spillage are satisfied when (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S with V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The set X defined by (35) is the state space of system (8)-(12), (14), (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In particular, we consider as state space the metric space X ⊂ R2 × H1 (0,L) × L2 (0,L) with metric induced by the norm of the underlying normed linear space R2 × H1 (0,L) × L2 (0,L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', ∥(ξ,w,h,v)∥X = � ξ2 + w2 + ∥h∥2 2 + ���h′���2 2 + ∥v∥2 2 �1/2 (45) However, we need to approximate the state space from its interior by using certain parameterized sets that allow us to obtain useful estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We define XV (r) := {(ξ,w,h,v) ∈ S : V (ξ,w,h,v) ≤ r }, for r ≥ 0 (46) Inequalities (44) imply that XV (r) ⊆ X, for all r ∈ [0,R) (47) As indicated by the following proposition the set XV (r) for r > 0 contains a neighborhood of � 0,0,h∗χ[0,L],0 � (in the topology of X with metric induced by the norm ∥ ∥X defined by (45)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants q,k,δ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Then for every (ξ,w,h,v) ∈ S satisfying the inequality ���(0,w,h − h∗χ[0,L],v) ���X ≤ ε (48) for some ε > 0 with ε < min(h∗,Hmax − h∗)/ √ L, (49) the following inequality holds: V (ξ,w,h,v) ≤ C1 ���(ξ,w,h − h∗χ[0,L],v) ���2 X + C2 ���(ξ,w,h − h∗χ[0,L],v) ���X (50) where C1 := max \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µ2 h∗ − ε √ L , δ + 1 2 g, (δ + 2)Hmax 2 ,q, 3qk2 2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8, (51) C2 := σ(δ + 1) √ L (52) and ∥·∥X is defined by (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='3 Stabilization results The following theorem guarantees exponential stabilization of the state of the system (8)-(12), (14), (16) by means of the nonlinear feedback law (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 11 Theorem 1 (Stabilization of the Tank-Liquid System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let arbitrary constants ω,k,q,δ > 0 be given and define R by means of (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let arbitrary r ∈ [0,R) be given and assume that k < qθ(r) (53) where θ(r) := ωgµδπ2Q1(r) gµδπ2Q1(r) + 2ωL(mgLHmax(δ + 1)2 + 2µ2δπ2Q1(r)) (54) where Q1 is defined by (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Then there exist constants M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='λ > 0 with the following property: (P) Every classical solution of the system (8)-(12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (16) and f (t) = −ω � (δ + 1) � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx + µ(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L) − h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='0)) − q(w(t) + kξ(t)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for t > 0 (55) with (ξ(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[0]) ∈ XV (r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' satisfies (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' v[t]) ∈ XV (r) and the following estimate for t ≥ 0: ���� � ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t] − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t] �����X ≤ M exp(−λt) ���� � ξ(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[0] − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[0] �����X (56) Remarks on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 1) The arbitrary quantities ω,k,q,δ > 0 are the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' We should point out that the ratio k/q must be sufficiently small due to (53), and this is the only restriction for the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 2) The set XV (r) is the set for which exponential stabilization is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' As indicated by Proposition 1, the set XV (r) for r > 0 contains a neighborhood of � 0,0,h∗χ[0,L],0 � (in the topology of X with metric induced by the norm ∥ ∥X defined by (45)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The size of the set XV (r) depends on r ∈ [0,R) and on δ,q,k (recall (36) and (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It is straightforward to see that the larger the parameter q (or k) the smaller the set XV (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' However, the dependence of XV (r) on δ (through the dependence of R on δ) is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' On the contrary it is a very complicated, non-monotonic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 3) The feedback law (55) only requires the measurement of the four following quantities: the position of the tank denoted by ξ(t), and the velocity of the tank denoted by w(t), the total momentum of the liquid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', the quantity � L 0 h(t,x)v(t,x)dx, and 12 the difference the liquid level at the tank walls, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=', the quantity h(t,L) − h(t,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It should be emphasized that the feedback law (55) does not require the mea- surement of the whole liquid level and liquid velocity profile whereas it is completely independent of the surface tension coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 4) The feedback law (55) is the same feedback law that was used in [23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' When the results in [23,25] and Theorem 1 are taken into account then it fol- lows that the feedback law (55) guarantees robustness with respect to surface tension as well as robustness with respect to wall friction forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' From a con- trol point of view, this is an ideal situation: the feedback law (55) is robust with respect to all possible perturbations of the basic model, its measurement requirements are minimal and guarantees exponential stabilization of the cor- responding closed-loop (nonlinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' not the linearized) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 5) In contrast with [25], Theorem 1 does not provide an estimate for the norm ∥vx[t]∥2, and consequently an estimate for the sup-norm of the fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' A topic for future research is the contruction of an appropriate CLF based on which an estimate for the norm ∥vx[t]∥2 can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 4 Proofs Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is exactly the same with the proof of Lemma 1 in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The only difference is that here we can obtain an additional estimate for ���h − h∗χ[0,L] ���∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Indeed, due to the fact that the function ϕ : R+ → R+defined by ϕ(s) = √ s2 + 1 − 1, for s ≥ 0 (57) is increasing and convex, we can use Jensen’s inequality (see page 120 in [9]) and get for all h ∈ C0 ([0,L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='(0,+∞)) ∩ H1(0,L) with � L 0 h(x)dx = m: ϕ �1 L ���h′���1 � = ϕ �1 L � L 0 ���h′(x) ���dx � ≤ 1 L � L 0 ϕ ����h′(x) ��� � dx = 1 L � L 0 � � (h′(x))2 + 1 − 1 � dx (58) Using (58), the inequality ���h − h∗χ[0,L] ���∞ ≤ ∥h′∥1 (which is a direct consequence of the fact that there exists x∗ ∈ [0,L] such that h(x∗) = h∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' a consequence of continuity of h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the mean value theorem and the facts that � L 0 h(x)dx = m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' h∗ = m/L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the fact that the function ϕ−1 : R+ → R+ (the inverse function of ϕ) is increasing with ϕ−1(s) = � (s + 1)2 − 1 for s ≥ 0 and the inequality � L 0 � � (h′(x))2 + 1 − 1 � dx ≤ V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) σ(δ + 1) (59) 13 which is a direct consequence of definitions (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we get for all (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ∈ S: ���h − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] ���∞ ≤ �� L + V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) σ(δ + 1) �2 − L2 (60) Using the additional estimate (60) in conjunction with the estimates shown in the proof of Lemma 1 in [25] and definitions (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (28) and (29) we get (27) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Consider arbitrary (ξ, w,h,v) ∈ S satisfying (48) and (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Definitions (22), (23), (24) and the inequalities (h(x)v(x) + µh′(x))2 ≤ 2h2(x)v2(x) + 2µ2 (h′(x))2 , (61) (w + kξ)2 ≤ 2w2 + 2k2ξ2, (62) � 1 + (h′(x))2 − 1 ≤ ���h′(x) ��� (63) imply: V (ξ,w,h,v) ≤ δ + 2 2 � L 0 h(x)v2(x)dx + µ2 � L 0 h−1(x)(h′(x))2 dx +δ + 1 2 g ���h − h∗χ[0,L] ���2 2 + 3qk2 2 ξ2 + qw2 + σ(δ + 1) ���h′���1 (64) Following the arguments of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='5 in [25] we obtain from (64) the following: V (ξ,w,h,v) ≤ δ + 2 2 Hmax ∥v∥2 2 + qw2 + 3qk2 2 ξ2 +µ2 � h∗ − ε √ L �−1 ���h′���2 2 + δ + 1 2 g ���h − h∗χ[0,L] ���2 2 + σ(δ + 1) √ L ���h′���2 (65) Inequality (50) is a direct consequence of (65) and definition (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ In order to give the proof of the main result of this study, we need to provide some preliminary lemmas along with their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Every classical solution of the system (8)-(12), (14), (16) satisfies the following equations for all t > 0: d dt E(h[t],v[t]) = −µ � L 0 h(t,x)v2 x(t,x)dx + f (t) � L 0 h(t,x)v(t,x)dx (66) 14 d dt W(h[t],v[t]) = −µg ∥hx[t]∥2 2 − µσ � L 0 h2xx(t,x)dx � 1 + h2x(t,x) �3/2 +f (t) � L 0 (h(t,x)v(t,x) + µhx(t,x))dx (67) where E,W are defined by (23), (24), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Due to (10) and (11) we get for t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' x ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L): vt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + ghx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) = σh−1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x +µh−1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))x + f (t) (68) Combining definition (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (10) and (68) we get for all t > 0 the following ex- pression for the time derivative of the functional (23) : d dt E(h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) = −1 2 � L 0 (h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xv2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx − � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx − g � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx +σ � L 0 v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x dx +µ � L 0 v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)vx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))x dx + f (t) � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx −g � L 0 (h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))x(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) − h∗)dx −σ � L 0 hx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) (h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xxdx (69) Using (69),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' integration by parts as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='11 in [25], (12), 15 (16) and the fact that for all t > 0 σ � L 0 v(t,x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2x(t,x) + h(t,x)hxx(t,x) � 1 + h2x(t,x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x dx = −σ � L 0 vx(t,x)1 + h2 x(t,x) + h(t,x)hxx(t,x) � 1 + h2x(t,x) �3/2 dx (70) − σ � L 0 hx(t,x) � 1 + h2x(t,x) (h(t,x)v(t,x))xxdx = σ � L 0 vx(t,x)1 + h2x(t,x) + hxx(t,x)h(t,x) � 1 + h2x(t,x) �3/2 dx (71) as a consequence of integration by parts as well, we obtain equation (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Next we define for all t ≥ 0 and x ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L]: ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) := h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + µhx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) (72) Definition (72),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (10) and (11) imply for all t > 0 and x ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L): ϕt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) = − \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8edv(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + 1 2gh2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) − σ 1 + h2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x +h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)f (t) (73) Using definition (24) along with (73) and (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we get for all t > 0 : d dt W(h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) = 1 2 � L 0 h−2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)ϕ2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xdx − � L 0 h−1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + 1 2gh2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � x dx +σ � L 0 h−1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x dx +f (t) � L 0 ϕ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx − g � L 0 (h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) − h∗)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xdx −σ � L 0 hx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xx � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) dx (74) Using (12) and integration by parts as in proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='11 in [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we obtain 16 from (74) and definition (72) for all t > 0: d dt W(h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) = −µg ∥hx[t]∥2 2 + f (t) � L 0 (h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + µhx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))dx +σ � L 0 v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x dx +µσ � L 0 h−1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x dx −σ � L 0 hx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)(h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x))xx � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) dx (75) Using (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (70),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (71) and the fact that h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x = \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 1 + h2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) + h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)hxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 x (76) we obtain from (75) equation (67) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants q,k,δ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Then there exists a non-decreasing function Λ : [0,R) → (0,+∞), where R > 0 is defined by (36) such that for every (ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) < R, the following inequality holds: V (ξ,w,h,v) Λ(V (ξ,w,h,v)) ≤ ���h′���2 2 + � L 0 (h′′(x))2 � 1 + (h′(x))2�3/2 dx + � L 0 h(x)(v′(x))2 dx + ξ2 + (w + kξ)2 (77) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let arbitrary (ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) < R be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Using the same arguments as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='12 in [25] and the fact that � L 0 � � 1 + (h′(x))2 − 1 � dx ≤ ���h′���2 2 (78) 17 we obtain the following estimate: V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ≤ L2 (δ + 2)Q2(V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) 2π2Q1(V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) � L 0 h(x)(v′(x))2 dx + \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed (δ + 1) � gL2 + 2σ � 2 + µ2 Q1 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ���h′���2 2 + qk2 2 ξ2 + q 2 (w + kξ)2 ≤ Λ(V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) × ����h′���2 2 + � L 0 (h′′(x))2 � 1 + (h′(x))2�3/2 dx + � L 0 h(x)(v′(x))2 dx + ξ2 + (w + kξ)2 � (79) where Λ(s) := 1 2 max � κ1 + 2µ2 Q1 (s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' κ2Q2(s) Q1(s) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='κ3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for s ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='R) (80) with κ1 := (δ + 1) � gL2 + 2σ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' κ2 := L2 (δ + 2)/π2 and κ3 := qmax(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Defini- tion (80) and the fact that Q2 : R+ → R is an increasing function and Q1 : R+ → R is a decreasing function imply that Λ : [0,R) → (0,+∞) is a non-decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Inequality (77) holds as a direct consequence of (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants q,k,δ > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Then there exist non-decreasing func- tions Gi : [0,R) → (0,+∞), i = 1,2, where R > 0 is defined by (36), such that for every (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R, the following inequalities hold: ���(ξ,w,h − h∗χ[0,L],v) ���2 X ≤ V (ξ,w,h,v)G1 (V (ξ,w,h,v)) (81) V (ξ,w,h,v) G2 (V (ξ,w,h,v)) ≤ ���(ξ,w,h − h∗χ[0,L],v) ���2 X (82) where ∥·∥X is defined by (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let arbitrary (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Using defini- tions (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' inequalities (61),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (62),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' the inequality � 1 + (h′(x))2 ≤ 1 + (h′(x))2 (83) and (44) we obtain V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v) ≤ δ + 2 2 Hmax ∥v∥2 2 + δ + 1 2 g ���h − h∗χ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L] ���2 2 + � µ2 Q1 (V (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v)) + σ (δ + 1) ����h′���2 2 + 3qk2 2 ξ2 + qw2 (84) 18 Inequality (84) implies inequality (82) with G2 (s) := max �δ + 2 2 Hmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' δ + 1 2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' µ2 Q1 (s) + σ (δ + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' 3qk2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='q � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' for s ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='R) (85) The fact that Q1 : R+ → R is a decreasing function and the above definition imply that G2 : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='R) → (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='+∞) is a non-decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof of inequality (81) is exactly the same with the proof of Lemma 4 in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants ω,k,q,δ > 0 and r ∈ [0,R) be given, where R > 0 is defined by (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Then every classical solution of the system (8)-(12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (16) and (55) satisfies the following inequality for all t > 0 for which V (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) < R: d dt V (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) ≤ −3µg 4 ∥hx[t]∥2 2 − qk3ξ2(t) − µδ 2Hmax � 2Hmax − Q1(r)Q2 (V (t)) Q1 (V (t)) �� L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx −µσ � L 0 h2xx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2 (86) where V (t) = V (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' θ(r) is defined by (54) and Qi : R+ → R (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='2) are the functions defined by (28) and (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let ω,k,q,δ > 0 be given constants and let r ∈ [0,R) be a constant, where R > 0 is defined by (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In addition to that we consider a classical solution of the system (8)-(12), (14), (16) and (55) at a time t > 0 for which V (ξ(t),w(t),h[t],v[t]) < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Using Lemma 2, (66), (67) and definition (22) and by following the same procedure as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='14 in [25] by assuming zero friction coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we establish the following inequality: d dt V (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) ≤ −3µg 4 ∥hx[t]∥2 2 − µδ � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx −µσ � L 0 h2xx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2 − qk3ξ2(t) +µδπ2Q1(r) 2L2Hmax � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx (87) Since v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='0) = v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='L) = 0 (recall (12)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' by virtue of Wirtinger’s inequality and (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we get: ∥v[t]∥2 2 ≤ L2 π2 ∥vx[t]∥2 2 ≤ L2 π2Q1 (V (t)) � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx (88) Combining (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (87) and (88),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we obtain (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ 19 We can now present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constants ω,q,k,δ > 0 satisfying (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let constant r ∈ [0,R) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Consider a classical solution of the system (8)-(12), (14), (16) with (55) that satisfies V (ξ(0),w(0),h[0],v[0]) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Let r ∈ (r,R) be a constant that satisfies: Q2 (r) Q1 (r) < 2Hmax Q1(r) (89) The existence of ¯r ∈ (r,R) is a direct consequence of the continuity of the func- tions involved in (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Due to (53), Lemma 5, (86) and (89) the following implication is true: If t > 0 and V (ξ(t),w(t),h[t],v[t]) ≤ r then d d t V (ξ(t),w(t),h[t],v[t]) ≤ 0 (90) A contradiction argument as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='6 in [25] implies that V (ξ(t),w(t), h[t],v[t]) ≤ r for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Implication (90) and the fact V (ξ(t),w(t),h[t],v[t]) ≤ r for all t ≥ 0 imply that d d t V (ξ(t),w(t),h[t],v[t]) ≤ 0 for all t > 0 (91) Due to the above and the continuity of the mapping t → V (ξ(t),w(t),h[t], v[t]), we get that V (ξ(t),w(t),h[t],v[t]) ≤ V (ξ(0),w(0),h[0],v[0]) ≤ r < R,for all t ≥ 0 (92) Consequently, (ξ(t),w(t),h[t],v[t]) ∈ XV (r) for all t ≥ 0 (recall (46)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Using (92) and Lemma 5, we conclude that (86) holds for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Using (92),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (86) and the fact that Q2 : R+ → R is an increasing function while Q1 : R+ → R is a decreasing function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' we obtain the following estimate for t > 0 d dt V (ξ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='w(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='h[t],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='v[t]) ≤ −β(r) � ∥hx[t]∥2 2 + � L 0 h(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)v2 x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x)dx + � L 0 h2xx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) � 1 + h2x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='x) �3/2 dx +ξ2(t) + (w(t) + kξ(t))2 � (93) where β(r) := min �3µg 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' µδ(2Hmax − Q2 (r)) 2Hmax ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='qk3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='q(qθ(r) − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='µσ � (94) Notice that (53) and the fact that r ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='R) in conjunction with definitions (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (36),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' (93) imply that β(r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It follows from Lemma 3, (77), the continuity of the mapping t → V (ξ(t),w(t),h[t],v[t]), (recall that v ∈ C0 (R+ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='H1 (0,L) � , 20 h ∈ C1 (R+ × [0,L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content='(0,+∞)) and v ∈ C0 (R+ × [0,L])), estimates (92), (93), Lemma 4, (81) and (82) that the following estimate holds for all t ≥ 0: ���(ξ(t),w(t),h[t] − h∗χ[0,L],v[t]) ���2 X ≤ Ω(r)exp � −β(r)t Λ(r) ����(ξ(0),w(0),h[0] − h∗χ[0,L],v[0]) ���2 X (95) with Ω(r) := G1 (r)G2 (r) (96) where Λ is the non-decreasing function involved in (77) and Gi : [0,R) → (0,+∞) (i = 1,2) are the non-decreasing functions involved in (81), (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' Es- timate (56) with M = � Ω(r) and λ = β(r) 2Λ(r) is a consequence of estimate (95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' □ 5 Concluding Remarks In this work we managed to show that the robust with respect to wall friction nonlinear feedback law proposed in [25] provides also robust stabilization re- sults with respect to surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' This shows even more the significance of the CLFs as stabilizing tools for the infinite-dimensional case of systems de- scribed by PDEs and illustrates the fact that robustness is inherent in the CLF methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' The present study deals with the case of viscous Saint-Venant system with surface tension and without wall friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' It is of interest to study the more challenging problem of the viscous Saint-Venant system with surface tension and with wall friction as well as the construction of an additional functional which provides a bound for the sup-norm of the fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQfuP2J/content/2301.01688v1.pdf'} +page_content=' In addition to that, other topics for future research are the study of existence and unique- ness of the solutions for the closed-loop system, the study of the problem with non constant (dynamic) contact angles, the study of the output feedback stabi- lization problem, the construction of appropriate numerical schemes and the derivation of stability estimates in stronger spatial norms.' metadata={'source': 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Not. R. Astron. Soc. 000, 000–000 (0000) +Printed 10 January 2023 +(MN LATEX style file v2.2) +Thermal hysteresis and front propagation in dense +planetary rings +R´emy Larue1,2,3, Henrik Latter1⋆, Hanno Rein4,5 +1DAMTP, University of Cambridge, CMS, Wilberforce Road, Cambridge CB3 0WA, UK +2ENS Paris-Saclay, 4 avenue des Sciences 91190 Gif-sur-Yvette, France +3Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS/IN2P3, Grenoble INP, 38000 Grenoble, France +4Department of Physical and Environmental Sciences, University of Toronto at Scarborough, Toronto, Ontario M1C 1A4, Canada +5David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, Toronto, Ontario, M5S 3H4, Canada +ABSTRACT +Saturn’s rings are composed of icy grains, most in the mm to m size ranges, under- +going several collisions per orbit. Their collective behaviour generates a remarkable +array of structure over many orders of magnitude, much of it not well understood. +On the other hand, the collisional properties and parameters of individual ring par- +ticles are poorly constrained; usually N-body simulations and kinetic theory employ +hard-sphere models with a coefficient of restitution ϵ that is constant or a decreasing +function of impact speed. Due to plastic deformation of surface regolith, however, it +is likely that ϵ will be more complicated, at the very least a non-monotonic function. +We undertake N-body simulations with the REBOUND code with non-monotonic ϵ +laws to approximate surfaces that are friable but not sticking. Our simulations reveal +that such ring models can support two thermally stable steady states for the same +(dynamical) optical depth: a cold and a warm state. If the ring breaks up into radial +bands of one or the other state, we find that warmer states tend to migrate into the +colder states via a coherent travelling front. We also find stationary ‘viscous’ fronts, +which connect states of different optical depth, but the same angular momentum flux. +We discuss these preliminary results and speculate on their implications for structure +formation in Saturn’s B and C-rings, especially with respect to structures that appear +in Cassini images but not in occultations. +Key words: instabilities – waves – planets and satellites: rings +1 +INTRODUCTION +Saturn’s rings flaunt an extraordinary array of axisymmetric +structure, both quasi-regular and chaotic, ranging over some +four orders of magnitude in length - from 10 m to 100 km +(Colwell et al. 2009, Cuzzi et al. 2018). Yet despite several +decades of theoretical effort, their origins are only partially +understood (Schmidt et al. 2009, Estrada et al. 2018, Salo et +al. 2018). In particular, the disjunct bands of high and low +optical depth in the B-ring (Horn and Cuzzi 1996, Colwell et +al. 2007), the plateaus in the C-ring (Tiscareno et al. 2019), +and the irregular intermediate scale striations in the A and +B-rings (Porco et al. 2005) are presently without plausible +explanations. Simply put, there is too much observed struc- +ture and too few suitable instabilities (or related processes) +in our theoretical models. Perhaps it is time to re-assess +⋆ E-mail: hl278@cam.ac.uk +some of our fundamental assumptions and explore a wider +range of alternative scenarios. +It is probable, though not assured, that much of the +ring’s unexplained structure arises spontaneously due to +its peculiar granular flow. Since the 1980s researchers have +turned to kinetic theory or N-body simulations to model +this flow, initially calculating the thermal balances under- +lying ring equilibria, and then the (viscous) instabilities +that might generate structure (e.g., H¨ameen-Anttila 1982, +Araki & Tremaine 1986, Wisdom & Tremaine 1988, Salo +1991, H¨ameen-Anttila & Salo 1993, Salo et al. 2001, Lat- +ter & Ogilve 2006, 2008). These studies have made several +strong assumptions, especially regarding the nature of the +ring particles and their collisional behaviour, for instance +rarely deviating from a hard-sphere model with either a con- +stant coefficient of restitution ϵ or a ‘Bridges law’ (Bridges +et al. 1984), whereby collisions below some critical impact +speed are perfectly elastic. In reality, ring particles are likely +to be irregularly shaped and coated in a regolith of small par- +© 0000 RAS +arXiv:2301.03289v1 [astro-ph.EP] 9 Jan 2023 + +2 +Larue, Latter, Rein +ticles ≲ 1 cm (e.g. Doyle et al.1989, Nicholson et al. 2008, +Morishima et al. 2012; Deau 2015) and, being irregular and +fluffy, their surfaces should produce an enhanced inelasticity +at low impact speeds, and indeed possible particle adhesion. +In light of this, the adoption of a constant ϵ, or a Bridges law, +may significantly misrepresent some of the ring’s collective +collisional dynamics. Our paper tests this idea by exploring +other, physically motivated, prescriptions for ϵ. We find, in +fact, that even very simple changes to the collision law can +give remarkably different outcomes. +Continuum mechanical models of viscoelastic collisions +that account for fluffy and/or sticky surfaces demonstrate +that ϵ is a non-monotonic function of impact speed vcoll. +Beneath some critical speed we have ϵ = 0, but on in- +creasing vcoll, ϵ rises, plateaus, and then decreases again +(Gorkavyi 1985, Hertzsch 2002, Albers & Spahn 2006, Bril- +liantov et al. 2007). Laboratory experiments appear to con- +firm this picture (Gorkavyi 1989, Hatzes et al. 1991, Bridges +et al. 1996). We implement collision laws of this basic form +in our paper and term them ‘regolith laws’. In addition, +at or below the critical speed colliding particles may stick, +but we neglect this important effect in order to avoid the +vexed and complicated issue of size-distribution dynamics +(e.g. Brilliantov et al. 2015). Our approach is mainly nu- +merical, via N-body simulations of monodisperse, spherical, +indestructible particles with the code REBOUND; but we +also employ a dense gas kinetic theory, where appropriate. +Note that we do not include self-gravity and thus our sim- +ulations fail to exhibit wakes, nor do they support viscous +overstability, both important phenomena we hope to test +in the future. Our study is distinct but complementary to +recent N-body simulations that explicitly test the role of +adhesion, especially on instabilities (Ballouz et al. 2017, Lu +et al. 2018; see also Section 16.7.1.7 in Salo et al. 2018). Our +main focus, in contrast, will be on disk thermodynamics. +Our first main result is that regolith laws permit a dense +ring to fall into one of two thermally stable states at the +same optical depth: (a) a very dense state with filling fac- +tors ∼ 0.3 and low temperatures, c ≲ aΩ (where c is velocity +dispersion, a is particle radius, and Ω is orbital frequency) +and (b) a moderately dense state with lower filling factors +(≲ 0.1) and a slightly warmer temperature, c ≳ 4aΩ. This +bistability generally favours optical depths less than 1, but +can be pushed up to higher values if we broaden our parame- +ter range. We also find in certain circumstance that the cold +state at low optical depth is metastable: shot noise permits +the ring to spontaneously jump into the hot state. +Our second set of results explores what happens when +different thermal states spatially adjoin. If two states of +the same optical depth but different temperature connect, a +travelling ‘thermal front’ develops that can reach speeds of +≲ aΩ, while maintaining a steady spatial structure. If the +front is too slow, the disparity in the angular momentum +flux between the two states reorganises the front profile so +that the flux is uniform but the optical depth undergoes a +jump, what we term a static ‘viscous front’. Some of the +latter behavior mirrors that witnessed by Salo and Schmidt +(2010) in their simulations of viscous instability. +The plan of the paper is as follows. The next section +begins with a review of the extant literature on low-impact +collisions between regolith covered and/or sticky particles, +moving on to a presentation of the model collision laws we +use, and then our numerical methods. Subsequently, we de- +tail out results: the calculation of thermal equilibria and hys- +teresis in smallish boxes (Section 3), potential metastability +(Section 4), and finally results on spatially adjoining states, +i.e. thermal and viscous front (Section 5). We conclude in +Section 6. +2 +BACKGROUND AND METHODS +This section presents the physical set-up and numerical +model by which we attack the thermal equilibria of rings +composed of regolith-coated particles. We first devote some +space to set the scene, by reviewing the theoretical and ex- +perimental literature and explaining the key ideas and pa- +rameters that underlie work in this area. The model collision +laws we adopt are then exhibited, followed by the details of +the N-body simulations with REBOUND we conduct. +2.1 +Collisional physics and the coefficient of +restitution +We aim to describe the collisional dynamics of many ring +particles in a local patch of a planetary ring. From the outset +we make several strong assumptions that we concede may +distort our results: the particles are taken to be identical, +spherical, and frictionless. Most of the ring mass is in metre- +sized particles, and thus it is that population that we track. +Only binary collisions are considered, and these are deemed +inelastic, so that g′ · k = −ϵ(g · k), where g is the relative +velocity of two colliding particles before the collision and g′ +afterwards, k is the unit vector pointing between the two +particles centres at the moment of collision, and ϵ is the +coefficient of restitution. This coefficient lies between 0 and +1 and is usually a function of the impact speed vcoll = |g · +k|. We neglect the possibility of two particles sticking and +assume that all the specifics of the particle surfaces can be +encapsulated in the functional behaviour of ϵ. Because we +find the ring dynamics are so sensitive to ϵ, we now spend +some time discussing this important physical input. +2.1.1 +Theoretical and experimental background +Research exploring the collisional behaviour of regolith- +covered particles can be separated into analytical calcula- +tions, drawing on continuum mechanics, and laboratory ex- +periments, approximating Saturnian conditions. We attempt +to review and synthesise this body of work. +The seminal experiments in this area were described +in Bridges et al. (1984) and collided smooth ice spheres +with an ice block at temperatures ∼ 170K. This work pro- +duced the collision law ϵ = min +� +1, (vcoll/vcrit)−0.234� +, for +vcrit = 0.008 cm s−1, a defining feature of which is perfect +elasticity at sufficiently low collision speeds (vcoll < vcrit). +This collision law became the standard for subsequent N- +body simulations and other theoretical work. Subsequently, +broken power laws of this type were shown to arise naturally +in generalisations of the Hertz theory to viscoelastic solids +(Dilley 1993, Hertzsch et al. 1995, Brilliantov et al. 1996, +Thornton 1997). However, such theoretical work must posit +that the surfaces of the colliding spheres are smooth and +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +3 +that irreversible energy losses arise solely from viscoelastic +deformations inside the spheres. +Shortly after the Bridges experiments, two neglected +but insightful papers by Gorkavyi (1985, 1989) highlighted +the importance of regolith and argued against perfectly elas- +tic restitution at low impact speed. Gorkavyi emphasised +that ϵ can be dramatically altered at small vcoll because (a) +impact energy can be used up when reshaping a soft fri- +able surface (leaving nothing left over for elastic rebound) +and/or (b) rebounding motion can be countered by surface +stickiness. Using energy arguments, the 1985 paper sketches +out three regimes: (a) at sufficiently low vcoll, there is total +energy loss and thus ϵ = 0 (sticking/adhesion is not consid- +ered); (b) at slightly larger vcoll, ϵ increases with vcoll; and +then (c) after a turning point, ϵ decreases with vcoll (tradi- +tional restitution). The collision law is hence non-monotonic. +Gorkavyi (1989) followed this up with simple experiments +using powders, metals, and marble at room temperature and +pressure, which agree with earlier lab work by Hartmann +(1978, 1985), in a different context, using rocks. +Subsequent papers from the Bridges research group ex- +amined how the state of the particle surface influenced colli- +sions, with a particular focus on the adhesive effect of frost, +a thin layer of microscopic structure that might behave simi- +larly to the thicker regolith layer expected on larger ring par- +ticles. Hatzes et al. (1991) showed frosty particles can stick +at impact speeds below some critical level (a few mm s−1), +but did not examine explicitly how it changed the form of +ϵ. Bridges et al. (1996) conducted a large set of experiments +for different kinds of ices and vcoll at relevant temperatures, +which further strengthened the case for sticking, and also +showed that ϵ exhibited the three main features predicted +by Gorkavyi. +On the theoretical side, the 2000s witnessed various ex- +tensions of Hertz contact mechanics, accounting for both +viscoelasticity and particle adhesion via JRK theory (Albers +and Spahn 2006, Brilliantov et al. 2007; see also Thornton +and Ning 1998, and Chokshi et al. 1993, the latter in the +context of ISM grains). Notable is the work by Hertzsch +(2002) who modelled the two effects of sticking and of pas- +sive regolith deformation, as discussed by Gorkavyi, treating +the passive regolith as a deformable viscous non-sticky ‘soft +layer’. Both physical effects appear to influence the form of +ϵ similarly. In all cases, non-monotonic ϵ laws were mathe- +matically derived. +Brilliantov et al. (2007) provides estimates for solid +water-ice particles of various sizes that, despite several +strong assumptions, help with Saturnian applications. For +metre-sized water-ice impactors, the theory predicts that the +maximum value ϵ takes is relatively large, potentially above +0.7. For cm sized particles, it drops to ≈ 0.3. On the other +hand, the critical vcoll for sticking is roughly 10−2 cm s−1 for +metre-sized ice impactors, and this rises to greater than 0.1 +cm s−1 for cm-sized particles. Because of the model assump- +tions care must be taken, however, when applying these es- +timates, and in fact the quoted critical collision speeds are +probably gross lower limits. The theory omits the energy +dissipation channel associated with irreversible regolith de- +formation (as well as internal fracture) by treating the par- +ticles as solid-ice non-spinning viscoelastic spheres. It also +sets the unknown dissipative constant A by fitting a (non- +sticking) viscoelastic model (Brilliantov et al. 1996) to the +(non-sticking) experimental data of Bridges et al. Nonethe- +less, the Brilliantov results provide a useful starting point +for our study. +Before moving on, we flag additional physics not yet +discussed. In applying the above ideas and prescriptions to +an ensemble of colliding particles, one must acknowledge +that, by virtue of the collisions themselves, particles’ surface +properties will evolve. Repeated collisions will presumably +‘compactify’ particle regolith and hence reduce the mean +critical sticking speed. On the other hand, bombardment by +micrometeoroids will disturb the surfaces and there will be +accretion of very small floating particles, processes that will +rejuvenate regolith. It follows that, in addition to the size +distribution dynamics (e.g. Longaretti 1989, Bodrova et al. +2012, Brilliantov et al. 2015), there will take place related +dynamics controlling the mean surface properties. We do +not attempt to construct a model for this interesting process +here. +2.1.2 +Important scales +This subsection briefly outlines the key velocity scales rel- +evant for our problem. We assume that there is a single +critical sticking speed vstick below which two impactors will +adhere. We also assume a second critical impact speed vcrit +below which ϵ = 0. It may be that these two speeds are the +same, though in general we expect vstick < vcrit, i.e. it is +possible for all the energy of the impact to be used up re- +shaping the surface and resisting the adhesive attraction of +the regolith, thereby allowing the impactors to roll clear of +each other. Particle spin and tidal shear may facilitate such +non-sticking ϵ = 0 encounters. +A third key speed is the velocity dispersion c, as impact +speeds will be distributed around it. Thus the relative size +of c relative to vcrit will determine which collisional regime +(sticking, non-sticking, etc.) the particles are in. Partly con- +trolling c is the orbital shear speed across a particle, aΩ +(recall a is particle radius and Ω the orbital frequency). The +importance of this scale issues from the fact that dense cold +rings adopt a velocity dispersion c ∼ aΩ, in the absence of +gravity wakes, and c ≲ 5aΩ, when gravity wakes are present +(e.g., Araki & Tremaine 1986, Salo et al. 2018)1. It follows +that if c ∼ aΩ ≫ vcrit then the regolith is not going to fea- +ture much in the mean thermal dynamics, and hence the +determination of c. On the other hand, if c ∼ aΩ ≪ vcrit +then the surface properties are going to be important. Com- +plicating this picture, of course, is the size dependence of +both aΩ and vcrit. In a polydisperse ring, however, the ve- +locity dispersion of smaller particles will be similar to the +metre-sized particles (Salo et al. 2018). We now obtain some +bounds on the important parameter vcrit/(aΩ). +First we situate ourselves at a representative location +in the C-ring, in which gravity wakes are likely absent, and +set Ω ≈ 10−4 s−1. If a = 1 m, the most dynamically im- +portant size, aΩ is roughly 0.01 cm/s. Next, applying the +estimates from Brilliantov et al. (2007) (cf. Section 2.1.1) +1 The second estimate can be obtained by assuming a gravita- +tionally unstable ring settles into a state where the Toomre Q is +∼ 1, and then taking typical values for the surface density (e.g. +Hedman & Nicholson 2013, 2016) +© 0000 RAS, MNRAS 000, 000–000 + +4 +Larue, Latter, Rein +and setting vcrit = vstick, we obtain vcrit/(aΩ) ∼ 1. For cm +sizes, vcrit/(aΩ) ≳ 10 (noting that the velocity dispersion of +this population is set by the metre sizes). As argued earlier, +the Brilliantov estimates for vcrit only provide lower bounds, +and hence we conclude that it is likely that the C-ring is in +a regime where surface regolith properties will matter. +At a representative location in the A or B-ring, we must +take into account gravity wakes. Thus we find ourselves in +a more ambiguous situation: the Brilliantov estimates yield +vcrit/c ≳ 0.1 for metre-sized particles, and vcrit/c ≳ 1 for +cm-sized particles. Depending on how badly the Brilliantov +results underestimate vcrit, we could be in a marginal regime +or in a regolith-dominated regime. Certainly, further work +on the collisional dynamics of ice would help decide on this +point. As we do not simulate self-gravity, for now we just +assume that aΩ < vcrit, and leave open its importance to +future work. +2.1.3 +Model coefficients of restitution +This section presents the two classes of non-monotonic ‘re- +golith’ ϵ-law we use in this paper. We have attempted to +paramaterise these laws in two readily understandable quan- +tities: vcrit, the impact speed at which collisions are perfectly +inelastic (cf. Section 2.1.2); and ϵmax, the turning point value +of ϵ (i.e., its maximum). +A broken power law (BPL) for ϵ, though somewhat +crude has the benefits that it has few input parameters and +some headway can be made with it using kinetic theory. We +define the law in the following way: +ϵ(vcoll) = +� +ϵ0, +if vcoll < vcrit. +ϵmax (vcoll/vcrit)−p , +if vcoll ⩾ vcrit. +(1) +We set the exponent p += +0.234, following Bridges et +al. (1984), though it could take other values. The quantity +ϵ0 we set equal to either 1, to obtain the Bridges et al. law +itself, or equal to 0, to get the opposite perfectly inelastic +law. The Bridges BPL is plotted in Fig. 1 in blue. +A more realistic non-monotonic ϵ law that is smoother +and exhibits something of a plateau near its maximum can +be defined in several ways. We choose the following: +ϵ(vcoll) = +� +0, +if vcoll < vcrit +1.625 ϵmax ζ/(1 + ζ1.234), +vcoll ⩾ vcrit, +(2) +where ζ = (vcoll−vcrit)/b and b is the plateau ‘width’, usually +set to aΩ. Constants have been chosen so that ϵ approaches +the Bridges law for large vcoll. To facilitate the discussion +later, when we compare the different models, we refer to +Eq. (2) as a ‘realistic’ law (though it is yet to be determined +how realistic it is). We plot it in Fig. 1 in red. +2.2 +The potential for bistability +Before presenting our numerical methods and the results +that ensue, we briefly explain why a non-monotonic colli- +sion law, such as given in Eq. (2) and displayed in Fig. 1, +potentially yields two stable states for the same parameters. +At lower optical depths, N-body simulations and ki- +netic theory show that the Bridges law yields equilibria with +c > aΩ, and thus most collisions sample the power-law de- +creasing segment of the ϵ curve (Salo 1991, Latter & Ogilvie +0 +1 +2 +3 +4 +5 +6 +vcoll / vcrit +0 +0.2 +0.4 +0.6 +0.8 +1 +Figure 1. Two forms of the coefficient of restitution ϵ as a func- +tion of impact speed vcoll. The solid blue curve is the Bridges +law, see Eq. (1), with ϵ0 = 1. The red solid curve is the ‘regolith’ +law, Eq. (2), with b = (1/4)vcrit and ϵmax = 0.75. In addition, +we have sketched two velocity distribution functions with black +dotted curves; see discussion in Section 2.2. +2008). As mentioned above, the realistic regolith law we +adopt approaches the Bridges law for impact speeds larger +than the turning point in ϵ, and is a reasonable approxi- +mation near the turning point. One might then expect that +collisions employing the regolith law would sample similar +values of ϵ and the resulting thermal equilibria will resemble +the Bridges equilibria, giving us a ‘warm’ ring. In Fig. 1 we +superimpose a mock impact velocity distribution at larger +vcoll to indicate such a state. +On the other hand, when ϵ is a constant and taken to +be equal to zero the thermal equilibria are especially cold, +with c ∼ aΩ (e.g. Araki & Tremaine 1986). It follows that +our regolith law might be capable of supporting these very +cold equilibria as well. This should certainly be the case +if vcrit is much larger than aΩ. In this circumstance, most +impact speeds will fall below vcrit and thus yield perfectly +inelastic collisions with ϵ = 0, never sampling the non-zero +segment of the ϵ curve. Fig. 1 indicates a schematic velocity +distribution for this state, centred on a value less than vcrit. +Both the warm state and the cold state are thermally +stable, as has been shown separately in N-body simulations. +And thus a non-monotonic law may yield bistability. The +disk may fall into either the cold or the warm homogeneous +state for exactly the same parameters (most notably optical +depth τ)2. Which is chosen depends on the initial conditions. +Moreover, it follows there must also be an intermediate ther- +mally unstable state separating the two stable states, though +this will not normally be observed. The argument for bista- +bility is strongest in a regime where vcrit ≫ aΩ. A question +then is: what is the minimum value of vcrit that yields bista- +2 This bistability is different to the ‘phase transitions’ associ- +ated with viscous instability, which drives the system to a non- +homogeneous state characterised by abutting radial regions of +high and low optical depth (e.g., Lukkari 1981, H¨ameen-Anttila +1982, Salo & Schmidt 2010). +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +5 +bility? Our simulations results in Section 3 aim to answer +this and other questions. +2.3 +N-body simulations +In this subsection we further outline the physical model we +adopt and the numerical methods used to calculate its non- +trivial thermal dynamics. We seek to determine the evolu- +tion of a large number of inelastically colliding particles, and +thus our main tool will be local N-body simulations. +2.3.1 +Equations of motion +We solve the equations of motion in the Hill approximation +(Hill 1878), a local coordinate system that is co-rotating +with a particle on a circular orbit. The gravity from the cen- +tral object is linearized in local coordinates and the orbital +frequency is a constant. This allows, but does not restrict, +us to use shear-periodic boundary conditions. In that case, +the Hill approximation is also referred to as the shearing +sheet. In our notation, the x, y, and z coordinates point in +the radial, azimuthal and vertical direction, respectively. +Treating the central object, Saturn, as a point source, +the equations of motion for a test particle can be written as +¨x = 2Ω ˙y + 3Ω2x + F coll +x +, +(3) +¨y = −2Ω ˙x + F coll +y +, +(4) +¨z = Ω2z + F coll +z +, +(5) +where Fcoll is the (intermittent) acceleration exerted on a +particle during a collision. In the absence of collisions, the +solution to these equations can be written as epicycles (e.g. +Rein & Tremaine 2011). +The particles move within a finite-size numerical do- +main/box. We denote the radial length of the box by Lx +and the azimuthal length by Ly. In all our experiments, the +vertical length of the box Lz has been chosen to be large +enough so that no particle ever crosses the vertical bound- +aries. Otherwise, the box is periodic in y and shear-periodic +in x. +The only further ingredients needed are the finite par- +ticle radius a and a collision model. We treat particles as +hard spheres (they are not permitted to overlap) and the +outcome of a collision is described using a normal coefficient +of restitution, as described in Section 2.1.3. The particles +have no spin. +2.3.2 +Numerical method +We use the freely available N-body code REBOUND (Rein +& Liu 2012) to perform all of the simulations presented in +the paper. To evolve the equations of motion forward in +time, we use the Symplectic Epicycle Integrator (SEI, Rein +& Tremaine 2011) which is well suited for simulations of +particle motion within the Hill approximation. +Collisions are detected using a nearest neighbour tree +search. We randomize the order in which collisions are re- +solved after each timestep. We found that this removes spu- +rious correlations which might otherwise be introduced when +choosing a specific order in which collisions are resolved (i.e. +resolving them from left to right, by a numerical particle +identifier, or by the position in memory). +2.3.3 +Diagnostics +In order to probe the collective behaviour of the granular +flow, we require a number of averaged quantities. We define +the mean normal geometrical optical depth τ as the total +projected area of the particles on the (x, y) plane divided by +the total area of the (x, y) plane. In other words, +τ = Nπa2/(LxLy), +(6) +where N is the number of particles. Thus, τ is stipulated +at the beginning of each run and does not change. We also +define the radially and temporally varying optical depths, +by subdividing the radial domain into thin strips of radial +length LS: +τ(xi, t) = Ni(t)πa2/(LSLy), +(7) +where xi is the radial location of, and Ni(t) is the number +of particles in, the i’th strip at time t. +The filling factor is defined as the proportion of volume +taken up by the particles. For spherical particles it can be +defined as FF = (4π/3)na3, where n is volumetric number +density. Particularly useful is the filling factor at the mid- +plane FF0, which requires the calculation of the number +density at z = 0. +The mean velocity dispersion tensor is computed via +Wij = ⟨ ˙xi ˙xj⟩ +(8) +where ( ˙x1, ˙x2, ˙x3) = ( ˙x, ˙y + 3 +2Ωx, ˙z) is the velocity relative +to the shear and the angle brackets indicate a suitable aver- +age over the particles and possibly over time. The velocity +dispersion c2 is then Wii/3. Note that this definition is only +correct if there are no mean flows additional to the Keple- +rian shear. If such flows are slow (as in viscous instability), +the error will be small, however. +The translational (local) component of the kinematic +viscosity is +νtrans = (2/3)Wxy/Ω. +(9) +The collisional (non-local) component of the viscosity is +νcoll = +2 +3ΩNδt +� +(x⟩ − x⟨)∆py +(10) +where the sum is taken over all binary collisions that occur +in a time interval δt. Here M is the total mass of all ring +particles, ∆py is the transfer of specific y momentum from +the inner to the outer particle in each collision, and x⟩ and +x⟨ are the radial locations of the two impacting particles +(Wisdom & Tremaine 1988; Daisaka, Tanaka & Ida 2001). +As we neglect self-gravity, there is no gravitational or wake +contribution to the overall momentum transport. The total +viscosity is hence νtot = νtrans + νcoll. +To determine the thermal conductivity of a given equi- +librium state, we follow the method of Salo et al. (2001) and +create a steady non-uniform temperature T profile in the +radial (x) direction, where T = c2. In our cold-state simula- +tions, we achieve this by making vcrit radially dependent in +the collision law. In our hot-state simulations, we vary ϵmax +by a small amount in the radial direction. In either case, +we end up with a steady-state sinusoidal radial temperature +profile, though some experimentation is required to find the +right amplitude for the variations in vcrit and ϵmax. The goal +is to keep the perturbations in the temperature ∆T small, +but not too small so that they are dominated by shot noise. +© 0000 RAS, MNRAS 000, 000–000 + +6 +Larue, Latter, Rein +We typically use a simulation with Lx = Ly = 200a and run +it for at least 1000 orbits. +After setting up the nonuniform temperature profiles, +we then measure specific translational (local) and collisional +(non-local) heat fluxes, +qtrans +i += 1 +2σ⟨c2ci⟩ +(11) +qcoll +i += σ � ∆xiδEs +Nδt +(12) +where σ = N/(LxLy) is the number surface density, δxi is +the absolute difference of the i-coordinates of the two parti- +cles involved in a collisions, and δEs is the change in trans- +ported energy (as opposed to dissipated energy) during the +collision for the particle with the larger xi coordinate. Fi- +nally, we assume the heat flux is linearly dependent on the +temperature gradient, +q = −κ∇T. +(13) +We can then correlate the measured qx and ∂xT and retrieve +the conductivity κ using a least squares fit. Finally, to verify +our set up was working properly, we successfully reproduced +Fig. 8 in Salo et al. (2001), though omit these results for the +sake of space. +2.3.4 +Parameters and initial conditions +In all our N-body simulations, we adopt units so that a = 1 +and Ω = 1, though in what follows a and Ω reappear oc- +casionally in order to make a point. As a consequence, the +main physically relevant input is the collision law. Specifi- +cally, we have some combination of vcrit/(aΩ), b, and ϵmax +for non-constant collision laws. We also have the sizes of the +numerical domain Lx and Ly and a constant dimensionless +time-step Ωdt. +We use initial conditions where particles are arranged +uniformly in the plane with a uniform optical depth τ. +Therefore an important initial input is particle number N +while keeping the computational domain fixed. Particles are +normally distributed in the z-direction. The initial velocities +are also normally distributed with an initial velocity disper- +sion c0. In most cases we initialize the particles close to the +thermal equilibrium we believe to be present. +We present convergence tests in Appendix A. These +tests shows that our simulations are converged as we vary +numerical parameters for both extremely high and low opti- +cal depth, as well as hot and cold equilibria. For the regimes +that we are interested in, we found that a dimensionless +timestep of 10−3 and a box size of 10s to 100s particle radii +are sufficient. The large box sizes are needed only for very +hot and dilute rings. +2.4 +Kinetic theory +Though not the focus of this paper, it is useful to have some +kinetic theoretical results, especially as they reveal the exis- +tence of the additional (thermally unstable) middle branch +of equilibrium solutions. The formalism adopted is Latter +and Ogilvie’s (2008) reformulation of Araki and Tremaine +(1986), which does not attempt to solve the Boltzmann- +Enskog equation but rather a truncated moment hierarchy +of continuum equations. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +100 +101 +C +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +10-2 +100 +total +1 +10 +5 +20 +0 +Figure 2. Velocity dispersion and total angular momentum flux +τνtot versus optical depth τ for various hard-sphere ϵ laws, cal- +culated from N-body simulations. In the top panel the appended +numbers ‘1-20’ describe the values of vcrit/(aΩ) when using the +standard Bridges law, whereas ‘0’ indicates runs with a constant +ϵ = 0. In the bottom panel, the ordering of the curves is retained. +The green symbols indicate that the viscous flux is decreasing and +the disk viscously unstable. +In previous deployments of this approach, the depen- +dence of ϵ on the impact speed was only approximately in- +corporated via a ‘pre-averaging’ procedure (see Section 2.2.7 +in Latter and Ogilvie 2008). Though convenient, this in- +troduces unacceptable errors when using complicated non- +monotonic laws as in Section 2.1.3. Thus the complete for- +malism is adopted. This does require completing three (in- +stead of two) integrations in the collision term. The other +main approximations adopted are ‘vertical locality’ and a +triaxial Gaussian for the velocity ellipsoid (see Araki and +Tremaine 1986 and Latter and Ogilvie 2008 for more de- +tails). +3 +HOMOGENEOUS STEADY STATES +In this section we simulate various thermodynamic equilibria +and demonstrate that a non-monotonic epsilon law supports +up to two equilibria for a given optical depth. We charac- +terise these several states with respect to not only their ve- +locity dispersion, but also their packing fraction FF0 and +transport properties, especially with respect to angular mo- +mentum and heat. +We begin by reproducing previous results in the litera- +ture with both a constant and monotonic epsilon law so as +to verify that our code is working properly. Moreover, as ar- +gued in Section 2.2, some of the equilibria obtained are lim- +iting cases of those appearing in the bistable circumstances +explored later and are thus useful in setting the scene. +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +7 +0 +1 +2 +3 +0 +5 +10 +15 +0 +1 +2 +3 +0 +5 +10 +0 +1 +2 +3 +0 +0.2 +0.4 +0.6 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +C +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +FF0 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +2 + tot +Realistic +max=0.75 +Realistic +max=0.923 +BPL +max=0.8 +Figure 3. Selected equilibrium properties as functions of τ for three regolith ϵ-laws (the three columns). The leftmost column shows +equilibria computed with the broken-power law model (BPL) with ϵ0 = 0 and ϵmax = 0.8, whereas the other two columns show the +realistic model with ϵmax = 0.75 and 0.923. In all cases vcrit = 5. In the top row the joined circles denote the velocity dispersion calculated +by N-body simulations, with the colours indicating hot (red) or cold (blue) branches. The second and third rows show the filling factor +and total angular momentum flux respectively. The dashed curve indicates equivalent solutions obtained from the kinetic theory (in the +BPL case only). In the bottom row, a green symbol indicates expected viscous instability. +Figure 4. The distribution of impact velocities in simulations +using the ‘realistic’ law at τ = 1 with parameters vcrit = 5, b = 1, +and ϵmax = 0.923. The left panel shows the system in the cold +state. The right panel shows the system in the hot state. The red +line corresponds to the ϵ law adopted. +3.1 +Comparison with previous calculations +Our reference cases include the simulations of Salo (1991), +who employed a Bridges law but with a variable scale ve- +locity, i.e. Eq.(1) with ϵ0 = 1 and vcrit = 1, 5, 10, 20 (see +Section 2.1), and also simulations with a constant ϵ = 0, +which brings about a very cold state. The results of our cal- +culations are plotted in Fig. 2, in which we show the velocity +dispersion c and angular momentum flux τνtot versus optical +depth τ. The simulations were run until they were collision- +ally relaxed, and then continued for the same length of time +to obtain averaged quantities. When τ was low, and colli- +sions relatively infrequent, the total run time was > 1000 +Ω−1; but at higher τ (∼ 2) runs could be as a short as 50-80 +Ω−1. +Direct comparison of Fig. 2 with the numerical results +of Salo (1991; cf. his Figs 3-5) shows good agreement, and +also consistency with the kinetic theory of Latter & Ogilvie +(2008) (note that both these works denote vcrit by vb). An +interesting feature of the ‘warmer’ solution branches is the +decreasing viscosity with τ. In fact, the hottest case, vcrit = +20, is viscously unstable because the gradient of the angular +momentum flux τν is negative in an interval of τ (green +markers). +By inflating vcrit in the Bridges law the velocity disper- +sion of the system can be controlled and, in particular, set +to ‘warm’ values greater than aΩ and, consequently, greater +than the temperature of the very cold ϵ = 0 states. These +warm and cold states help illustrate the arguments presented +in Section 2.2. If we take one of the two non-monotonic col- +lision laws and set vcrit ten or more times aΩ, then start +the simulation with a hot initial condition, we might expect +the subsequent spread of impact speeds to be sufficiently +far from ϵ’s turning point (cf. Fig. 1) so that the system +settles into a warm ‘Bridges equilibrium’, similar to those +plotted in Fig. 2. On the other hand, if we begin the same +simulation but with very cold initial velocities (≪ vcrit), the +subsequent spread of impact speeds will remain less than +vcrit and ϵ will almost always take the value of 0; the system +© 0000 RAS, MNRAS 000, 000–000 + +LD +LD +0.B +0.B +0.6 +0.6 +0.4 +t0 +0.2 +0.2 +20 +0.0 +-0.0 +0 +0 +24 +mpact velociby y8 +Larue, Latter, Rein +will then converge to the appropriate constant ϵ = 0 state in +Fig. 2. Note that the Bridges law produces a velocity disper- +sion c that decreases with τ and we may then expect that +for sufficiently large τ the upper ‘hot state’ will be too close +to the ‘cold state’ and bistability may disappear. +3.2 +Non-monotonic collision laws +In this section we calculate equilibria for ‘regolith’ epsilon +laws that are non-monotonic: either the broken power law +(BPL) with ϵ0 = 0 or the realistic law (2). The parameters +are ϵmax = 0.75, 0.8 or 0.923, vcrit = 5aΩ, and b = 1, though +we examine a broader spread of values in Section 3.2.2. We +first examine in some detail the thermal properties of the +states, then their transport of angular momentum and heat. +3.2.1 +Thermal hysteresis +Figure 3 constitutes the first main results of the paper. Here +we plot the equilibrium velocity dispersions (top row), filling +factors (middle row), and total radial angular momentum +fluxes (τνtot; bottom row) obtained in a sequence of simu- +lations at different optical depths and for different ϵ mod- +els and parameters. Each circular marker corresponds to a +different simulation. These values are obtained by time av- +eraging a quantity once the system has become collisionally +mature, as earlier. For example, τ = 0.1 runs were run for +1600Ω−1 and averaged for the last 800Ω−1, while at τ = 2 +the total run length was 80Ω−1, with the averaging taking +place over the last 40Ω−1. +As is clear, in the three models presented, two steady +state branches (distinguished by red and blue) are possi- +ble within a certain range of optical depth. Which of the +two the system selects depends on the initial condition: a +‘cold start’ (low initial c) usually (but not always) takes the +system to the nearby cold state, whereas a ‘hot start’ (ini- +tial c sufficiently high) settles on the hot state. Typically, +runs starting with c = 0.5aΩ converged to the nearby cold +state, while runs beginning with c = 10aΩ migrated to the +hot state, if one was available, even if that state’s veloc- +ity dispersion was significantly larger than the initial c. The +direction of migration is discussed further in Section 4. +The apparent bistability extends over a range of small +to intermediate optical depths. Beyond a special τ the hot +state disappears, and all hot start simulations landed on the +cold branch. At small τ we never found that the cold state +disappeared, except in the case of the realistic model with +ϵmax = 0.923 and τ = 0.1; this equilibrium was metastable +(explored in more detail in Section 4). The bistable regime’s +width (in τ) depends on the parameters. From Fig. 3, in- +creasing the ϵmax in the realistic model from 0.75 to 0.923 +moved the special τ from roughly 0.5 to 1.6 (cf. middle and +right columns). +The cold equilibria take c values very much in agree- +ment with the constant ϵ = 0 states simulated in the previ- +ous subsection, while the hot state resembles a Bridges law, +with c decreasing with τ. In fact, the hot simulations of the +realistic model with ϵmax = 0.923 take a similar c as the +Bridges vcrit = 10 runs, while those with ϵmax = 0.75 resem- +ble a Bridges law with (roughly) vcrit = 5. These similarities +bolster our interpretation of the two states as ‘separated’ by +Figure 5. Grids of simulations undertaken with different vcrit +and ϵmax using the realistic regolith law with widths b = 1 (top) +and b = 2 (bottom). Colours correspond to values of |chot −ccold| +(see text). The contour is a conservative boundary between cases +that support bistability (to the right and above) and those that +do not. +the turning point of the ϵ curve: only a minority of collisions +in the hot state occur with the low impact speeds that would +trigger ϵ = 0, while collisions in the cold state rarely occur +with impact speeds sufficiently large to trigger larger ϵ. To +flesh out this point further we plot in Fig. 4 the distribution +function of impact speed for a hot state (right panel) and +a cold state (left panel) for the same τ = 1 (and other pa- +rameters). Superimposed in red is the ϵ law used. As the left +panel indicates, cold state collisions are almost completely +inelastic; the narrow spread in impact speeds barely overlaps +the portion of the curve for which ϵ ̸= 0. In contrast, the hot +state (shown in the right panel) is much broader and thus +samples a wide range of ϵ, but importantly peaks at speeds +which yield collisions with a small dissipation of energy. +The filling factors in the middle row of Fig. 3 reveal that +the hot branches are far less dense than the cold branches. +For example, in the realistic model with ϵ = 0.923, at τ = 1 +the hot state possesses a filling factor of 0.08, while the cold +state has 0.35. The difference, of course, is not due to the +surface number density (which is the same) but because the +disk semi-thickness is so different between these two states: +in the hot state it is ≈ 6a, compared to ∼ a in the cold state. +The ratio of the two filling factors should scale roughly with +the ratio of semi-thicknesses and that is indeed what we see. +The hot state branch terminates when its velocity dis- +persion approaches a critical value ∼ 3. In reality the sys- +tem here encounters a saddle-node bifurcation and the solu- +tion curve bends ‘backwards’ thus forming an intermediate +branch of thermally unstable solutions. Because these solu- +tions are unstable they cannot manifest in N-body simula- +tions3, but they can be calculated by kinetic theory. Kinetic +3 See Salo et al. (1988) for a numerical exploration of a thermally +unstable state. +© 0000 RAS, MNRAS 000, 000–000 + +b=1 +30 +0.9 +25 +0.8 +20 +max +15 +0.7 +10 +0.6 +5 +0.5 +0 +0 +1 +2 +3 +4 +5 +9 +7 +8 +9 +b=2 +50 +0.9 +40 +0.8 +max +30 +0.7 +20 +0.6 +10 +0.5 +0 +0 +1 +2 +3 +4 +5 +9 +7 +8 +9 +critThermal hysteresis in rings +9 +theoretical equilibria are plotted in the leftmost column with +a dashed black curve; the top and middle panels show clearly +an intermediate cool, semi-dense branch. The agreement be- +tween theory and simulations is qualitative good, with the +biggest deviation in the translational viscosity in the hot +state, a discrepancy that has been noted in previous com- +parisons (Latter and Ogilvie 2008, Rein and Latter 2013).4 +3.2.2 +Parameter survey +In the preceding subsection we examined only three param- +eter sets/models; in this subsection we adopt the realistic ϵ +law and scan through vcrit and ϵmax for two different widths +b. Our aim is to determine how representative the thermal +hysteresis explored in the previous subsection really is. Of +particular interest are the lowest values of vcrit and ϵmax that +yield bistability. +In Fig. 5 we present ‘bistability plots’ for b = 1 and +2. Each square in the grid corresponds to a parameter pair +(vcrit, ϵmax), and for each square we conduct two simulations +with τ = 0.1, one with a hot initial condition and the other +with a cold initial condition. Each simulation has been run +until thermal equilibrium has been obtained, and the dif- +ference in final velocity dispersion calculated, |chot − ccold|. +Finally, the square is coloured accordingly (cf. the colour +bar). If the difference in final c is between 0 and 5, we in- +terpret that the two simulations are converging on to the +same (cold) equilibrium. Values larger than 5 (admittedly, a +rather large value, given Fig. 3) we assume correspond to a +bistable situation: the two simulations are settling on differ- +ent thermal states. In both panels we have superimposed the +contour of |chot − ccold| = 5. The reader should then assign +bistability to regions of the parameter plane above and/or +to the right of this curve. +The plots indicate, as expected, that bistability is +favoured by larger values of vcrit and ϵmax. Increasing both +parameters helps to separate the typical impact speeds of +the hot state from those of the cold state. Interestingly, +the bistable region is quite rectangular. Thus when b = 1, +bistability is guaranteed (roughly) if both vcrit > 4 and +ϵmax > 0.7. We expect that these parameter restrictions +should hold roughly for other non-monotonic laws. Finally, +the range of bistability is also sensitive to the width of the +epsilon law, as the b = 2 plot demonstrates. Increasing the +width also helps separate out the two states. In the b = 2 +case bistability occurs when vcrit > 3 and ϵmax > 0.65. +3.2.3 +Viscous properties +The equilibrium states discussed in the previous subsection +support a viscous stress that, by acting on the background +orbital shear, transports angular momentum radially across +the numerical domain. The viscous properties of the flow are +important thermodynamically because the stress extracts +free energy from the shear, thus providing the heating source +in the thermal balances undergirding these states. But the +viscous stress is also important dynamically because it can +4 Unfortunately, numerical difficulties prevented us calculating +kinetic solutions for the realistic model. +τ +κL (C) +κNL (C) +κL (H) +κNL (H) +0.1 +4.75 +0.42 +77.94 +0.72 +0.2 +5.92 +0.62 +119.26 +2.01 +0.3 +7.42 +1.15 +111.88 +3.39 +0.4 +6.83 +1.61 +89.95 +3.59 +Table 1. Calculated translational (local) thermal conductivities +κL and collisional (non-local) thermal conductivities κNL in cold +(C) and hot (H) equilibria at various optical depths τ. A realistic +collision law is adopted with ϵmax = 0.75, vcrit = 5, and b = 1. +beget instabilities, such as the viscous overstability and in- +stability (Schmidt et al. 2009). In particular, if d(τνtot)/dτ +is negative then viscous instability occurs (Lin and Boden- +heimer 1981, Lukkari 1981, Ward 1981). +The angular momentum flux is plotted in the bottom +row of Fig. 3. Note that a subset of hot states possess a +decreasing flux and are thus viscously unstable; these are +marked in green. In the BPL model, the unstable interval +encompasses τ of 0.4 and 0.5, whereas in the realistic model +only the ϵmax = 0.923 case yields instability and then for +τ between approximately 0.8 and 1.6. Instability here is as- +sociated with a dominant translational viscosity, which can +decline at sufficiently large τ. Growing modes do not appear +in these simulations, however, because the numerical do- +main size is smaller than the shortest unstable wavelength; +in Section 5.2.2 we simulate larger domains and recover the +instability. +3.2.4 +Thermal conductivity +Anticipating later sections which explore different thermal +states that spatially adjoin, we compute the radial flux of +thermal energy. In the absence of any mean spatial gradients, +such as in the homogeneous equilibria calculated, the flux +must be zero. But if two states connect in radius the flux +must control, in part, how their interface evolves. +As explained in Section 2.3.3, we adopt the approach of +Salo et al (2001) and impose a radial sinusoidal temperature +structure upon the box, through the parameters vcrit and +ϵmax. In Fig. 7 we show calculations of the radial thermal +flux qx and the thermal conductivity κ for a fixed set of +parameters (ϵmax = 0.75, vcrit = 5, b = 1) and for the same +optical depth τ = 0.2. The left four panels correspond to the +cold state (c ≈ 1), and the right to the hot state (c ≈ 6). +The top left panel in each case describes the temperature +profile across the box, while the top right panel shows the +temperature gradient (solid blue), the translational (local, +‘L’) heat flux (dashed gold), and the collisional (nonlocal, +‘NL’) heat flux (dotted green). The latter two are plotted +separately as functions of the temperature gradient in the +bottom panels; a best-fit line extracts the conductivities. +In both the hot and cold cases, the translational heat +flux dominates the collisional flux. This means that the heat +flux in the two states differs significantly, despite possessing +the same τ. In Table I we list κ for a range of τ and otherwise +with the same parameters as in Fig. 6. +© 0000 RAS, MNRAS 000, 000–000 + +10 +Larue, Latter, Rein +100 +0 +100 +x +0.75 +0.80 +0.85 +0.90 +0.95 +T +100 +0 +100 +x +0.02 +0.00 +0.02 +dT/dx +qx (L) +qx (NL) +100 +0 +100 +x +40 +45 +50 +T +100 +0 +100 +x +20 +10 +0 +10 +20 +0.003 +0.000 +0.003 +- dT/dx +0.02 +0.00 +0.02 +qx (L) +L=5.92 +0.003 +0.000 +0.003 +- dT/dx +0.004 +0.002 +0.000 +0.002 +0.004 +qx (NL) +NL=0.62 +0.2 +0.0 +0.2 +- dT/dx +20 +10 +0 +10 +20 +30 +qx (L) +L=119.26 +0.2 +0.0 +0.2 +- dT/dx +0.4 +0.2 +0.0 +0.2 +0.4 +qx (NL) +NL=2.01 +COLD, max = 0.75, vc = 5, b = 1, += 0.2 +HOT, max = 0.75, vc = 5, b = 1, += 0.2 +Figure 6. Thermal diffusivity measurements for τ = 0.2 in the cold state (left panels) and hot state (right panels) for the realistic model +with ϵmax = 0.75, vcrit = 5, and b = 1. +Figure 7. Velocity dispersion as a function of time for runs with τ = 0.1 (top panel) and τ = 0.2 (bottom panel). The realistic model is +adopted with ϵmax = 0.923, vcrit = 5, and b = 1. +4 +METASTABILITY +In the last section we calculated steady states that appear to +be thermally stable, at least linearly according to a contin- +uum interpretation. However, N-body systems are replete +with small but finite amplitude shot noise that continually +tests the nonlinear stability of any steady state. If the basin +of attraction of a linearly stable state is small relative to the +amplitude of these fluctuations, the system can potentially +jump out of the state and migrate elsewhere. Many phys- +ical and biological systems offer similar examples of noise +destabilising what should be linearly stable fixed points (e.g. +Mel’nikov 1991, May 1973, De Swart and Grasman 1987, +Majda, Timofeyev and Vanden-Eijinden 1999, 2003). In this +section we investigate this possibility. +Our focus will be on cold states of low-optical depth +and on the hot states near the saddle node bifurcation. The +reason is that these states are close to the unstable mid- +dle branch which can serve as the boundary of the basin +of attraction in each case. We find that, for the parameters +and models we employ, metastability is relatively uncom- +mon, only occurring in certain dilute and cold states. In +particular, states near the saddle node are generally stable +to shot noise perturbations. +Before presenting our results we emphasise that we +only explore the effect of intrinsic shot noise, but in real +rings there are several other sources of finite amplitude +disturbances that may work similarly, e.g. meteoroid bom- +bardment, embedded moonlets, density waves, and gravity +wakes. +© 0000 RAS, MNRAS 000, 000–000 + +20 +15 +10 +5 +0 +0 +100 +200 +300 +400 +500 +time [orbits]1.1 +1.0 +0.9 +0.8 +0 +25 +50 +75 +100 +125 +150 +175 +200 +time [orbits]Thermal hysteresis in rings +11 +Figure 8. Velocity dispersion as a function of time for runs of different initial conditions with τ = 1.61 (top panel) and τ = 1.64 (bottom +panel). The realistic model is adopted with ϵmax = 0.923, vcrit = 5, and b = 1. +4.1 +Cold to hot transitions +We find spontaneous transitions from the cold lower branch +to the hot upper branch in only a few low τ cases when +adopting a realistic collision law and ϵmax = 0.923. Specif- +ically, when τ = 0.1 the system can hover about the cold +steady state for several hundred orbits before jumping to +the hot state. +To probe this behaviour we ran 24 runs with slightly +different initial conditions (varying both particles’ locations +and velocities) but all starting with the same low c. To make +doubly certain that the system is as close to the cold equi- +librium as possible, and that any future transition is not +the result of a wayward initial condition, we force ϵ = 0 (a +constant) for several orbits at the start. +The evolution of these runs are plotted in the top panel +of Fig. 7, with the shaded region indicating when ϵ = 0. +As is clear from the figure, all but three runs jumped to +the hot state by 500 orbits (roughly > 25 collision times), +though there was a wide spread of transition times, indica- +tive that the process is stochastic and issues from the noise: +ultimately, after some period, an overenthusiastic collision, +dissipating insufficient velocity dispersion, seeds a patch of +more energetic particles, which then spreads spatially and +takes over the system. +Of course, this is only part of the story, because en- +ergetic events must happen at slightly larger τ but do not +appear to instigate runaway heating. Indeed, we undertake +a similar experiment at τ = 0.2, plotted in the lower panel +of Fig. 7, and witness no transitions at all. What is key is +the overall basin of attraction of the cold state; as shown by +the kinetic curves in the top left panel of Fig. 3, the middle +unstable branch and the cold lower branch become closest +at low τ. The middle branch acts as the boundary of the +lower state’s basin of attraction (at least in this simple phase +space projection); thus at low τ it becomes more likely that +a finite amplitude perturbation can tip the system over this +boundary. That said, it is not straightforward to firmly con- +nect microphysical fluctuations (shot noise) to such a mean +finite-amplitude perturbation in this phase space. +4.2 +Hot to cold transitions +We now check if it is possible to obtain spontaneous hot to +cold transitions. We focus on states near the tip of the saddle +node, i.e. the termination of the hot branch (see top row in +Fig. 3), and examine a range of τ between 1.61 to 1.65 in +the realistic model with ϵmax = 0.923. We simulate several +runs with slightly different initial conditions, as before, and +plot the results in Fig. 8, top and bottom panels. As in the +previous subsection, to ensure that we start the simulations +in a hot state we set vcrit to a very small value initially. Over +several orbits (indicated by the shaded area in the figures), +we slowly increase vcrit to the nominal value. +Unlike cold to hot transitions, the systems either imme- +diately drop to the cold state or relax into the hot state on +a timescale of 10 orbits or so (a handful of collision times). +At τ = 1.65 all the simulations ended up in the cold state, +while at 1.64, some stayed in the hot state, while at lower +tau again (1.61) most stay in the hot state. Putting aside +the percentages in one or the other, the system transitions +promptly or not at all. We attribute this more to the initial +condition at the end of the blue phase, rather than having +to wait for a more sluggish group of collisions that lead to a +‘chain reaction’ and a switching of states. +The difference with the low τ runs explored earlier may +partially be explained by the separation between the middle +and hot branches, which is relatively large, even near the +tip of the saddle node (see kinetic theory curves in top left +panel of Fig. 3). Once a system settles on to the hot state, +and its initial conditions mostly forgotten, its intrinsic shot +noise is insufficient to tip it out of its basin of attraction and +into the cold state. +© 0000 RAS, MNRAS 000, 000–000 + +5 +4 +3 +2 +1 +0 +50 +100 +0 +150 +200 +250 +time [orbits]5 +4 +3 +2 +1 +0 +100 +0 +50 +150 +200 +250 +time [orbits]12 +Larue, Latter, Rein +600 +400 +200 +0 +200 +400 +600 +x +20 +10 +0 +10 +20 +z +t=0 orbits +Figure 9. Initial condition for the fiducial thermal-front simula- +tion described in Section 5.1.1 in the form of an (x, z) projection +of the particle positions. +5 +THERMAL AND VISCOUS FRONTS +Having computed several homogeneous states, we now ex- +plore the dynamics when different states spatially adjoin. If +a ring region is bistable, then it is likely that such situa- +tions occur, given the varying dynamical histories at differ- +ent radii. Our main focus is on the structure and evolution +of the transition (or front) between two states. We will con- +sider two cases: (a) thermal fronts, which join two states +of the same τ but different c, and (b) viscous fronts, which +connect two states of the same angular momentum flux τν, +but different τ and c +Thermal fronts involve a hot and a cold state, with the +pair joined by a vertical line in the top panels of Fig. 3. +Though sharing the same optical depth, they possess dis- +tinct vertical thicknesses that may produce a photometric +variation, and thus observable structure (e.g. Salo and Kar- +jalainen 2003). However, the two states will support different +angular momentum fluxes τν, and thus mass may pile up or +evacuate near the thermal front, potentially leading to non- +steadiness and a complete break down of the structure. We +find that this is avoided if the front itself moves sufficiently +fast. +One might expect radial mass redistribution is negated +if two adjoining states possess the same angular momentum +flux, with the pair joined by a horizontal line in the bot- +tom panels of Fig. 3. In fact, similar structures have already +been witnessed in simulations of the viscous instability with +monotonic ϵ laws (Salo and Schmidt 2010). We find, how- +ever, that the finite width of the front itself spoils the ex- +act matching of fluxes and makes the establishment of such +fronts more complicated. +5.1 +Thermal fronts +In order to explore the structure and dynamics of fronts +connecting equilibria of different temperatures but the same +surface density, we concentrate on a single parameter set. +The behaviour obtained is then interpreted using a simple +continuum model, before other parameters are trialled. +5.1.1 +Fiducial case +Our fiducial run employs a realistic ϵ law with the following +parameters: ϵmax = 0.75, vcrit = 5, and b = 1. We examine +a hot and cold state of the same τ = 0.2, with the former +possessing c = 6.7 and the latter c = 0.87. We adopt a wide +box of radial size 1000a and insert a strip of particles from +the (previously computed) hot state in the centre (with ra- +dial extent 100a), while distributing particles from the cold +state throughout the rest of the numerical domain. Figure +9 plots this initial condition as a projection of the particle +locations in the (x, z) plane. Away from the borders of the +hot/cold zones, the ring is in thermal equilibrium. +The subsequent evolution of the ring is shown in Fig. 10, +which presents four snapshots at different times on each row. +The left panels describe the (x, z) projections of the parti- +cles, while the right panels plot the radial variation of τ +(blue) and c (red). As is clear, the two fronts move radially +into the cold state, until the hot state takes over the box +entirely. Meanwhile, τ remain roughly constant throughout, +except for some minor deviations around the front itself. +The front speed is constant until the moment that the +cold state evaporates. This is demonstrated in Figure 11, +which plots the location of the rightmost front as a func- +tion of time. A c intermediate between the c in the hot and +cold states was selected (here c = 4) and its x location was +determined at each time-step, which provided a means to +capture the movement of the front as a whole. The front +speed is 0.685aΩ, thus slightly less than c in the cold state. +Generally, in bistable systems, the conductivity controls +the structure of fronts; a small conductivity yields a narrow +transition, while a large conductivity gives a more diffuse +transition (e.g. Latter and Balbus 2012). In our granular gas, +the thermal conductivity κ depends on c, and thus jumps +by at least an order of magnitude as we go from the cold +to the hot state (see Table I). This explains why the front +structure is sharp near the cold state (though always longer +than the ‘granularity scale’, a), while broader and smoother +near the hot state. The overall width of the front (≳ 100a) +is hence determined approximately by κ in the hot phase. +5.1.2 +Physics of front motion; a simple continuum model +The basic mechanism driving the movement of a thermal +front relies on the finite-amplitude perturbations arising +from the proximity of the different states. These perturba- +tions can only be communicated via thermal diffusion. For +example, near a front, the cold state will receive thermal en- +ergy (via diffusion) from the adjacent hot state. If the energy +received is sufficient to push the cold ring material out of the +cold state’s basin of attraction, then one might expect it to +heat up and settle on the hot state; as a consequence, the +front advances into the cold phase. But, by the same token, +on the other side of the front, material in the hot state will +also be perturbed by the heat flux and will cool down. If this +cooled material is pushed beyond the hot state’s basin of at- +traction, then it will undergo a runaway cooling and then +we might expect the front to advance into the hot state. +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +13 +600 +400 +200 +0 +200 +400 +600 +x +20 +10 +0 +10 +20 +z +t=0.8 orbits +500 +250 +0 +250 +500 +x +0 +2 +4 +6 +8 +c +0.0 +0.1 +0.2 +0.3 +0.4 +t=0.80 orbits +600 +400 +200 +0 +200 +400 +600 +x +20 +10 +0 +10 +20 +z +t=8 orbits +500 +250 +0 +250 +500 +x +0 +2 +4 +6 +8 +c +0.0 +0.1 +0.2 +0.3 +0.4 +t=8 orbits +600 +400 +200 +0 +200 +400 +600 +x +20 +10 +0 +10 +20 +z +t=80 orbits +500 +250 +0 +250 +500 +x +0 +2 +4 +6 +8 +c +0.0 +0.1 +0.2 +0.3 +0.4 +t=80 orbits +600 +400 +200 +0 +200 +400 +600 +x +20 +10 +0 +10 +20 +z +t=191 orbits +500 +250 +0 +250 +500 +x +0 +2 +4 +6 +8 +c +0.0 +0.1 +0.2 +0.3 +0.4 +t=191 orbits +Figure 10. Snapshots of a thermal front at t = 0.8, 8, 80 and 191 orbits. Panels on the left describe a projection of ring particles on to +the (x, z) plane. Panels on the right depict the x-dependent velocity dispersion c (red) and optical depth τ (blue). +© 0000 RAS, MNRAS 000, 000–000 + +14 +Larue, Latter, Rein +0 +200 +400 +600 +800 +1000 +t +0 +100 +200 +300 +400 +500 +600 +700 +x-coordinate of front +Figure 11. Outer front radial location as a function of time in +the simulation shown in Fig. 10. +Which thermal runaway is favoured on average depends on +the relative sizes of the hot and cold state’s basins of attrac- +tion, which can be approximated (roughly) by how close the +intermediate unstable state is to either state (see discussion +in the section on metastability, and also Latter and Balbus +2012). +These ideas can be illustrated by a continuum model. +The energy equation of the gas may be written as +∂tE = Λ(E) + ∂x(k∂xE), +where E = (3/2)c2, Λ combines viscous heating and col- +lisional cooling, and k is thermal diffusivity (= 2κ/(3σ)). +Thus Λ = 0 when E is equal to the stable hot, cold, and +unstable intermediate steady states, EH, EC, and EI, re- +spectively. Moreover, dΛ/dE < 0 when E = EH or EC. We +assume a steady front, moving at speed vf, with the hot +state to the right and the cold state to the left, and thus +introduce the comoving variable ξ = x − vft, which trans- +forms the energy equation into a type of Stefan problem for +the front shape E(ξ) and speed vf, +∂ξ(k∂ξE) + vf∂ξE + Λ(E) = 0. +(14) +The boundary conditions are E → EH as ξ → ∞ and +E → EC as ξ → −∞ (hot to the right and cold to the left). +This is a nonlinear eigenvalue problem that, after specify- +ing the functional forms of Λ(E) and k(E), would normally +require a numerical solution. In Appendix B we adopt sim- +ple prescriptions for these functions and solve the equation, +thereby illustrating some of the main features discussed be- +low and qualitatively reproducing our N-body results. +An illuminating expression for the speed vf can be ob- +tained by multiplying Eq. (14) by dE/dξ and integrating +between −∞ and ∞. After some manipulation, one gets +vf = − +� EH +EC Λ dE +� ∞ +−∞(dE/dξ)2dξ − +� ∞ +−∞(dk/dE)(dE/dξ)3dξ +2 +� ∞ +−∞(dE/dξ)2dξ +. (15) +If the thermal diffusivity is a constant, the second term is +zero. In this case, the sign of vf is determined solely by the +integral of the heating/cooling term Λ. Because Λ(EH) = +Λ(EI) = Λ(EC) = 0, the integral can be subdivided into (a) +a positive part (between EC and EI) that measures the ‘size’ +of the cold state’s basin of attraction, and (b) a negative +part (between EI and EH) that measures the hot state’s +basin of attraction. The proximity of EI to either EC or +EH indicates the basins’ relative sizes. If EI is closer to EC, +then the integral is dominated by the positive area, vf < 0, +and the front moves into the cold state. Physically, cold ring +material near a front finds it easier to undergo a heating +runaway, when perturbed by the front, than hot material +finds a cooling runaway; thus, the front advances into the +cold material. If EI is closer to EH, then the converse holds +and the front moves into the hot state. Turning now to the +top row of Fig. 3 (first panel especially), one naively expects +that at low τ fronts initially move into the cold state, but at +higher τ fronts are slower and then at some critical τ may +reverse direction. +If k depends on E then things are more complicated. +The second term in Eq. (15) is a weighted average of dk/dE, +and shows that a non-uniformity in the transport of heat +moderates the effect discussed above. If the front shape is +monotonic in ξ, then dE/dξ > 0 throughout and the sign of +the second term is determined by dk/dE. As demonstrated +in Section 3.2.3 and Table I, dk/dE > 0, and so the second +term in Eq. (15) is always positive, thus biasing the front’s +movement into the cold state. The underlying mechanism +here rests not on the system’s bistability but on exacerbating +the imbalance in the heat flux throughout the front struc- +ture: at any given point more heat is arriving from the hot +state than is being evacuated. +The discussion above suggests that the sharp region at +the foot of the front controls the front speed. Taking an +order of magnitude approach and equating the three terms +in Eq. (14) yields the estimate vf ∼ +� +kC/tth, where the +thermal timescale is defined as tth = E/Λ ∼ c2/(νΩ2), and +kC is the diffusivity evaluated in the cold state. Putting in +values for the cold state gives us vf ∼ aΩ, which is consistent +with the value calculated numerically. The width λ of the +front extending through the hot phase can then be estimated +by balancing the first two terms in Eq. (14); we find λ ∼ +kH/vf ≳ 500a, which is also consistent with the simulation. +5.1.3 +Front stability +We conducted a short survey of fronts at different τ and +calculated their speeds. When τ = 0.1 we found vf = 0.518, +and when τ = 0.3, vf = 0.591. While no clear trend could be +observed between τ = 0.1 − 0.3, we expected at larger τ, as +we approached the saddle node, that the front speed should +decrease. In fact, what we found for τ = 0.4 or larger is that +the front would slow to a halt and then viscously reshape; +i.e. τ would evolve away from a uniform profile. Ultimately, +the system moves to a state of constant angular momentum +flux τν, and the thermal front dissolves. +As mentioned earlier, the issue here is that across a +thermal front τ is constant, but τν is not. As a consequence, +mass can potentially build-up/evacuate. If the front moves +faster than τ can be viscously redistributed, then we expect +the front to remain coherent and to travel unimpeded. If the +front speed is too slow, then it will be viscously reshaped and +will collapse. For the model chosen, τ ⩽ 0.3 corresponds to +the first case, and τ > 0.3 to the latter. +A rough criterion for the ‘stability’ of the front to vis- +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +15 +cous redistribution would tension the relative sizes of the +front speed vf and the viscous diffusion speed. To deter- +mine an estimate on the latter, we employ the lengthscale +of the abrupt transition at the foot of the structure and +thus estimate the diffusion speed as ∼ (νC/κC)vf. A sim- +ple criterion for front dissolution requires that this speed +is greater than vf, and hence depends solely on the size of +the Prandtl number Pr = ν/κ in the cold state: when Pr is +greater than a critical value Prc, we expect the front to dis- +solve. Indeed, Pr increases monotonically between τ = 0.1 +and 0.4, though takes relatively small values. At τ = 0.4, we +find that Pr ∼ 0.04, which must be near Prc. +5.2 +Viscous fronts and viscous instability +Given the issue of the unbalanced angular momentum in +thermal fronts, it is natural to explore fronts that join states +with the same viscous transport properties, specifically ντ. +We present simulations of such joined states in this subsec- +tion, in addition to a short treatment of viscous instability. +A simple continuum model can guide our expectations. +In the shearing sheet, the one-dimensional diffusion equation +for viscous Keplerian disks is +∂tτ = 3∂2 +x(ντ) +(e.g. Lynden-Bell and Pringle 1973). Suppose a viscous front +moves with speed vf with τ → τA as x → −∞ and τ → τB as +x → ∞. As earlier, we adopt a comoving variable ξ = x−vft, +which permits the complete integration of the problem. We +find that vf = 0 (the structure must be stationary) and +ντ (= νAτA = νBτB) is a constant throughout the entirety +of the front. The last constraint is a potential difficulty: while +it is possible to find two homogeneous steady states of the +same ντ (cf. panels in the bottom row of Fig. 3), a realis- +tic front will have a finite width in which τ will vary and +thus ντ will deviate from the required constant value. Our +simulations show, in fact, that the system can overcome this +problem by settling on a front structure in which the average +ντ equals νAτA = νBτB. +5.2.1 +Fronts +We present a fiducial simulation with the realistic law, and +parameters b = 1, ϵmax = 0.923, and vcrit = 5. To construct a +suitable initial condition that might produce a viscous front, +we select two thermally and viscously stable states with the +same ντ from the bottom right panel of Fig. 3. Such pairs +are joined by horizontal lines. We select two states of the +same angular momentum flux ντ ≈ 2, with optical depths +τ = 1.5 and τ = 0.16. The numerical domain is chosen to +be sufficiently large (L = 800) to accommodate relatively +undisturbed expanses of the two states, in addition to the +front itself; the low τ state is placed between x = −100 and +100, with the high τ state taking up the remainder of the +box. +Figure 12 shows eight snapshots of the resulting simu- +lation at different times. In each panel we plot τ (red) and +τν (blue). At t = 0, the angular momentum flux τν is a +constant, but τ undergoes two jumps (at x = ±100a). As +the system evolves, the two jumps/fronts relax and exhibit a +characteristic width, with τ taking values between those of +the two steady states. An immediate consequence is that the +angular momentum flux within the fronts begins to deviate +from the fixed value ≈ 2. In fact, the first four panels show +that it takes significantly larger values than 2, in agreement +with the bottom right panel of Fig. 3, which shows that +states with τ between 0.16 and 1.5 exhibit ντ > 2. Because +of the enhanced flux in the fronts, mass is being transported +out of the fronts, which then appear to move as the system +evolves far way from the initial condition. +Ultimately, we find that the system redistributes the +mass throughout the numerical domain so that τν is roughly +constant (≈ 7), but still allows for strong variations in τ. +This outcome is not a constant τ state, but consists of +two static viscous fronts joining two homogeneous states +of τ ≈ 0.4 and 2.7, which according to Fig. 3 possess the +same angular momentum flux (∼ 7). Evidently, the front +that joins the two states also possesses a similar approxi- +mate flux, though this is difficult to determine from Fig. 3. +A similar final state was found by Salo and Schmidt (2010) +when simulating the viscous instability directly (see next +subsection). +This static structure is an interesting outcome for the +system, but we stress that it is possible only because of +the periodicity of the numerical domain. Owing to those +boundary conditions, mass in the whole domain can be re- +distributed until the desired constant ντ state can be found. +In a more realistic setting, the system is unlikely to come +to steady state and the front will continue to move until it +encounters large-scale variations in background disk proper- +ties, etc. +5.2.2 +Viscous instability +In the previous subsection we explored two adjoined vis- +cously stable states, but the lower right panel of Fig. 3 in- +dicates that there is a branch of viscously unstable states +of intermediate τ between roughly 0.8 and 1.6. An obvious +question is: to where does the system evolve if started from +one of these states? We thus present a simulation with the +same collisional parameters as earlier, but with a homoge- +neous τ of 1.4. According to Fig. 3, this state is viscously +unstable. Figure 13 shows 5 snapshots of the system’s evo- +lution. +Despite possessing a constant τν, the system moves +slowly away from this state and begins to develop grow- +ing patches of high and low τ. Unlike the previous subsec- +tion, where the evolution is being driven by large-scale flux +imbalances, here there is an instability mechanism, in which +small-scale fluctuations in the flux self-reinforce (Lin and Bo- +denheimer 1981, Lukkari 1981, Ward 1981). Ultimately, the +system settles on a sequence of distinct high-τ islands sur- +rounded by relatively dilute regions, but both with roughly +the same flux (≈ 6, in this case), as is necessary for a steady +state. +These results are very similar to those predicted by +H¨ameen-Anttila (1982) and witnessed in Lukkari (1981) and +Salo and Schmidt (2010), though they use a monotonic col- +lision law. A key difference is that in the monotonic ϵ simu- +lations, the final outcome joins states from the same branch, +while in our non-monotonic simulations states from different +branches adjoin. An interesting consequence of this is that it +is still possible for the system to separate into a sequence of +© 0000 RAS, MNRAS 000, 000–000 + +16 +Larue, Latter, Rein +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=5 orbits +400 +300 +200 +100 +0 +100 +200 +300 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=20 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=30 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=50 orbits +400 +300 +200 +100 +0 +100 +200 +300 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=100 orbits +400 +300 +200 +100 +0 +100 +200 +300 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=500 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=1000 orbits +400 +300 +200 +100 +0 +100 +200 +300 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=2000 orbits +Figure 12. Snapshots of an example viscous front, showing optical depth and angular momentum flux as a function of x. The initial +condition connects two states of different τ but the same angular momentum flux τν. Despite this balance, the system evolves, redis- +tributing mass and angular momentum until a steady state is achieved characterised by a different constant τν. The collision law employs +the realistic model with vcrit = 5, ϵmax = 0.923, b = 1. Snapshots are at t = 5, 20, 30, 50, 100, 500, 1000, and 2000 orbits. +high and low τ states (of the same ντ), even when there is no +intermediate viscously unstable state. In particular, this ap- +pears achievable for the parameters of the middle column in +Fig. 3. More generally, systems with non-monotonic collision +laws have more freedom to exhibit viscous phase-separation +in radius. +6 +DISCUSSION AND CONCLUSION +Most previous work describing the local collisional dynam- +ics of Saturn’s rings uses relatively simple collision models. +Given the poorly constrained nature of the collisions, and +the numerical challenges involved, this is understandable, +and indeed some success has been achieved in certain appli- +cations (e.g. self-gravity wakes, viscous overstability). How- +ever, current models still fail to describe much (if not most) +of the irregular axisymmetric structure exhibited in Saturn’s +B and C rings. This invites us to experiment with other more +complicated collision laws, in particular those that account +(in a basic way) for surface regolith on ring particles, which +is deemed to be present and important (e.g. Nicholson et +al. 2008, Morishima et al. 2012, Deau 2015). +We conduct N-body simulations with the REBOUND +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +17 +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=50 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=750 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=1000 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=1050 orbits +400 +200 +0 +200 +400 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +2 +4 +6 +8 +10 +12 +t=2000 orbits +Figure 13. Snapshots showing the progress of viscous instability starting from an unstable state of τ = 1.4. The collisional parameters +are vcrit = 5, ϵmax = 0.923, b = 1. The panels describe the x dependent optical depth τ (red) and the angular momentum flux τν (blue). +Snapshots are at 50, 750, 1000, 1050, and 2000 orbits. +code of a local patch of Saturn’s rings in which particles un- +dergo collisions with a prescribed coefficient of restitution +ϵ depending on impact speed. The main novelty of our ap- +proach is to employ an ϵ that is a non-monotonic function +of impact speed, as is suggested by theoretical and experi- +mental studies of regolith-coated particles (cf. Section 2.1). +Below a critical impact speed we set ϵ = 0, though neglect +particle sticking. This relatively minor change in the phys- +ical set-up immediately introduces major thermodynamical +changes. For the same optical depth, the rings yield two +thermally stable steady states, a hot c ≳ 4aΩ and a cold +c < aΩ state. Which is selected depends on the local ther- +mal and/or dynamical history, and thus different ring radii +might fall into one or the other. +An obvious follow up question is to ask what happens +at the boundaries of two adjoining different states? We run +additional simulations in larger domains and find that in +general the hot state will engulf the cold state, with the +transition front moving at a speed ≈ 0.5aΩ. Slower mov- +ing fronts break down because of the imbalance in angular +momentum flux across the transition. Stationary ‘viscous +fronts’ are also simulated which join states of different opti- +cal depth and c but the same angular momentum flux. Note +that it need not necessarily be the case that hot states always +take over: smooth variations in the ring’s background prop- +erties may change propagation, and large amplitude pertur- +bations (meteoroids, density waves, gravity wakes, etc.) will +also complicate the picture. +Our simulation results are exploratory, and should be +taken as a demonstration of what happens when one relaxes +the strong modelling assumptions of previous work. They are +perhaps not yet ready for direct application to structure for- +mation in Saturn’s rings, not least because of the parameters +in our regolith laws are poorly constrained. Nonetheless, it is +irresistible to speculate. We anticipate that a thermal front, +connecting a warm and cold state of the same dynamical +optical depth, gives rise to photometric variation (which the +Cassini cameras may have picked up) but no variation de- +tectable by occultation experiments. This is precisely the sit- +uation in the C-ring plateaus (Hedman and Nicholson 2013), +and indeed, there is evidence of size segregation across these +structures which may tie in to the greater chance of sticking +in the colder phase (Marouf et al. 2013, Colwell et al. 2018). +It may also be relevant for the 10km striations shown by +Cassini’s cameras in the A and B-rings (cf. Figs 5A and 5B +in Porco et al. 2005). On the other hand, the steady viscous +fronts our simulations support, which connect states of high +and moderate optical depth, bear some resemblance to the +disjunct bands in the middle B-ring (Colwell et al. 2009). +A great deal more theoretical work and modelling is needed +before these associations can be made secure. In particu- +lar, applications to ring regions exhibiting self-gravity wakes +must remain tentative until we produce better constrained +estimates on typical sticking speeds. +Other areas of future work could explore the interplay +between the hysteresis and self-gravity wakes, on one hand, +and viscous overstability, on the other. For example, we +might anticipate wakes appear only in the cold state, chang- +ing its viscous properties, and providing energy to jump into +the hot state. More generally wake activity will produce en- +hanced heating and thus a change in the thermodynamic +balances calculated in this paper. Viscous overstability gen- +erates nonlinear travelling wavetrains which may also favour +the cold phase; these waves will reflect off the boundaries be- +tween states, hence complicating the nonlinear saturation of +the wave turbulence. Simulations including realistic photom- +etry of thermal fronts might help establish if they might cor- +respond to any observable structure (Salo and Karjalainen +2003). Finally, the robustness of bistability must be estab- +lished when particle sticking is permitted, as in recent sim- +ulations by Ballouz et al. (2017) and Lu et al. (2018). +© 0000 RAS, MNRAS 000, 000–000 + +18 +Larue, Latter, Rein +ACKNOWLEDGMENTS +The authors thank the reviewer Heikki Salo and Juergen +Schmidt, who generously provided a set of helpful and thor- +ough comments that markedly improved the paper. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable +request to the corresponding author. +REFERENCES +Albers, N., Spahn, F., 2006. Icarus, 181, 292. +Araki, S., Tremaine, S., 1986. Icarus, 65, 83. +Ballouz, R.-L., Richardson, D. C., Morishima, R., 2017. ApJ, 153, +146. +Bodrova, A., Schmidt, J., Spahn, F., Brilliantov, N., 2012. Icarus, +218, 60. +Bridges, F. 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Planetary Ring Sys- +tems. Properties, Structure, and Evolution, Edited by M.S. +Tiscareno and C.D. Murray. Cambridge University Press, p. +434-493 +Schmidt, J., Ohtsuki, K., Rappaport, N., Salo, H., Spahn, F., +2009. In: Dougherty, M. K., Esposito, L. W., Krimigis, S. M. +(eds.), Saturn from Cassini-Huygens, Springer, Dordrecht +Netherlands, p413. +Thornton, C., 1997. Journal of Applied Mechanics, 64, 383. +Thornton, C., Ning, Z., 1998. Powder Technology, 99, 154. +Tiscareno, M.S., and 24 coauthors, 2019. Science, 364, 6445, id. +aau1017. +Ward, W.R., 1981. GRL, 8, 641. +Wisdom, J., Tremaine, S., 1988. The Astronomical Journal, 95, +925. +APPENDIX A: CONVERGENCE TESTS +We present some results showing the behaviour of a subset +of our equilibrium solutions as the numerical parameters are +varied. In particular, we explore their dependence on the size +of the time-step dt and the numerical domain, showing that +convergence is achieved when the former is sufficiently small +and the latter sufficiently large. To simplify the study, we +adopt a standard Bridges law for two different vcrit (yielding +hot and warm equilibria) and also a constant ϵ = 0 (yielding +cold equilibria. We examine very dilute cases τ = 0.1 and +very dense cases τ = 2.5, thereby determining the numerical +requirements at the physical ‘boundaries’ of our main set of +results, and thus for the main results themselves. +Our convergence results are plotted in Figs A1 and A2, +the former showing the velocity dispersion c as a function of +dt, the latter c as a function of box size. Time steps of 10−2 +© 0000 RAS, MNRAS 000, 000–000 + +Thermal hysteresis in rings +19 +10-3 +10-2 +10-1 +dt +100 +101 +102 +C +Bridges laws +vcrit=20, = 0.1 +vcrit=20, = 2.5 +vcrit=1, = 0.1 +vcrit=1, = 2.5 +10-4 +10-3 +10-2 +10-1 +dt +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +C + = 0 + = 0.1 + = 2.5 +Figure A1. Convergence tests in time step for several set-ups +spanning dilute and cold, dense and hot, etc. +or less and a box size of 30 or greater appear to be sufficient +in most cases. In our main equilibrium runs in Section 3, we +use dt = 10−3 and a box size of 100. +APPENDIX B: ILLUSTRATIVE TOY FRONTS +In this appendix we calculate thermal fronts using the simple +continuum model of Section 5.1.2 with prescribed functions +for Λ and k. Noting the bistability at low τ, we adopt a lo- +gistic reaction term and a linear diffusivity, which in suitable +units take the form +Λ = (E − EC)(E − EI)(E − EH), +k = αE, +where EC < EI < EH are constant parameters denoting the +cold, intermediate, and hot states (respectively), and α is an +additional constant. Both EC and EH are thermally stable, +but EI is unstable. The basins of attraction of EC and EH, +however, are controlled by their proximity to EI. +These functional choices simplify the integrals in the nu- +merator of (15). The integral of Λ becomes simply −(EC − +EH)3(EC −2EI +EH)/12, and is negative when the interme- +diate state is less than the arithmetic mean of the hot and +cold states, EI < (EC + EH)/2, and positive otherwise. In +other words, the front will tend to move into the cold state +when the intermediate state is closer to the cold state, i.e. +when its basin of attraction is smaller. Similarly, the front +will tend to move into the hot state when EI is closer to +EH. If the three thermal states are equidistant and k is a +101 +102 +Box size +100 +101 +C +Bridges laws +vcrit=20, = 0.1 +vcrit=20, = 2.5 +vcrit=1, = 0.1 +vcrit=1, = 2.5 +101 +102 +Box size +0.5 +1 +1.5 +2 +C + = 0 + = 0.1 + = 2.5 +Figure A2. Convergence tests in box size for several set-ups +spanning dilute and cold, dense and hot, etc. +constant, then c = 0 and the front profile can be expressed +in terms of elliptic integrals. +The second term in (15) cannot be evaluated without +knowledge of the front profile. It nonetheless simplifies to +−α +� ∞ +−∞(dE/dξ)3dξ, which is clearly negative for monotonic +front profiles. Thus the linear k law favours the front’s move- +ment into the cold state, as discussed in Section 5.1.2. +Finally, we numerically solved (14) using a relaxation +method (Press et al. 1986), due to the problem’s charac- +teristic stiffness. We plot a representative front solution in +Fig. B1. As in Fig. 10, the front is sharp at the cold tran- +sition, where k is small, and diffuse at the hot transition, +where it is an order of magnitude larger. +© 0000 RAS, MNRAS 000, 000–000 + +20 +Larue, Latter, Rein +-2 +-1 +0 +1 +2 +3 +4 +0 +2 +4 +6 +8 +10 +12 +E +Figure B1. Illustrative front calculated numerically, with pa- +rameters EC = 1, EI = 1.5, EH = 12, and α = 0.5. The front +moves to the left into the cold state with a speed c = −12.3728. +© 0000 RAS, MNRAS 000, 000–000 + diff --git a/3NE1T4oBgHgl3EQflwRz/content/tmp_files/load_file.txt b/3NE1T4oBgHgl3EQflwRz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ced9466f17acee869b597a9560c076fb7874d5b4 --- /dev/null +++ b/3NE1T4oBgHgl3EQflwRz/content/tmp_files/load_file.txt @@ -0,0 +1,1658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf,len=1657 +page_content='Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 000, 000–000 (0000) Printed 10 January 2023 (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2) Thermal hysteresis and front propagation in dense planetary rings R´emy Larue1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Henrik Latter1⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Hanno Rein4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1DAMTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' CMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Wilberforce Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Cambridge CB3 0WA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' UK 2ENS Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 4 avenue des Sciences 91190 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' France 3Laboratoire de Physique Subatomique et de Cosmologie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Universit´e Grenoble-Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' CNRS/IN2P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Grenoble INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' France 4Department of Physical and Environmental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' University of Toronto at Scarborough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Ontario M1C 1A4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Canada 5David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Dunlap Department of Astronomy and Astrophysics, University of Toronto, Toronto, Ontario, M5S 3H4, Canada ABSTRACT Saturn’s rings are composed of icy grains, most in the mm to m size ranges, under- going several collisions per orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Their collective behaviour generates a remarkable array of structure over many orders of magnitude, much of it not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, the collisional properties and parameters of individual ring par- ticles are poorly constrained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' usually N-body simulations and kinetic theory employ hard-sphere models with a coefficient of restitution ϵ that is constant or a decreasing function of impact speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Due to plastic deformation of surface regolith, however, it is likely that ϵ will be more complicated, at the very least a non-monotonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We undertake N-body simulations with the REBOUND code with non-monotonic ϵ laws to approximate surfaces that are friable but not sticking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our simulations reveal that such ring models can support two thermally stable steady states for the same (dynamical) optical depth: a cold and a warm state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the ring breaks up into radial bands of one or the other state, we find that warmer states tend to migrate into the colder states via a coherent travelling front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We also find stationary ‘viscous’ fronts, which connect states of different optical depth, but the same angular momentum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We discuss these preliminary results and speculate on their implications for structure formation in Saturn’s B and C-rings, especially with respect to structures that appear in Cassini images but not in occultations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Key words: instabilities – waves – planets and satellites: rings 1 INTRODUCTION Saturn’s rings flaunt an extraordinary array of axisymmetric structure, both quasi-regular and chaotic, ranging over some four orders of magnitude in length - from 10 m to 100 km (Colwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2009, Cuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Yet despite several decades of theoretical effort, their origins are only partially understood (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2009, Estrada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018, Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particular, the disjunct bands of high and low optical depth in the B-ring (Horn and Cuzzi 1996, Colwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2007), the plateaus in the C-ring (Tiscareno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2019), and the irregular intermediate scale striations in the A and B-rings (Porco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2005) are presently without plausible explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Simply put, there is too much observed struc- ture and too few suitable instabilities (or related processes) in our theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Perhaps it is time to re-assess ⋆ E-mail: hl278@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='uk some of our fundamental assumptions and explore a wider range of alternative scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It is probable, though not assured, that much of the ring’s unexplained structure arises spontaneously due to its peculiar granular flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Since the 1980s researchers have turned to kinetic theory or N-body simulations to model this flow, initially calculating the thermal balances under- lying ring equilibria, and then the (viscous) instabilities that might generate structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', H¨ameen-Anttila 1982, Araki & Tremaine 1986, Wisdom & Tremaine 1988, Salo 1991, H¨ameen-Anttila & Salo 1993, Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2001, Lat- ter & Ogilve 2006, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These studies have made several strong assumptions, especially regarding the nature of the ring particles and their collisional behaviour, for instance rarely deviating from a hard-sphere model with either a con- stant coefficient of restitution ϵ or a ‘Bridges law’ (Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1984), whereby collisions below some critical impact speed are perfectly elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In reality, ring particles are likely to be irregularly shaped and coated in a regolith of small par- © 0000 RAS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='03289v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='EP] 9 Jan 2023 2 Larue, Latter, Rein ticles ≲ 1 cm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Doyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1989, Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2008, Morishima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Deau 2015) and, being irregular and fluffy, their surfaces should produce an enhanced inelasticity at low impact speeds, and indeed possible particle adhesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In light of this, the adoption of a constant ϵ, or a Bridges law, may significantly misrepresent some of the ring’s collective collisional dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our paper tests this idea by exploring other, physically motivated, prescriptions for ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We find, in fact, that even very simple changes to the collision law can give remarkably different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Continuum mechanical models of viscoelastic collisions that account for fluffy and/or sticky surfaces demonstrate that ϵ is a non-monotonic function of impact speed vcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Beneath some critical speed we have ϵ = 0, but on in- creasing vcoll, ϵ rises, plateaus, and then decreases again (Gorkavyi 1985, Hertzsch 2002, Albers & Spahn 2006, Bril- liantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Laboratory experiments appear to con- firm this picture (Gorkavyi 1989, Hatzes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1991, Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We implement collision laws of this basic form in our paper and term them ‘regolith laws’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In addition, at or below the critical speed colliding particles may stick, but we neglect this important effect in order to avoid the vexed and complicated issue of size-distribution dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our approach is mainly nu- merical, via N-body simulations of monodisperse, spherical, indestructible particles with the code REBOUND;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' but we also employ a dense gas kinetic theory, where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Note that we do not include self-gravity and thus our sim- ulations fail to exhibit wakes, nor do they support viscous overstability, both important phenomena we hope to test in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our study is distinct but complementary to recent N-body simulations that explicitly test the role of adhesion, especially on instabilities (Ballouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2017, Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' see also Section 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 in Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our main focus, in contrast, will be on disk thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our first main result is that regolith laws permit a dense ring to fall into one of two thermally stable states at the same optical depth: (a) a very dense state with filling fac- tors ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 and low temperatures, c ≲ aΩ (where c is velocity dispersion, a is particle radius, and Ω is orbital frequency) and (b) a moderately dense state with lower filling factors (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1) and a slightly warmer temperature, c ≳ 4aΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This bistability generally favours optical depths less than 1, but can be pushed up to higher values if we broaden our parame- ter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We also find in certain circumstance that the cold state at low optical depth is metastable: shot noise permits the ring to spontaneously jump into the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our second set of results explores what happens when different thermal states spatially adjoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If two states of the same optical depth but different temperature connect, a travelling ‘thermal front’ develops that can reach speeds of ≲ aΩ, while maintaining a steady spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the front is too slow, the disparity in the angular momentum flux between the two states reorganises the front profile so that the flux is uniform but the optical depth undergoes a jump, what we term a static ‘viscous front’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Some of the latter behavior mirrors that witnessed by Salo and Schmidt (2010) in their simulations of viscous instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The plan of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The next section begins with a review of the extant literature on low-impact collisions between regolith covered and/or sticky particles, moving on to a presentation of the model collision laws we use, and then our numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Subsequently, we de- tail out results: the calculation of thermal equilibria and hys- teresis in smallish boxes (Section 3), potential metastability (Section 4), and finally results on spatially adjoining states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' thermal and viscous front (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2 BACKGROUND AND METHODS This section presents the physical set-up and numerical model by which we attack the thermal equilibria of rings composed of regolith-coated particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We first devote some space to set the scene, by reviewing the theoretical and ex- perimental literature and explaining the key ideas and pa- rameters that underlie work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The model collision laws we adopt are then exhibited, followed by the details of the N-body simulations with REBOUND we conduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Collisional physics and the coefficient of restitution We aim to describe the collisional dynamics of many ring particles in a local patch of a planetary ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' From the outset we make several strong assumptions that we concede may distort our results: the particles are taken to be identical, spherical, and frictionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Most of the ring mass is in metre- sized particles, and thus it is that population that we track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Only binary collisions are considered, and these are deemed inelastic, so that g′ · k = −ϵ(g · k), where g is the relative velocity of two colliding particles before the collision and g′ afterwards, k is the unit vector pointing between the two particles centres at the moment of collision, and ϵ is the coefficient of restitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This coefficient lies between 0 and 1 and is usually a function of the impact speed vcoll = |g · k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We neglect the possibility of two particles sticking and assume that all the specifics of the particle surfaces can be encapsulated in the functional behaviour of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Because we find the ring dynamics are so sensitive to ϵ, we now spend some time discussing this important physical input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Theoretical and experimental background Research exploring the collisional behaviour of regolith- covered particles can be separated into analytical calcula- tions, drawing on continuum mechanics, and laboratory ex- periments, approximating Saturnian conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We attempt to review and synthesise this body of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The seminal experiments in this area were described in Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1984) and collided smooth ice spheres with an ice block at temperatures ∼ 170K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This work pro- duced the collision law ϵ = min � 1, (vcoll/vcrit)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='234� , for vcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='008 cm s−1, a defining feature of which is perfect elasticity at sufficiently low collision speeds (vcoll < vcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This collision law became the standard for subsequent N- body simulations and other theoretical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Subsequently, broken power laws of this type were shown to arise naturally in generalisations of the Hertz theory to viscoelastic solids (Dilley 1993, Hertzsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1995, Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1996, Thornton 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' However, such theoretical work must posit that the surfaces of the colliding spheres are smooth and © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 3 that irreversible energy losses arise solely from viscoelastic deformations inside the spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Shortly after the Bridges experiments, two neglected but insightful papers by Gorkavyi (1985, 1989) highlighted the importance of regolith and argued against perfectly elas- tic restitution at low impact speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Gorkavyi emphasised that ϵ can be dramatically altered at small vcoll because (a) impact energy can be used up when reshaping a soft fri- able surface (leaving nothing left over for elastic rebound) and/or (b) rebounding motion can be countered by surface stickiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Using energy arguments, the 1985 paper sketches out three regimes: (a) at sufficiently low vcoll, there is total energy loss and thus ϵ = 0 (sticking/adhesion is not consid- ered);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (b) at slightly larger vcoll, ϵ increases with vcoll;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' and then (c) after a turning point, ϵ decreases with vcoll (tradi- tional restitution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The collision law is hence non-monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Gorkavyi (1989) followed this up with simple experiments using powders, metals, and marble at room temperature and pressure, which agree with earlier lab work by Hartmann (1978, 1985), in a different context, using rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Subsequent papers from the Bridges research group ex- amined how the state of the particle surface influenced colli- sions, with a particular focus on the adhesive effect of frost, a thin layer of microscopic structure that might behave simi- larly to the thicker regolith layer expected on larger ring par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Hatzes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1991) showed frosty particles can stick at impact speeds below some critical level (a few mm s−1), but did not examine explicitly how it changed the form of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1996) conducted a large set of experiments for different kinds of ices and vcoll at relevant temperatures, which further strengthened the case for sticking, and also showed that ϵ exhibited the three main features predicted by Gorkavyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the theoretical side, the 2000s witnessed various ex- tensions of Hertz contact mechanics, accounting for both viscoelasticity and particle adhesion via JRK theory (Albers and Spahn 2006, Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' see also Thornton and Ning 1998, and Chokshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1993, the latter in the context of ISM grains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Notable is the work by Hertzsch (2002) who modelled the two effects of sticking and of pas- sive regolith deformation, as discussed by Gorkavyi, treating the passive regolith as a deformable viscous non-sticky ‘soft layer’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Both physical effects appear to influence the form of ϵ similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In all cases, non-monotonic ϵ laws were mathe- matically derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2007) provides estimates for solid water-ice particles of various sizes that, despite several strong assumptions, help with Saturnian applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For metre-sized water-ice impactors, the theory predicts that the maximum value ϵ takes is relatively large, potentially above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For cm sized particles, it drops to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, the critical vcoll for sticking is roughly 10−2 cm s−1 for metre-sized ice impactors, and this rises to greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 cm s−1 for cm-sized particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Because of the model assump- tions care must be taken, however, when applying these es- timates, and in fact the quoted critical collision speeds are probably gross lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The theory omits the energy dissipation channel associated with irreversible regolith de- formation (as well as internal fracture) by treating the par- ticles as solid-ice non-spinning viscoelastic spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It also sets the unknown dissipative constant A by fitting a (non- sticking) viscoelastic model (Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1996) to the (non-sticking) experimental data of Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Nonethe- less, the Brilliantov results provide a useful starting point for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Before moving on, we flag additional physics not yet discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In applying the above ideas and prescriptions to an ensemble of colliding particles, one must acknowledge that, by virtue of the collisions themselves, particles’ surface properties will evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Repeated collisions will presumably ‘compactify’ particle regolith and hence reduce the mean critical sticking speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, bombardment by micrometeoroids will disturb the surfaces and there will be accretion of very small floating particles, processes that will rejuvenate regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It follows that, in addition to the size distribution dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Longaretti 1989, Bodrova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2012, Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2015), there will take place related dynamics controlling the mean surface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We do not attempt to construct a model for this interesting process here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Important scales This subsection briefly outlines the key velocity scales rel- evant for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We assume that there is a single critical sticking speed vstick below which two impactors will adhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We also assume a second critical impact speed vcrit below which ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It may be that these two speeds are the same, though in general we expect vstick < vcrit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' it is possible for all the energy of the impact to be used up re- shaping the surface and resisting the adhesive attraction of the regolith, thereby allowing the impactors to roll clear of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Particle spin and tidal shear may facilitate such non-sticking ϵ = 0 encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A third key speed is the velocity dispersion c, as impact speeds will be distributed around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus the relative size of c relative to vcrit will determine which collisional regime (sticking, non-sticking, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=') the particles are in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Partly con- trolling c is the orbital shear speed across a particle, aΩ (recall a is particle radius and Ω the orbital frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The importance of this scale issues from the fact that dense cold rings adopt a velocity dispersion c ∼ aΩ, in the absence of gravity wakes, and c ≲ 5aΩ, when gravity wakes are present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', Araki & Tremaine 1986, Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It follows that if c ∼ aΩ ≫ vcrit then the regolith is not going to fea- ture much in the mean thermal dynamics, and hence the determination of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, if c ∼ aΩ ≪ vcrit then the surface properties are going to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Com- plicating this picture, of course, is the size dependence of both aΩ and vcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In a polydisperse ring, however, the ve- locity dispersion of smaller particles will be similar to the metre-sized particles (Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We now obtain some bounds on the important parameter vcrit/(aΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' First we situate ourselves at a representative location in the C-ring, in which gravity wakes are likely absent, and set Ω ≈ 10−4 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If a = 1 m, the most dynamically im- portant size, aΩ is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='01 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Next, applying the estimates from Brilliantov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2007) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1) 1 The second estimate can be obtained by assuming a gravita- tionally unstable ring settles into a state where the Toomre Q is ∼ 1, and then taking typical values for the surface density (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Hedman & Nicholson 2013, 2016) © 0000 RAS, MNRAS 000, 000–000 4 Larue, Latter, Rein and setting vcrit = vstick, we obtain vcrit/(aΩ) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For cm sizes, vcrit/(aΩ) ≳ 10 (noting that the velocity dispersion of this population is set by the metre sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As argued earlier, the Brilliantov estimates for vcrit only provide lower bounds, and hence we conclude that it is likely that the C-ring is in a regime where surface regolith properties will matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At a representative location in the A or B-ring, we must take into account gravity wakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus we find ourselves in a more ambiguous situation: the Brilliantov estimates yield vcrit/c ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 for metre-sized particles, and vcrit/c ≳ 1 for cm-sized particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Depending on how badly the Brilliantov results underestimate vcrit, we could be in a marginal regime or in a regolith-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Certainly, further work on the collisional dynamics of ice would help decide on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As we do not simulate self-gravity, for now we just assume that aΩ < vcrit, and leave open its importance to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 Model coefficients of restitution This section presents the two classes of non-monotonic ‘re- golith’ ϵ-law we use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We have attempted to paramaterise these laws in two readily understandable quan- tities: vcrit, the impact speed at which collisions are perfectly inelastic (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' and ϵmax, the turning point value of ϵ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', its maximum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A broken power law (BPL) for ϵ, though somewhat crude has the benefits that it has few input parameters and some headway can be made with it using kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We define the law in the following way: ϵ(vcoll) = � ϵ0, if vcoll < vcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' ϵmax (vcoll/vcrit)−p , if vcoll ⩾ vcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1) We set the exponent p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='234, following Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1984), though it could take other values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The quantity ϵ0 we set equal to either 1, to obtain the Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' law itself, or equal to 0, to get the opposite perfectly inelastic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The Bridges BPL is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A more realistic non-monotonic ϵ law that is smoother and exhibits something of a plateau near its maximum can be defined in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We choose the following: ϵ(vcoll) = � 0, if vcoll < vcrit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='625 ϵmax ζ/(1 + ζ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='234), vcoll ⩾ vcrit, (2) where ζ = (vcoll−vcrit)/b and b is the plateau ‘width’, usually set to aΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Constants have been chosen so that ϵ approaches the Bridges law for large vcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To facilitate the discussion later, when we compare the different models, we refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2) as a ‘realistic’ law (though it is yet to be determined how realistic it is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We plot it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1 in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 The potential for bistability Before presenting our numerical methods and the results that ensue, we briefly explain why a non-monotonic colli- sion law, such as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2) and displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1, potentially yields two stable states for the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At lower optical depths, N-body simulations and ki- netic theory show that the Bridges law yields equilibria with c > aΩ, and thus most collisions sample the power-law de- creasing segment of the ϵ curve (Salo 1991, Latter & Ogilvie 0 1 2 3 4 5 6 vcoll / vcrit 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Two forms of the coefficient of restitution ϵ as a func- tion of impact speed vcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The solid blue curve is the Bridges law, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1), with ϵ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The red solid curve is the ‘regolith’ law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2), with b = (1/4)vcrit and ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In addition, we have sketched two velocity distribution functions with black dotted curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' see discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As mentioned above, the realistic regolith law we adopt approaches the Bridges law for impact speeds larger than the turning point in ϵ, and is a reasonable approxi- mation near the turning point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' One might then expect that collisions employing the regolith law would sample similar values of ϵ and the resulting thermal equilibria will resemble the Bridges equilibria, giving us a ‘warm’ ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1 we superimpose a mock impact velocity distribution at larger vcoll to indicate such a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, when ϵ is a constant and taken to be equal to zero the thermal equilibria are especially cold, with c ∼ aΩ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Araki & Tremaine 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It follows that our regolith law might be capable of supporting these very cold equilibria as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This should certainly be the case if vcrit is much larger than aΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In this circumstance, most impact speeds will fall below vcrit and thus yield perfectly inelastic collisions with ϵ = 0, never sampling the non-zero segment of the ϵ curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1 indicates a schematic velocity distribution for this state, centred on a value less than vcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Both the warm state and the cold state are thermally stable, as has been shown separately in N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' And thus a non-monotonic law may yield bistability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The disk may fall into either the cold or the warm homogeneous state for exactly the same parameters (most notably optical depth τ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Which is chosen depends on the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Moreover, it follows there must also be an intermediate ther- mally unstable state separating the two stable states, though this will not normally be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The argument for bista- bility is strongest in a regime where vcrit ≫ aΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A question then is: what is the minimum value of vcrit that yields bista- 2 This bistability is different to the ‘phase transitions’ associ- ated with viscous instability, which drives the system to a non- homogeneous state characterised by abutting radial regions of high and low optical depth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', Lukkari 1981, H¨ameen-Anttila 1982, Salo & Schmidt 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 5 bility?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our simulations results in Section 3 aim to answer this and other questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 N-body simulations In this subsection we further outline the physical model we adopt and the numerical methods used to calculate its non- trivial thermal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We seek to determine the evolu- tion of a large number of inelastically colliding particles, and thus our main tool will be local N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Equations of motion We solve the equations of motion in the Hill approximation (Hill 1878), a local coordinate system that is co-rotating with a particle on a circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The gravity from the cen- tral object is linearized in local coordinates and the orbital frequency is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This allows, but does not restrict, us to use shear-periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In that case, the Hill approximation is also referred to as the shearing sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In our notation, the x, y, and z coordinates point in the radial, azimuthal and vertical direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Treating the central object, Saturn, as a point source, the equations of motion for a test particle can be written as ¨x = 2Ω ˙y + 3Ω2x + F coll x , (3) ¨y = −2Ω ˙x + F coll y , (4) ¨z = Ω2z + F coll z , (5) where Fcoll is the (intermittent) acceleration exerted on a particle during a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the absence of collisions, the solution to these equations can be written as epicycles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Rein & Tremaine 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The particles move within a finite-size numerical do- main/box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We denote the radial length of the box by Lx and the azimuthal length by Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In all our experiments, the vertical length of the box Lz has been chosen to be large enough so that no particle ever crosses the vertical bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Otherwise, the box is periodic in y and shear-periodic in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The only further ingredients needed are the finite par- ticle radius a and a collision model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We treat particles as hard spheres (they are not permitted to overlap) and the outcome of a collision is described using a normal coefficient of restitution, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The particles have no spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Numerical method We use the freely available N-body code REBOUND (Rein & Liu 2012) to perform all of the simulations presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To evolve the equations of motion forward in time, we use the Symplectic Epicycle Integrator (SEI, Rein & Tremaine 2011) which is well suited for simulations of particle motion within the Hill approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Collisions are detected using a nearest neighbour tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We randomize the order in which collisions are re- solved after each timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We found that this removes spu- rious correlations which might otherwise be introduced when choosing a specific order in which collisions are resolved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' resolving them from left to right, by a numerical particle identifier, or by the position in memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 Diagnostics In order to probe the collective behaviour of the granular flow, we require a number of averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We define the mean normal geometrical optical depth τ as the total projected area of the particles on the (x, y) plane divided by the total area of the (x, y) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In other words, τ = Nπa2/(LxLy), (6) where N is the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus, τ is stipulated at the beginning of each run and does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We also define the radially and temporally varying optical depths, by subdividing the radial domain into thin strips of radial length LS: τ(xi, t) = Ni(t)πa2/(LSLy), (7) where xi is the radial location of, and Ni(t) is the number of particles in, the i’th strip at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The filling factor is defined as the proportion of volume taken up by the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For spherical particles it can be defined as FF = (4π/3)na3, where n is volumetric number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Particularly useful is the filling factor at the mid- plane FF0, which requires the calculation of the number density at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The mean velocity dispersion tensor is computed via Wij = ⟨ ˙xi ˙xj⟩ (8) where ( ˙x1, ˙x2, ˙x3) = ( ˙x, ˙y + 3 2Ωx, ˙z) is the velocity relative to the shear and the angle brackets indicate a suitable aver- age over the particles and possibly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The velocity dispersion c2 is then Wii/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Note that this definition is only correct if there are no mean flows additional to the Keple- rian shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If such flows are slow (as in viscous instability), the error will be small, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The translational (local) component of the kinematic viscosity is νtrans = (2/3)Wxy/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (9) The collisional (non-local) component of the viscosity is νcoll = 2 3ΩNδt � (x⟩ − x⟨)∆py (10) where the sum is taken over all binary collisions that occur in a time interval δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Here M is the total mass of all ring particles, ∆py is the transfer of specific y momentum from the inner to the outer particle in each collision, and x⟩ and x⟨ are the radial locations of the two impacting particles (Wisdom & Tremaine 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Daisaka, Tanaka & Ida 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As we neglect self-gravity, there is no gravitational or wake contribution to the overall momentum transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The total viscosity is hence νtot = νtrans + νcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To determine the thermal conductivity of a given equi- librium state, we follow the method of Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2001) and create a steady non-uniform temperature T profile in the radial (x) direction, where T = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In our cold-state simula- tions, we achieve this by making vcrit radially dependent in the collision law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In our hot-state simulations, we vary ϵmax by a small amount in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In either case, we end up with a steady-state sinusoidal radial temperature profile, though some experimentation is required to find the right amplitude for the variations in vcrit and ϵmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The goal is to keep the perturbations in the temperature ∆T small, but not too small so that they are dominated by shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 6 Larue, Latter, Rein We typically use a simulation with Lx = Ly = 200a and run it for at least 1000 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' After setting up the nonuniform temperature profiles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' we then measure specific translational (local) and collisional (non-local) heat fluxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' qtrans i = 1 2σ⟨c2ci⟩ (11) qcoll i = σ � ∆xiδEs Nδt (12) where σ = N/(LxLy) is the number surface density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' δxi is the absolute difference of the i-coordinates of the two parti- cles involved in a collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' and δEs is the change in trans- ported energy (as opposed to dissipated energy) during the collision for the particle with the larger xi coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Fi- nally, we assume the heat flux is linearly dependent on the temperature gradient, q = −κ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (13) We can then correlate the measured qx and ∂xT and retrieve the conductivity κ using a least squares fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Finally, to verify our set up was working properly, we successfully reproduced Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 8 in Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2001), though omit these results for the sake of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 Parameters and initial conditions In all our N-body simulations, we adopt units so that a = 1 and Ω = 1, though in what follows a and Ω reappear oc- casionally in order to make a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As a consequence, the main physically relevant input is the collision law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Specifi- cally, we have some combination of vcrit/(aΩ), b, and ϵmax for non-constant collision laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We also have the sizes of the numerical domain Lx and Ly and a constant dimensionless time-step Ωdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We use initial conditions where particles are arranged uniformly in the plane with a uniform optical depth τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Therefore an important initial input is particle number N while keeping the computational domain fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Particles are normally distributed in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The initial velocities are also normally distributed with an initial velocity disper- sion c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In most cases we initialize the particles close to the thermal equilibrium we believe to be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We present convergence tests in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These tests shows that our simulations are converged as we vary numerical parameters for both extremely high and low opti- cal depth, as well as hot and cold equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For the regimes that we are interested in, we found that a dimensionless timestep of 10−3 and a box size of 10s to 100s particle radii are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The large box sizes are needed only for very hot and dilute rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 Kinetic theory Though not the focus of this paper, it is useful to have some kinetic theoretical results, especially as they reveal the exis- tence of the additional (thermally unstable) middle branch of equilibrium solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The formalism adopted is Latter and Ogilvie’s (2008) reformulation of Araki and Tremaine (1986), which does not attempt to solve the Boltzmann- Enskog equation but rather a truncated moment hierarchy of continuum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 2 100 101 C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 2 10-2 100 total 1 10 5 20 0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Velocity dispersion and total angular momentum flux τνtot versus optical depth τ for various hard-sphere ϵ laws, cal- culated from N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the top panel the appended numbers ‘1-20’ describe the values of vcrit/(aΩ) when using the standard Bridges law, whereas ‘0’ indicates runs with a constant ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the bottom panel, the ordering of the curves is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The green symbols indicate that the viscous flux is decreasing and the disk viscously unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In previous deployments of this approach, the depen- dence of ϵ on the impact speed was only approximately in- corporated via a ‘pre-averaging’ procedure (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 in Latter and Ogilvie 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Though convenient, this in- troduces unacceptable errors when using complicated non- monotonic laws as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus the complete for- malism is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This does require completing three (in- stead of two) integrations in the collision term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The other main approximations adopted are ‘vertical locality’ and a triaxial Gaussian for the velocity ellipsoid (see Araki and Tremaine 1986 and Latter and Ogilvie 2008 for more de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3 HOMOGENEOUS STEADY STATES In this section we simulate various thermodynamic equilibria and demonstrate that a non-monotonic epsilon law supports up to two equilibria for a given optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We charac- terise these several states with respect to not only their ve- locity dispersion, but also their packing fraction FF0 and transport properties, especially with respect to angular mo- mentum and heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We begin by reproducing previous results in the litera- ture with both a constant and monotonic epsilon law so as to verify that our code is working properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Moreover, as ar- gued in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2, some of the equilibria obtained are lim- iting cases of those appearing in the bistable circumstances explored later and are thus useful in setting the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 7 0 1 2 3 0 5 10 15 0 1 2 3 0 5 10 0 1 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 0 5 10 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 FF0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 2 tot Realistic max=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 Realistic max=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923 BPL max=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Selected equilibrium properties as functions of τ for three regolith ϵ-laws (the three columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The leftmost column shows equilibria computed with the broken-power law model (BPL) with ϵ0 = 0 and ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8, whereas the other two columns show the realistic model with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In all cases vcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the top row the joined circles denote the velocity dispersion calculated by N-body simulations, with the colours indicating hot (red) or cold (blue) branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The second and third rows show the filling factor and total angular momentum flux respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The dashed curve indicates equivalent solutions obtained from the kinetic theory (in the BPL case only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the bottom row, a green symbol indicates expected viscous instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The distribution of impact velocities in simulations using the ‘realistic’ law at τ = 1 with parameters vcrit = 5, b = 1, and ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The left panel shows the system in the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The right panel shows the system in the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The red line corresponds to the ϵ law adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Comparison with previous calculations Our reference cases include the simulations of Salo (1991), who employed a Bridges law but with a variable scale ve- locity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1) with ϵ0 = 1 and vcrit = 1, 5, 10, 20 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1), and also simulations with a constant ϵ = 0, which brings about a very cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The results of our cal- culations are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2, in which we show the velocity dispersion c and angular momentum flux τνtot versus optical depth τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The simulations were run until they were collision- ally relaxed, and then continued for the same length of time to obtain averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' When τ was low, and colli- sions relatively infrequent, the total run time was > 1000 Ω−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' but at higher τ (∼ 2) runs could be as a short as 50-80 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Direct comparison of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2 with the numerical results of Salo (1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' his Figs 3-5) shows good agreement, and also consistency with the kinetic theory of Latter & Ogilvie (2008) (note that both these works denote vcrit by vb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An interesting feature of the ‘warmer’ solution branches is the decreasing viscosity with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In fact, the hottest case, vcrit = 20, is viscously unstable because the gradient of the angular momentum flux τν is negative in an interval of τ (green markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' By inflating vcrit in the Bridges law the velocity disper- sion of the system can be controlled and, in particular, set to ‘warm’ values greater than aΩ and, consequently, greater than the temperature of the very cold ϵ = 0 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These warm and cold states help illustrate the arguments presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If we take one of the two non-monotonic col- lision laws and set vcrit ten or more times aΩ, then start the simulation with a hot initial condition, we might expect the subsequent spread of impact speeds to be sufficiently far from ϵ’s turning point (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1) so that the system settles into a warm ‘Bridges equilibrium’, similar to those plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, if we begin the same simulation but with very cold initial velocities (≪ vcrit), the subsequent spread of impact speeds will remain less than vcrit and ϵ will almost always take the value of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the system © 0000 RAS, MNRAS 000, 000–000 LD LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0 0 24 mpact velociby y8 Larue, Latter, Rein will then converge to the appropriate constant ϵ = 0 state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Note that the Bridges law produces a velocity disper- sion c that decreases with τ and we may then expect that for sufficiently large τ the upper ‘hot state’ will be too close to the ‘cold state’ and bistability may disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Non-monotonic collision laws In this section we calculate equilibria for ‘regolith’ epsilon laws that are non-monotonic: either the broken power law (BPL) with ϵ0 = 0 or the realistic law (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The parameters are ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, vcrit = 5aΩ, and b = 1, though we examine a broader spread of values in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We first examine in some detail the thermal properties of the states, then their transport of angular momentum and heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Thermal hysteresis Figure 3 constitutes the first main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Here we plot the equilibrium velocity dispersions (top row), filling factors (middle row), and total radial angular momentum fluxes (τνtot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' bottom row) obtained in a sequence of simu- lations at different optical depths and for different ϵ mod- els and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Each circular marker corresponds to a different simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These values are obtained by time av- eraging a quantity once the system has become collisionally mature, as earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For example, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 runs were run for 1600Ω−1 and averaged for the last 800Ω−1, while at τ = 2 the total run length was 80Ω−1, with the averaging taking place over the last 40Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As is clear, in the three models presented, two steady state branches (distinguished by red and blue) are possi- ble within a certain range of optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Which of the two the system selects depends on the initial condition: a ‘cold start’ (low initial c) usually (but not always) takes the system to the nearby cold state, whereas a ‘hot start’ (ini- tial c sufficiently high) settles on the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Typically, runs starting with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5aΩ converged to the nearby cold state, while runs beginning with c = 10aΩ migrated to the hot state, if one was available, even if that state’s veloc- ity dispersion was significantly larger than the initial c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The direction of migration is discussed further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The apparent bistability extends over a range of small to intermediate optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Beyond a special τ the hot state disappears, and all hot start simulations landed on the cold branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At small τ we never found that the cold state disappeared, except in the case of the realistic model with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' this equilibrium was metastable (explored in more detail in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The bistable regime’s width (in τ) depends on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3, in- creasing the ϵmax in the realistic model from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923 moved the special τ from roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' middle and right columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The cold equilibria take c values very much in agree- ment with the constant ϵ = 0 states simulated in the previ- ous subsection, while the hot state resembles a Bridges law, with c decreasing with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In fact, the hot simulations of the realistic model with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923 take a similar c as the Bridges vcrit = 10 runs, while those with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 resem- ble a Bridges law with (roughly) vcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These similarities bolster our interpretation of the two states as ‘separated’ by Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Grids of simulations undertaken with different vcrit and ϵmax using the realistic regolith law with widths b = 1 (top) and b = 2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Colours correspond to values of |chot −ccold| (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The contour is a conservative boundary between cases that support bistability (to the right and above) and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the turning point of the ϵ curve: only a minority of collisions in the hot state occur with the low impact speeds that would trigger ϵ = 0, while collisions in the cold state rarely occur with impact speeds sufficiently large to trigger larger ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To flesh out this point further we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 4 the distribution function of impact speed for a hot state (right panel) and a cold state (left panel) for the same τ = 1 (and other pa- rameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Superimposed in red is the ϵ law used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As the left panel indicates, cold state collisions are almost completely inelastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the narrow spread in impact speeds barely overlaps the portion of the curve for which ϵ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In contrast, the hot state (shown in the right panel) is much broader and thus samples a wide range of ϵ, but importantly peaks at speeds which yield collisions with a small dissipation of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The filling factors in the middle row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3 reveal that the hot branches are far less dense than the cold branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For example, in the realistic model with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, at τ = 1 the hot state possesses a filling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='08, while the cold state has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The difference, of course, is not due to the surface number density (which is the same) but because the disk semi-thickness is so different between these two states: in the hot state it is ≈ 6a, compared to ∼ a in the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The ratio of the two filling factors should scale roughly with the ratio of semi-thicknesses and that is indeed what we see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The hot state branch terminates when its velocity dis- persion approaches a critical value ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In reality the sys- tem here encounters a saddle-node bifurcation and the solu- tion curve bends ‘backwards’ thus forming an intermediate branch of thermally unstable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Because these solu- tions are unstable they cannot manifest in N-body simula- tions3, but they can be calculated by kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Kinetic 3 See Salo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (1988) for a numerical exploration of a thermally unstable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 b=1 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='9 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 20 max 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 0 0 1 2 3 4 5 9 7 8 9 b=2 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='9 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 max 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 0 0 1 2 3 4 5 9 7 8 9 critThermal hysteresis in rings 9 theoretical equilibria are plotted in the leftmost column with a dashed black curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the top and middle panels show clearly an intermediate cool, semi-dense branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The agreement be- tween theory and simulations is qualitative good, with the biggest deviation in the translational viscosity in the hot state, a discrepancy that has been noted in previous com- parisons (Latter and Ogilvie 2008, Rein and Latter 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Parameter survey In the preceding subsection we examined only three param- eter sets/models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' in this subsection we adopt the realistic ϵ law and scan through vcrit and ϵmax for two different widths b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our aim is to determine how representative the thermal hysteresis explored in the previous subsection really is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Of particular interest are the lowest values of vcrit and ϵmax that yield bistability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5 we present ‘bistability plots’ for b = 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Each square in the grid corresponds to a parameter pair (vcrit, ϵmax), and for each square we conduct two simulations with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1, one with a hot initial condition and the other with a cold initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Each simulation has been run until thermal equilibrium has been obtained, and the dif- ference in final velocity dispersion calculated, |chot − ccold|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Finally, the square is coloured accordingly (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the colour bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the difference in final c is between 0 and 5, we in- terpret that the two simulations are converging on to the same (cold) equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Values larger than 5 (admittedly, a rather large value, given Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3) we assume correspond to a bistable situation: the two simulations are settling on differ- ent thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In both panels we have superimposed the contour of |chot − ccold| = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The reader should then assign bistability to regions of the parameter plane above and/or to the right of this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The plots indicate, as expected, that bistability is favoured by larger values of vcrit and ϵmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Increasing both parameters helps to separate the typical impact speeds of the hot state from those of the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Interestingly, the bistable region is quite rectangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus when b = 1, bistability is guaranteed (roughly) if both vcrit > 4 and ϵmax > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We expect that these parameter restrictions should hold roughly for other non-monotonic laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Finally, the range of bistability is also sensitive to the width of the epsilon law, as the b = 2 plot demonstrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Increasing the width also helps separate out the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the b = 2 case bistability occurs when vcrit > 3 and ϵmax > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 Viscous properties The equilibrium states discussed in the previous subsection support a viscous stress that, by acting on the background orbital shear, transports angular momentum radially across the numerical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The viscous properties of the flow are important thermodynamically because the stress extracts free energy from the shear, thus providing the heating source in the thermal balances undergirding these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' But the viscous stress is also important dynamically because it can 4 Unfortunately, numerical difficulties prevented us calculating kinetic solutions for the realistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' τ κL (C) κNL (C) κL (H) κNL (H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='42 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='62 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='15 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='61 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='59 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Calculated translational (local) thermal conductivities κL and collisional (non-local) thermal conductivities κNL in cold (C) and hot (H) equilibria at various optical depths τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A realistic collision law is adopted with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vcrit = 5, and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' beget instabilities, such as the viscous overstability and in- stability (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particular, if d(τνtot)/dτ is negative then viscous instability occurs (Lin and Boden- heimer 1981, Lukkari 1981, Ward 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The angular momentum flux is plotted in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Note that a subset of hot states possess a decreasing flux and are thus viscously unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' these are marked in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the BPL model, the unstable interval encompasses τ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5, whereas in the realistic model only the ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923 case yields instability and then for τ between approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Instability here is as- sociated with a dominant translational viscosity, which can decline at sufficiently large τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Growing modes do not appear in these simulations, however, because the numerical do- main size is smaller than the shortest unstable wavelength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 we simulate larger domains and recover the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 Thermal conductivity Anticipating later sections which explore different thermal states that spatially adjoin, we compute the radial flux of thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the absence of any mean spatial gradients, such as in the homogeneous equilibria calculated, the flux must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' But if two states connect in radius the flux must control, in part, how their interface evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3, we adopt the approach of Salo et al (2001) and impose a radial sinusoidal temperature structure upon the box, through the parameters vcrit and ϵmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 7 we show calculations of the radial thermal flux qx and the thermal conductivity κ for a fixed set of parameters (ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vcrit = 5, b = 1) and for the same optical depth τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The left four panels correspond to the cold state (c ≈ 1), and the right to the hot state (c ≈ 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The top left panel in each case describes the temperature profile across the box, while the top right panel shows the temperature gradient (solid blue), the translational (local, ‘L’) heat flux (dashed gold), and the collisional (nonlocal, ‘NL’) heat flux (dotted green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The latter two are plotted separately as functions of the temperature gradient in the bottom panels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' a best-fit line extracts the conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In both the hot and cold cases, the translational heat flux dominates the collisional flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This means that the heat flux in the two states differs significantly, despite possessing the same τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In Table I we list κ for a range of τ and otherwise with the same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 10 Larue, Latter, Rein 100 0 100 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='95 T 100 0 100 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='02 dT/dx qx (L) qx (NL) 100 0 100 x 40 45 50 T 100 0 100 x 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='003 dT/dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='02 qx (L) L=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='003 dT/dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='004 qx (NL) NL=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 dT/dx 20 10 0 10 20 30 qx (L) L=119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 dT/dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 qx (NL) NL=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='01 COLD, max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vc = 5, b = 1, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 HOT, max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vc = 5, b = 1, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thermal diffusivity measurements for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 in the cold state (left panels) and hot state (right panels) for the realistic model with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vcrit = 5, and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Velocity dispersion as a function of time for runs with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 (top panel) and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The realistic model is adopted with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, vcrit = 5, and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 4 METASTABILITY In the last section we calculated steady states that appear to be thermally stable, at least linearly according to a contin- uum interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' However, N-body systems are replete with small but finite amplitude shot noise that continually tests the nonlinear stability of any steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the basin of attraction of a linearly stable state is small relative to the amplitude of these fluctuations, the system can potentially jump out of the state and migrate elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Many phys- ical and biological systems offer similar examples of noise destabilising what should be linearly stable fixed points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Mel’nikov 1991, May 1973, De Swart and Grasman 1987, Majda, Timofeyev and Vanden-Eijinden 1999, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In this section we investigate this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our focus will be on cold states of low-optical depth and on the hot states near the saddle node bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The reason is that these states are close to the unstable mid- dle branch which can serve as the boundary of the basin of attraction in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We find that, for the parameters and models we employ, metastability is relatively uncom- mon, only occurring in certain dilute and cold states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particular, states near the saddle node are generally stable to shot noise perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Before presenting our results we emphasise that we only explore the effect of intrinsic shot noise, but in real rings there are several other sources of finite amplitude disturbances that may work similarly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' meteoroid bom- bardment, embedded moonlets, density waves, and gravity wakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 20 15 10 5 0 0 100 200 300 400 500 time [orbits]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 0 25 50 75 100 125 150 175 200 time [orbits]Thermal hysteresis in rings 11 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Velocity dispersion as a function of time for runs of different initial conditions with τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='61 (top panel) and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='64 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The realistic model is adopted with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, vcrit = 5, and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Cold to hot transitions We find spontaneous transitions from the cold lower branch to the hot upper branch in only a few low τ cases when adopting a realistic collision law and ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Specif- ically, when τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 the system can hover about the cold steady state for several hundred orbits before jumping to the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To probe this behaviour we ran 24 runs with slightly different initial conditions (varying both particles’ locations and velocities) but all starting with the same low c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To make doubly certain that the system is as close to the cold equi- librium as possible, and that any future transition is not the result of a wayward initial condition, we force ϵ = 0 (a constant) for several orbits at the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The evolution of these runs are plotted in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 7, with the shaded region indicating when ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As is clear from the figure, all but three runs jumped to the hot state by 500 orbits (roughly > 25 collision times), though there was a wide spread of transition times, indica- tive that the process is stochastic and issues from the noise: ultimately, after some period, an overenthusiastic collision, dissipating insufficient velocity dispersion, seeds a patch of more energetic particles, which then spreads spatially and takes over the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Of course, this is only part of the story, because en- ergetic events must happen at slightly larger τ but do not appear to instigate runaway heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Indeed, we undertake a similar experiment at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2, plotted in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 7, and witness no transitions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' What is key is the overall basin of attraction of the cold state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' as shown by the kinetic curves in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3, the middle unstable branch and the cold lower branch become closest at low τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The middle branch acts as the boundary of the lower state’s basin of attraction (at least in this simple phase space projection);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' thus at low τ it becomes more likely that a finite amplitude perturbation can tip the system over this boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' That said, it is not straightforward to firmly con- nect microphysical fluctuations (shot noise) to such a mean finite-amplitude perturbation in this phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Hot to cold transitions We now check if it is possible to obtain spontaneous hot to cold transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We focus on states near the tip of the saddle node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the termination of the hot branch (see top row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3), and examine a range of τ between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='61 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='65 in the realistic model with ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We simulate several runs with slightly different initial conditions, as before, and plot the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 8, top and bottom panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As in the previous subsection, to ensure that we start the simulations in a hot state we set vcrit to a very small value initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Over several orbits (indicated by the shaded area in the figures), we slowly increase vcrit to the nominal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Unlike cold to hot transitions, the systems either imme- diately drop to the cold state or relax into the hot state on a timescale of 10 orbits or so (a handful of collision times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='65 all the simulations ended up in the cold state, while at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='64, some stayed in the hot state, while at lower tau again (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='61) most stay in the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Putting aside the percentages in one or the other, the system transitions promptly or not at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We attribute this more to the initial condition at the end of the blue phase, rather than having to wait for a more sluggish group of collisions that lead to a ‘chain reaction’ and a switching of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The difference with the low τ runs explored earlier may partially be explained by the separation between the middle and hot branches, which is relatively large, even near the tip of the saddle node (see kinetic theory curves in top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Once a system settles on to the hot state, and its initial conditions mostly forgotten, its intrinsic shot noise is insufficient to tip it out of its basin of attraction and into the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 5 4 3 2 1 0 50 100 0 150 200 250 time [orbits]5 4 3 2 1 0 100 0 50 150 200 250 time [orbits]12 Larue, Latter, Rein 600 400 200 0 200 400 600 x 20 10 0 10 20 z t=0 orbits Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Initial condition for the fiducial thermal-front simula- tion described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 in the form of an (x, z) projection of the particle positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5 THERMAL AND VISCOUS FRONTS Having computed several homogeneous states, we now ex- plore the dynamics when different states spatially adjoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If a ring region is bistable, then it is likely that such situa- tions occur, given the varying dynamical histories at differ- ent radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our main focus is on the structure and evolution of the transition (or front) between two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We will con- sider two cases: (a) thermal fronts, which join two states of the same τ but different c, and (b) viscous fronts, which connect two states of the same angular momentum flux τν, but different τ and c Thermal fronts involve a hot and a cold state, with the pair joined by a vertical line in the top panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Though sharing the same optical depth, they possess dis- tinct vertical thicknesses that may produce a photometric variation, and thus observable structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Salo and Kar- jalainen 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' However, the two states will support different angular momentum fluxes τν, and thus mass may pile up or evacuate near the thermal front, potentially leading to non- steadiness and a complete break down of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We find that this is avoided if the front itself moves sufficiently fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' One might expect radial mass redistribution is negated if two adjoining states possess the same angular momentum flux, with the pair joined by a horizontal line in the bot- tom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In fact, similar structures have already been witnessed in simulations of the viscous instability with monotonic ϵ laws (Salo and Schmidt 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We find, how- ever, that the finite width of the front itself spoils the ex- act matching of fluxes and makes the establishment of such fronts more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Thermal fronts In order to explore the structure and dynamics of fronts connecting equilibria of different temperatures but the same surface density, we concentrate on a single parameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The behaviour obtained is then interpreted using a simple continuum model, before other parameters are trialled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Fiducial case Our fiducial run employs a realistic ϵ law with the following parameters: ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='75, vcrit = 5, and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We examine a hot and cold state of the same τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2, with the former possessing c = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 and the latter c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We adopt a wide box of radial size 1000a and insert a strip of particles from the (previously computed) hot state in the centre (with ra- dial extent 100a), while distributing particles from the cold state throughout the rest of the numerical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figure 9 plots this initial condition as a projection of the particle locations in the (x, z) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Away from the borders of the hot/cold zones, the ring is in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The subsequent evolution of the ring is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 10, which presents four snapshots at different times on each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The left panels describe the (x, z) projections of the parti- cles, while the right panels plot the radial variation of τ (blue) and c (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As is clear, the two fronts move radially into the cold state, until the hot state takes over the box entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Meanwhile, τ remain roughly constant throughout, except for some minor deviations around the front itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The front speed is constant until the moment that the cold state evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This is demonstrated in Figure 11, which plots the location of the rightmost front as a func- tion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A c intermediate between the c in the hot and cold states was selected (here c = 4) and its x location was determined at each time-step, which provided a means to capture the movement of the front as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The front speed is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='685aΩ, thus slightly less than c in the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Generally, in bistable systems, the conductivity controls the structure of fronts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' a small conductivity yields a narrow transition, while a large conductivity gives a more diffuse transition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Latter and Balbus 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In our granular gas, the thermal conductivity κ depends on c, and thus jumps by at least an order of magnitude as we go from the cold to the hot state (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This explains why the front structure is sharp near the cold state (though always longer than the ‘granularity scale’, a), while broader and smoother near the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The overall width of the front (≳ 100a) is hence determined approximately by κ in the hot phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Physics of front motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' a simple continuum model The basic mechanism driving the movement of a thermal front relies on the finite-amplitude perturbations arising from the proximity of the different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These perturba- tions can only be communicated via thermal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For example, near a front, the cold state will receive thermal en- ergy (via diffusion) from the adjacent hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the energy received is sufficient to push the cold ring material out of the cold state’s basin of attraction, then one might expect it to heat up and settle on the hot state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' as a consequence, the front advances into the cold phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' But, by the same token, on the other side of the front, material in the hot state will also be perturbed by the heat flux and will cool down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If this cooled material is pushed beyond the hot state’s basin of at- traction, then it will undergo a runaway cooling and then we might expect the front to advance into the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 13 600 400 200 0 200 400 600 x 20 10 0 10 20 z t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 orbits 500 250 0 250 500 x 0 2 4 6 8 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='80 orbits 600 400 200 0 200 400 600 x 20 10 0 10 20 z t=8 orbits 500 250 0 250 500 x 0 2 4 6 8 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 t=8 orbits 600 400 200 0 200 400 600 x 20 10 0 10 20 z t=80 orbits 500 250 0 250 500 x 0 2 4 6 8 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 t=80 orbits 600 400 200 0 200 400 600 x 20 10 0 10 20 z t=191 orbits 500 250 0 250 500 x 0 2 4 6 8 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 t=191 orbits Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Snapshots of a thermal front at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8, 8, 80 and 191 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Panels on the left describe a projection of ring particles on to the (x, z) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Panels on the right depict the x-dependent velocity dispersion c (red) and optical depth τ (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 14 Larue, Latter, Rein 0 200 400 600 800 1000 t 0 100 200 300 400 500 600 700 x-coordinate of front Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Outer front radial location as a function of time in the simulation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Which thermal runaway is favoured on average depends on the relative sizes of the hot and cold state’s basins of attrac- tion, which can be approximated (roughly) by how close the intermediate unstable state is to either state (see discussion in the section on metastability, and also Latter and Balbus 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These ideas can be illustrated by a continuum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The energy equation of the gas may be written as ∂tE = Λ(E) + ∂x(k∂xE), where E = (3/2)c2, Λ combines viscous heating and col- lisional cooling, and k is thermal diffusivity (= 2κ/(3σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus Λ = 0 when E is equal to the stable hot, cold, and unstable intermediate steady states, EH, EC, and EI, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Moreover, dΛ/dE < 0 when E = EH or EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We assume a steady front, moving at speed vf, with the hot state to the right and the cold state to the left, and thus introduce the comoving variable ξ = x − vft, which trans- forms the energy equation into a type of Stefan problem for the front shape E(ξ) and speed vf, ∂ξ(k∂ξE) + vf∂ξE + Λ(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (14) The boundary conditions are E → EH as ξ → ∞ and E → EC as ξ → −∞ (hot to the right and cold to the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This is a nonlinear eigenvalue problem that, after specify- ing the functional forms of Λ(E) and k(E), would normally require a numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In Appendix B we adopt sim- ple prescriptions for these functions and solve the equation, thereby illustrating some of the main features discussed be- low and qualitatively reproducing our N-body results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An illuminating expression for the speed vf can be ob- tained by multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (14) by dE/dξ and integrating between −∞ and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' After some manipulation, one gets vf = − � EH EC Λ dE � ∞ −∞(dE/dξ)2dξ − � ∞ −∞(dk/dE)(dE/dξ)3dξ 2 � ∞ −∞(dE/dξ)2dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (15) If the thermal diffusivity is a constant, the second term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In this case, the sign of vf is determined solely by the integral of the heating/cooling term Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Because Λ(EH) = Λ(EI) = Λ(EC) = 0, the integral can be subdivided into (a) a positive part (between EC and EI) that measures the ‘size’ of the cold state’s basin of attraction, and (b) a negative part (between EI and EH) that measures the hot state’s basin of attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The proximity of EI to either EC or EH indicates the basins’ relative sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If EI is closer to EC, then the integral is dominated by the positive area, vf < 0, and the front moves into the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Physically, cold ring material near a front finds it easier to undergo a heating runaway, when perturbed by the front, than hot material finds a cooling runaway;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' thus, the front advances into the cold material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If EI is closer to EH, then the converse holds and the front moves into the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Turning now to the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3 (first panel especially), one naively expects that at low τ fronts initially move into the cold state, but at higher τ fronts are slower and then at some critical τ may reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If k depends on E then things are more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (15) is a weighted average of dk/dE, and shows that a non-uniformity in the transport of heat moderates the effect discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the front shape is monotonic in ξ, then dE/dξ > 0 throughout and the sign of the second term is determined by dk/dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As demonstrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 and Table I, dk/dE > 0, and so the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (15) is always positive, thus biasing the front’s movement into the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The underlying mechanism here rests not on the system’s bistability but on exacerbating the imbalance in the heat flux throughout the front struc- ture: at any given point more heat is arriving from the hot state than is being evacuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The discussion above suggests that the sharp region at the foot of the front controls the front speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Taking an order of magnitude approach and equating the three terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (14) yields the estimate vf ∼ � kC/tth, where the thermal timescale is defined as tth = E/Λ ∼ c2/(νΩ2), and kC is the diffusivity evaluated in the cold state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Putting in values for the cold state gives us vf ∼ aΩ, which is consistent with the value calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The width λ of the front extending through the hot phase can then be estimated by balancing the first two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' we find λ ∼ kH/vf ≳ 500a, which is also consistent with the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 Front stability We conducted a short survey of fronts at different τ and calculated their speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' When τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 we found vf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='518, and when τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3, vf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' While no clear trend could be observed between τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3, we expected at larger τ, as we approached the saddle node, that the front speed should decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In fact, what we found for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 or larger is that the front would slow to a halt and then viscously reshape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' τ would evolve away from a uniform profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Ultimately, the system moves to a state of constant angular momentum flux τν, and the thermal front dissolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As mentioned earlier, the issue here is that across a thermal front τ is constant, but τν is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As a consequence, mass can potentially build-up/evacuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the front moves faster than τ can be viscously redistributed, then we expect the front to remain coherent and to travel unimpeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the front speed is too slow, then it will be viscously reshaped and will collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For the model chosen, τ ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 corresponds to the first case, and τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3 to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A rough criterion for the ‘stability’ of the front to vis- © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 15 cous redistribution would tension the relative sizes of the front speed vf and the viscous diffusion speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To deter- mine an estimate on the latter, we employ the lengthscale of the abrupt transition at the foot of the structure and thus estimate the diffusion speed as ∼ (νC/κC)vf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A sim- ple criterion for front dissolution requires that this speed is greater than vf, and hence depends solely on the size of the Prandtl number Pr = ν/κ in the cold state: when Pr is greater than a critical value Prc, we expect the front to dis- solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Indeed, Pr increases monotonically between τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4, though takes relatively small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4, we find that Pr ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='04, which must be near Prc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Viscous fronts and viscous instability Given the issue of the unbalanced angular momentum in thermal fronts, it is natural to explore fronts that join states with the same viscous transport properties, specifically ντ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We present simulations of such joined states in this subsec- tion, in addition to a short treatment of viscous instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A simple continuum model can guide our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In the shearing sheet, the one-dimensional diffusion equation for viscous Keplerian disks is ∂tτ = 3∂2 x(ντ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Lynden-Bell and Pringle 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Suppose a viscous front moves with speed vf with τ → τA as x → −∞ and τ → τB as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As earlier, we adopt a comoving variable ξ = x−vft, which permits the complete integration of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We find that vf = 0 (the structure must be stationary) and ντ (= νAτA = νBτB) is a constant throughout the entirety of the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The last constraint is a potential difficulty: while it is possible to find two homogeneous steady states of the same ντ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' panels in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3), a realis- tic front will have a finite width in which τ will vary and thus ντ will deviate from the required constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our simulations show, in fact, that the system can overcome this problem by settling on a front structure in which the average ντ equals νAτA = νBτB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 Fronts We present a fiducial simulation with the realistic law, and parameters b = 1, ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, and vcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To construct a suitable initial condition that might produce a viscous front, we select two thermally and viscously stable states with the same ντ from the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Such pairs are joined by horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We select two states of the same angular momentum flux ντ ≈ 2, with optical depths τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The numerical domain is chosen to be sufficiently large (L = 800) to accommodate relatively undisturbed expanses of the two states, in addition to the front itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' the low τ state is placed between x = −100 and 100, with the high τ state taking up the remainder of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figure 12 shows eight snapshots of the resulting simu- lation at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In each panel we plot τ (red) and τν (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' At t = 0, the angular momentum flux τν is a constant, but τ undergoes two jumps (at x = ±100a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As the system evolves, the two jumps/fronts relax and exhibit a characteristic width, with τ taking values between those of the two steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An immediate consequence is that the angular momentum flux within the fronts begins to deviate from the fixed value ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In fact, the first four panels show that it takes significantly larger values than 2, in agreement with the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3, which shows that states with τ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='16 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 exhibit ντ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Because of the enhanced flux in the fronts, mass is being transported out of the fronts, which then appear to move as the system evolves far way from the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Ultimately, we find that the system redistributes the mass throughout the numerical domain so that τν is roughly constant (≈ 7), but still allows for strong variations in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This outcome is not a constant τ state, but consists of two static viscous fronts joining two homogeneous states of τ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7, which according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3 possess the same angular momentum flux (∼ 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Evidently, the front that joins the two states also possesses a similar approxi- mate flux, though this is difficult to determine from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A similar final state was found by Salo and Schmidt (2010) when simulating the viscous instability directly (see next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This static structure is an interesting outcome for the system, but we stress that it is possible only because of the periodicity of the numerical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Owing to those boundary conditions, mass in the whole domain can be re- distributed until the desired constant ντ state can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In a more realistic setting, the system is unlikely to come to steady state and the front will continue to move until it encounters large-scale variations in background disk proper- ties, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 Viscous instability In the previous subsection we explored two adjoined vis- cously stable states, but the lower right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3 in- dicates that there is a branch of viscously unstable states of intermediate τ between roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An obvious question is: to where does the system evolve if started from one of these states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We thus present a simulation with the same collisional parameters as earlier, but with a homoge- neous τ of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3, this state is viscously unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figure 13 shows 5 snapshots of the system’s evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Despite possessing a constant τν, the system moves slowly away from this state and begins to develop grow- ing patches of high and low τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Unlike the previous subsec- tion, where the evolution is being driven by large-scale flux imbalances, here there is an instability mechanism, in which small-scale fluctuations in the flux self-reinforce (Lin and Bo- denheimer 1981, Lukkari 1981, Ward 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Ultimately, the system settles on a sequence of distinct high-τ islands sur- rounded by relatively dilute regions, but both with roughly the same flux (≈ 6, in this case), as is necessary for a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These results are very similar to those predicted by H¨ameen-Anttila (1982) and witnessed in Lukkari (1981) and Salo and Schmidt (2010), though they use a monotonic col- lision law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A key difference is that in the monotonic ϵ simu- lations, the final outcome joins states from the same branch, while in our non-monotonic simulations states from different branches adjoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An interesting consequence of this is that it is still possible for the system to separate into a sequence of © 0000 RAS, MNRAS 000, 000–000 16 Larue, Latter, Rein 400 200 0 200 400 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Snapshots of an example viscous front, showing optical depth and angular momentum flux as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The initial condition connects two states of different τ but the same angular momentum flux τν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Despite this balance, the system evolves, redis- tributing mass and angular momentum until a steady state is achieved characterised by a different constant τν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The collision law employs the realistic model with vcrit = 5, ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Snapshots are at t = 5, 20, 30, 50, 100, 500, 1000, and 2000 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' high and low τ states (of the same ντ), even when there is no intermediate viscously unstable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particular, this ap- pears achievable for the parameters of the middle column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' More generally, systems with non-monotonic collision laws have more freedom to exhibit viscous phase-separation in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 6 DISCUSSION AND CONCLUSION Most previous work describing the local collisional dynam- ics of Saturn’s rings uses relatively simple collision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Given the poorly constrained nature of the collisions, and the numerical challenges involved, this is understandable, and indeed some success has been achieved in certain appli- cations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' self-gravity wakes, viscous overstability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' How- ever, current models still fail to describe much (if not most) of the irregular axisymmetric structure exhibited in Saturn’s B and C rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This invites us to experiment with other more complicated collision laws, in particular those that account (in a basic way) for surface regolith on ring particles, which is deemed to be present and important (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2008, Morishima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2012, Deau 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We conduct N-body simulations with the REBOUND © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 17 400 200 0 200 400 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1.' 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+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='0 0 2 4 6 8 10 12 t=2000 orbits Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Snapshots showing the progress of viscous instability starting from an unstable state of τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The collisional parameters are vcrit = 5, ϵmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='923, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The panels describe the x dependent optical depth τ (red) and the angular momentum flux τν (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Snapshots are at 50, 750, 1000, 1050, and 2000 orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' code of a local patch of Saturn’s rings in which particles un- dergo collisions with a prescribed coefficient of restitution ϵ depending on impact speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The main novelty of our ap- proach is to employ an ϵ that is a non-monotonic function of impact speed, as is suggested by theoretical and experi- mental studies of regolith-coated particles (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Below a critical impact speed we set ϵ = 0, though neglect particle sticking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This relatively minor change in the phys- ical set-up immediately introduces major thermodynamical changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For the same optical depth, the rings yield two thermally stable steady states, a hot c ≳ 4aΩ and a cold c < aΩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Which is selected depends on the local ther- mal and/or dynamical history, and thus different ring radii might fall into one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' An obvious follow up question is to ask what happens at the boundaries of two adjoining different states?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We run additional simulations in larger domains and find that in general the hot state will engulf the cold state, with the transition front moving at a speed ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5aΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Slower mov- ing fronts break down because of the imbalance in angular momentum flux across the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Stationary ‘viscous fronts’ are also simulated which join states of different opti- cal depth and c but the same angular momentum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Note that it need not necessarily be the case that hot states always take over: smooth variations in the ring’s background prop- erties may change propagation, and large amplitude pertur- bations (meteoroids, density waves, gravity wakes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=') will also complicate the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our simulation results are exploratory, and should be taken as a demonstration of what happens when one relaxes the strong modelling assumptions of previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' They are perhaps not yet ready for direct application to structure for- mation in Saturn’s rings, not least because of the parameters in our regolith laws are poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Nonetheless, it is irresistible to speculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We anticipate that a thermal front, connecting a warm and cold state of the same dynamical optical depth, gives rise to photometric variation (which the Cassini cameras may have picked up) but no variation de- tectable by occultation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' This is precisely the sit- uation in the C-ring plateaus (Hedman and Nicholson 2013), and indeed, there is evidence of size segregation across these structures which may tie in to the greater chance of sticking in the colder phase (Marouf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2013, Colwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It may also be relevant for the 10km striations shown by Cassini’s cameras in the A and B-rings (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Figs 5A and 5B in Porco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' On the other hand, the steady viscous fronts our simulations support, which connect states of high and moderate optical depth, bear some resemblance to the disjunct bands in the middle B-ring (Colwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' A great deal more theoretical work and modelling is needed before these associations can be made secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particu- lar, applications to ring regions exhibiting self-gravity wakes must remain tentative until we produce better constrained estimates on typical sticking speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Other areas of future work could explore the interplay between the hysteresis and self-gravity wakes, on one hand, and viscous overstability, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' For example, we might anticipate wakes appear only in the cold state, chang- ing its viscous properties, and providing energy to jump into the hot state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' More generally wake activity will produce en- hanced heating and thus a change in the thermodynamic balances calculated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Viscous overstability gen- erates nonlinear travelling wavetrains which may also favour the cold phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' these waves will reflect off the boundaries be- tween states, hence complicating the nonlinear saturation of the wave turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Simulations including realistic photom- etry of thermal fronts might help establish if they might cor- respond to any observable structure (Salo and Karjalainen 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Finally, the robustness of bistability must be estab- lished when particle sticking is permitted, as in recent sim- ulations by Ballouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2017) and Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 18 Larue, Latter, Rein ACKNOWLEDGMENTS The authors thank the reviewer Heikki Salo and Juergen Schmidt, who generously provided a set of helpful and thor- ough comments that markedly improved the paper.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Planetary Ring Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Properties, Structure, and Evolution, Edited by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Tiscareno and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Journal of Applied Mechanics, 64, 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thornton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', Ning, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Powder Technology, 99, 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Tiscareno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', and 24 coauthors, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Science, 364, 6445, id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' aau1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Ward, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' GRL, 8, 641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Wisdom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', Tremaine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=', 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The Astronomical Journal, 95, 925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' APPENDIX A: CONVERGENCE TESTS We present some results showing the behaviour of a subset of our equilibrium solutions as the numerical parameters are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In particular, we explore their dependence on the size of the time-step dt and the numerical domain, showing that convergence is achieved when the former is sufficiently small and the latter sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' To simplify the study, we adopt a standard Bridges law for two different vcrit (yielding hot and warm equilibria) and also a constant ϵ = 0 (yielding cold equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We examine very dilute cases τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 and very dense cases τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5, thereby determining the numerical requirements at the physical ‘boundaries’ of our main set of results, and thus for the main results themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Our convergence results are plotted in Figs A1 and A2, the former showing the velocity dispersion c as a function of dt, the latter c as a function of box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Time steps of 10−2 © 0000 RAS, MNRAS 000, 000–000 Thermal hysteresis in rings 19 10-3 10-2 10-1 dt 100 101 102 C Bridges laws vcrit=20, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 vcrit=20, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 vcrit=1, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 vcrit=1, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 10-4 10-3 10-2 10-1 dt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='9 1 C = 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Convergence tests in time step for several set-ups spanning dilute and cold, dense and hot, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' or less and a box size of 30 or greater appear to be sufficient in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In our main equilibrium runs in Section 3, we use dt = 10−3 and a box size of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' APPENDIX B: ILLUSTRATIVE TOY FRONTS In this appendix we calculate thermal fronts using the simple continuum model of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2 with prescribed functions for Λ and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Noting the bistability at low τ, we adopt a lo- gistic reaction term and a linear diffusivity, which in suitable units take the form Λ = (E − EC)(E − EI)(E − EH), k = αE, where EC < EI < EH are constant parameters denoting the cold, intermediate, and hot states (respectively), and α is an additional constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Both EC and EH are thermally stable, but EI is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The basins of attraction of EC and EH, however, are controlled by their proximity to EI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' These functional choices simplify the integrals in the nu- merator of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The integral of Λ becomes simply −(EC − EH)3(EC −2EI +EH)/12, and is negative when the interme- diate state is less than the arithmetic mean of the hot and cold states, EI < (EC + EH)/2, and positive otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' In other words, the front will tend to move into the cold state when the intermediate state is closer to the cold state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' when its basin of attraction is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Similarly, the front will tend to move into the hot state when EI is closer to EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' If the three thermal states are equidistant and k is a 101 102 Box size 100 101 C Bridges laws vcrit=20, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 vcrit=20, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 vcrit=1, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 vcrit=1, = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 101 102 Box size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 2 C = 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Convergence tests in box size for several set-ups spanning dilute and cold, dense and hot, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' constant, then c = 0 and the front profile can be expressed in terms of elliptic integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The second term in (15) cannot be evaluated without knowledge of the front profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' It nonetheless simplifies to −α � ∞ −∞(dE/dξ)3dξ, which is clearly negative for monotonic front profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Thus the linear k law favours the front’s move- ment into the cold state, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Finally, we numerically solved (14) using a relaxation method (Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 1986), due to the problem’s charac- teristic stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' We plot a representative front solution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' 10, the front is sharp at the cold tran- sition, where k is small, and diffuse at the hot transition, where it is an order of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000 20 Larue, Latter, Rein 2 1 0 1 2 3 4 0 2 4 6 8 10 12 E Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' Illustrative front calculated numerically, with pa- rameters EC = 1, EI = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5, EH = 12, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' The front moves to the left into the cold state with a speed c = −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content='3728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} +page_content=' © 0000 RAS, MNRAS 000, 000–000' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQflwRz/content/2301.03289v1.pdf'} diff --git a/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/2301.11769v1.pdf.txt b/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/2301.11769v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3130c5888d84f3d8d4a61868306a548838fc3187 --- /dev/null +++ b/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/2301.11769v1.pdf.txt @@ -0,0 +1,1394 @@ +arXiv:2301.11769v1 [astro-ph.EP] 27 Jan 2023 +MNRAS 000, 1–13 (2023) +Preprint 30 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Formation of polar circumstellar discs in binary star systems +Jeremy L. Smallwood,1,2★ Rebecca G. Martin2 and Stephen H. Lubow3 +1Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan +2Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA +3Space Telescope Science Institute, Baltimore, MD 21218, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We investigate the flow of material from highly misaligned and polar circumbinary discs that feed the formation of circumstellar +discs around each binary component. With three-dimensional hydrodynamic simulations we consider equal mass binaries with +low eccentricity. We also simulate inclined test particles and highly-misaligned circumstellar discs around one binary component +for comparison.During Kozai-Lidov (KL) cycles, the circumstellar disc structure is altered through exchangesof disc eccentricity +with disc tilt. Highly inclined circumstellar discs and test particles around individual binary components can experience very +strong KL oscillations. The continuous accretion of highly misaligned material from the circumbinary disc allows the KL +oscillations of circumstellar discs to be long-lived. In this process, the circumbinary material is continuously delivered with a +high inclination to the lower inclination circumstellar discs. We find that the simulation resolution is important for modeling +the longevity of the KL oscillations. An initially polar circumbinary disc forms nearly polar, circumstellar discs that undergo +KL cycles. The gas steams accreting onto the polar circumstellar discs vary in tilt during each binary orbital period, which +determines how much material is accreted onto the discs. The long-lived KL cycles in polar circumstellar discs may lead to the +formation of polar S-type planets in binary star systems. +Key words: binaries: general – circumstellar matter– accretion, accretion discs +1 INTRODUCTION +The majority of stars born in dense stellar clusters are part +of binary star systems (Duquennoy & Mayor 1991; Ghez et al. +1993; Duchêne & Kraus 2013). The observed orbital eccentrici- +ties of binaries vary with orbital separation (Raghavan et al. 2010; +Tokovinin & Kiyaeva 2016). For tight binaries, the eccentricities are +small, which implies that there has been circularization of the bi- +nary orbit caused by stellar tidal dissipation (Zahn 1977). More +widely-separated binaries have observed eccentricities ranging from +푒b = 0.39 to 0.59, with a considerable number of highly eccentric +systems with 푒b > 0.8. The interactions of the binary with sur- +rounding gas may be responsible for the present-day observed binary +eccentricities (Goldreich & Tremaine 1980; Artymowicz et al. 1991; +Artymowicz 1992; Armitage & Natarajan 2005; Cuadra et al. 2009; +Roedig et al. 2011; Muñoz et al. 2019; Zrake et al. 2021). Circumbi- +nary discs of gas and dust are sometimes observed to be responsible +to be providing accreting material onto the binary (e.g., Alves et al. +2019). The gas flow dynamics from the circumbinary disc onto the +binary components has significant implications for planet formation +scenarios in binary systems. +Circumbinary discs are commonly observed to be moderately to +highly misaligned to the binary orbital plane. For example, the pre- +main sequence binary KH 15D has a circumbinary disc inclined +by 5 − 16◦ (Chiang & Murray-Clay 2004; Smallwood et al. 2019; +Poon et al. 2021). The radial extent of the disc is narrow and pre- +★ E-mail: jlsmallwood@asiaa.sinica.edu.tw +sumed to be rigidly precessing to explain the unique periodic light +curve. A ∼ 60◦ inclined circumbinary disc is found around the +main-sequence binary IRS 43 (Brinch et al. 2016), along with mis- +aligned circumstellar discs around each binary component. There is +an observed misalignment of about 70◦ between the circumbinary +disc and the circumprimary disc in HD 142527 (Marino et al. 2015; +Owen & Lai 2017). Another young binary, HD 98800 BaBb, has the +only observed polar (inclined by ∼ 90◦) gaseous circumbinary disc +(Kennedy et al. 2019). The 6–10 Gyr old binary system, 99 Herculis, +has a nearly polar (about 87◦) debris ring (Kennedy et al. 2012; +Smallwood et al. 2020). Apart from binaries, stars may also form +in higher-order systems (Tokovinin 2014a,b). The circumtriple disc +around the hierarchical triple star system, GW Ori, is tilted by about +38◦ (Bi et al. 2020; Kraus et al. 2020; Smallwood et al. 2021a). +The observations of inclined circumbinary discs have implications +on planet formation models. Observations from space and ground- +based telescopes reveal that ∼ 50 per cent of the confirmed exoplan- +ets reside in binary systems (Horch et al. 2014; Deacon et al. 2016; +Ziegler et al. 2018). For example, the binary system 훾 Cep AB hosts a +giant planet around the primary star, 훾 Cep Ab (Hatzes et al. 2003). It +is crucial to study the structure and evolution of protoplanetary discs +since these are the sites for planet formation (D’Angelo & Lissauer +2018). A forming planet’s orbital properties are directly related to +the orientation of the protoplanetary disc. For example, the observed +young binary system XZ Tau shows both the circumprimary and +circumsecondary discs are misaligned to the binary orbital plane +(Ichikawa et al. 2021). The binary system HD 142527 shows the +presence of a misaligned inner disc around one of the stellar com- +© 2023 The Authors + +2 +Smallwood et al. +ponents, presumably fed from the circumbinary disc (Price et al. +2018b). Furthermore, IRAS 04158+2805 is a binary system where +the two circumstellar discs and the circumbinary discs have been +observed to be misaligned (Ragusa et al. 2021). Therefore, highly- +inclined circumstellar discs may give birth to planets on highly-tilted +orbits. +Due to viscous dissipation, a misaligned circumbinary disc un- +dergoes nodal precession and evolves towards either a coplanar or +polar alignment. For an initially low-inclination circumbinary disc, +the disc precesses about the angular momentum vector of the bi- +nary and eventually evolves to be coplanar to the binary orbital +plane (Facchini et al. 2013; Foucart & Lai 2014). Slightly misaligned +discs around an eccentric binary undergo tilt oscillations as they +align, due to the nonaxisymmetric potential produced by the ec- +centric binary (Smallwood et al. 2019, 2020). For highly inclined +discs around eccentric orbit binaries, the angular momentum vec- +tor of the disc precesses about the eccentricity vector of the bi- +nary (e.g. Aly et al. 2015), which leads the disc to align perpen- +dicular (i.e., polar) to the binary orbital plane (Martin & Lubow +2017; Lubow & Martin 2018; Zanazzi & Lai 2018; Martin & Lubow +2018; Cuello & Giuppone 2019). A massive circumbinary disc that +is undergoing polar alignment aligns to a generalized polar state +which is less than 90◦ (Zanazzi & Lai 2018; Martin & Lubow 2019; +Chen et al. 2019). +Circumbinary gas discs contain a central cavity around the bi- +nary where little material is present. The cavity size is determined +by where the tidal torque is balanced with the viscous torque +(Artymowicz & Lubow 1994; Lubow et al. 2015; Miranda & Lai +2015; Franchini et al. 2019b; Hirsh et al. 2020; Ragusa et al. 2020). +The strength of the binary torque on the disc is dependent on +the tilt of the circumbinary disc and binary eccentricity. The +tidal torque at a given radius is zero when the circumbinary +disc is polar and the binary eccentricity approaches 푒b += 1 +(Lubow & Martin 2018) or if the disc is retrograde (e.g., Nixon et al. +2013). In the simplest models, the production of an outward +forcing torque by the binary can prevent circumbinary material +from flowing through the cavity (Lynden-Bell & Pringle 1974; +Pringle 1991). However, material from the circumbinary disc flows +through the binary cavity in the form of gaseous streams (e.g. +Artymowicz & Lubow 1996; Günther & Kley 2002; Nixon & King +2012; Shi et al. 2012; D’Orazio et al. 2013; Farris et al. 2014; +Muñoz et al. 2019; Alves et al. 2019). These streams are respon- +sible for forming and replenishing circumstellar discs around each +binary component. The accretion of material onto the circumstellar +discs may aid in the formation of 푆–type planets, those that orbit one +component of a binary. Accretion of material onto the central binary +may be suppressed for small disc aspect ratios. +The structure of a circumstellar disc around one star is +strongly affected by the tidal field of the binary compan- +ion +(Papaloizou & Pringle +1977; +Artymowicz & Lubow +1994; +Pichardo et al. 2005; Jang-Condell 2015). Circumstellar discs around +each binary component undergo tidal truncation. A circumstellar disc +in a circular orbit binary is typically truncated to about one-third to +one-half of the binary orbital separation The tidal truncation radius +is expected to decrease with increasing binary eccentricity. +Kozai-Lidov (KL) oscillations (Kozai 1962; Lidov 1962) have +been studied extensively to analyze several astronomical pro- +cesses involving bodies that orbit a member of a binary sys- +tem that begin on highly misaligned orbits. During KL os- +cillations, the object’s inclination is exchanged for eccentricity, +and vice versa. These processes include asteroids and irregular +satellites (Kozai 1962; Nesvorný et al. 2003), artificial satellites +(Lidov 1962), tidal disruption events (Chen et al. 2011), forma- +tion of Type Ia supernovae (Kushnir et al. 2013), triple star systems +(Eggleton & Kiseleva-Eggleton 2001; Fabrycky & Tremaine 2007), +planet formation with inclined stellar companions (Wu & Murray +2003; Takeda & Rasio 2005), giant outbursts in Be/X-ray bi- +naries (Martin et al. 2014a; Martin & Franchini 2019), inclined +planetary companions (Nagasawa et al. 2008), mergers of bina- +ries in galactic nuclei (Blaes et al. 2002; Antonini & Perets 2012; +Hamers et al. 2018; Hoang et al. 2018; Fragione et al. 2019a,b), stel- +lar compact objects (Thompson 2011), and blue straggler stars +(Perets & Fabrycky 2009). +A highly misaligned initially circular disc around one compo- +nent of a binary undergoes KL cycles in which its inclination is +exchanged for eccentricity, and vice versa (Martin et al. 2014a). Due +to disc dissipation by viscosity and shocks, these oscillations are +typically significantly damped after a few oscillations. KL oscilla- +tions can occur in a fluid disc with a wide variety of disc and binary +parameters (Fu et al. 2015a). When the disc becomes eccentric, it +overflows its Roche lobe and transfers material to the companion +star (Franchini et al. 2019a). Self-gravity of a disc can suppress disc +KL oscillations if the disc is close to being gravitationally unstable +(Fu et al. 2015b). KL oscillations in a circumstellar disc may have +significant consequences for planet formation since strong shocks in +the gas are produced during high eccentricity phases (Fu et al. 2017). +A misaligned circumbinary disc may form misaligned circumstel- +lar discs around the individual binary components (e.g., Nixon et al. +2013; Smallwood et al. 2021b). A highly misaligned disc around +one component of a binary may be unstable to the Kozai-Lidov (KL) +mechanism (Martin et al. 2014a). Smallwood et al. (2021b) simu- +lated the flow of gas originating from an initially misaligned cir- +cumbinary disc by 60◦. The misaligned gas streams that flow into the +binary cavity result in formation of highly tilted circumstellar discs +around each binary component. The inclined circumstellar discs in +turn undergo KL oscillations. However, the KL oscillations are long- +lived, due to the continuous accretion of inclined material from the +circumbinary disc. Long-lived KL cycles have important implica- +tions for planet formation in binary systems. +In this work, we extend the previous study Smallwood et al. +(2021b) and consider more highly inclined circumbinary discs. We +first revisit the dynamics of highly inclined test particle orbits around +one component of a binary in Section 2. In Section 3, we describe +the setup for our hydrodynamical simulations. In Section 4, we dis- +cuss the results of our circumprimary disc simulations. We simulate +a highly inclined circumprimary disc in a binary to explore the dy- +namics of the KL cycles. Previous studies have only dealt with cir- +cumprimary disc inclinations ≲ 60◦, while we consider higher tilts, +including a polar circumprimary disc. In Section 5, we show the re- +sults of our hydrodynamical simulations with an initial circumbinary +disc, where we consider the flow of material from discs with various +initial misalignments, including a polar circumbinary disc. Finally, a +summary is given in Section 6. +2 KOZAI-LIDOV OSCILLATIONS OF TEST PARTICLES +Before considering discs, we consider the properties of test parti- +cle orbits that undergo KL oscillations. As a consequence of the +conservation of the component of the angular momentum that is +perpendicular to the binary orbital plane, the test particle’s inclina- +tion is recurrently exchanged for eccentricity. This conservation is +MNRAS 000, 1–13 (2023) + +Formation of polar circumstellar discs +3 +Figure 1. Eccentricity (upper panel) and inclination (lower panel) evolution +of circumprimary test particles under the influence of a circular binary for +initially circular orbit particles. We vary the initial particle orbital tilt, 푖0, +beginning with 30◦ (black), 45◦ (blue), 60◦ (red), 75◦ (green), 80◦ (yellow), +85◦ (purple), and 90◦ (pink). The initial orbital radius of the particle is set at +푟0 = 0.06푎, where 푎 is the separation of the binary. The time is in units of +binary orbital period 푃orb. +expressed as +� +1 − 푒2p cos 푖p ≈ const, +(1) +where 푖p is the particle inclination with respect to the binary orbital +plane and 푒p is the eccentricity of the test particle. A initially circular +orbit particle initially gains eccentricity while reducing its orbital tilt +(i.e. going towards alignment which means higher values of | cos 푖p|) +and then circularizes while gaining orbital tilt back to its original +inclination. For an initially circular orbit particle, KL oscillations +only occur if the initial tilt of the test particle 푖p0 satisfies cos2 푖p0 < +cos2 푖cr = 3/5 (Innanen et al. 1997), which requires that 39◦ ≲ 푖p0 ≲ +141◦. From Eq. (1), an initially circular particle orbit can achieve a +maximum eccentricity given by +푒max = +� +1 − 5 +3 cos2 푖p0. +(2) +The increase in a circular particle’s eccentricity can be quite signif- +icant. For example, if the particle’s initial orbit is tilted by 60◦, the +maximum eccentricity reached during a KL cycle is about 0.75. +For eccentric binaries, stronger effects from KL oscillations +have been found to exist (Ford et al. 2000; Lithwick & Naoz 2011; +Naoz et al. 2011, 2013a,b; Teyssandier et al. 2013; Li et al. 2014; +Liu et al. 2015). The KL oscillation period for a particle in the po- +tential of an eccentric binary is approximately given by +휏KL +푃b +≈ 푀1 + 푀2 +푀2 +푃b +푃 (1 − 푒2 +b)3/2 +(3) +(Holman et al. 1997; Innanen et al. 1997; Kiseleva et al. 1998), +where 푀1 and 푀2 are the masses of the primary and secondary +components of the binary, respectively, 푃 = 2휋/ +� +퐺푀1/푎3p is the +orbital period of the particle with semimajor axis 푎p, 푃b = 2휋/Ωb +is the orbital period of the binary, 푒b is the binary eccentricity, and +Ωb = +� +퐺(푀1 + 푀2)/푎3 +b is the binary orbital frequency for binary +semimajor axis 푎b. +To simulate an inclined circumprimary test particle in a binary, +we use the 푁–body integrator, MERCURY (Chambers 1999). The +test particle is orbiting the primary companion with an initial tilt +푖0 relative to the binary orbital plane. The binary components have +equal mass so that 푀1 = 푀2 = 푀/2, where 푀 is the total mass +of the binary. Fu et al. (2015b) ran numerous test particle orbits +showing the effects the particle and binary parameters have on the +induced KL oscillations. Following their work, we model an eccentric +inclined particle around one component of an eccentric binary, more +applicable to binary systems. +We first simulate an inclined particle in a circular binary to match +previous results. Fig. 1 shows the eccentricity and inclination of a +circumprimary particle as a function of time that begins on a cir- +cular orbit. The analytic solution for these test particle orbits in the +quadrupole approximation is given in Lubow (2021). We consider +various initial tilts of the test particle orbit. The critical inclination +that the test particle orbit must have to induce KL cycles is ∼ 39◦. +Thus, a particle tilt of 30◦ (black line) does not undergo KL oscil- +lations. As the initial inclination of the particle increases, the KL +oscillations become more frequent, and the growth in the eccentric- +ity becomes more prominent (in agreement with Fig. 1 in Fu et al. +(2015b)). The trough in the inclination profile of a test particle be- +comes narrower with initial inclination. An initial particle orbit tilt +of 90◦ becomes unstable and collides with the primary star during +the first KL oscillation because the particles eccentricity exceeds +1.0. The eccentricity of the polar particle increases almost up to its +maximum eccentricity before the tilt begins to change. +Next, we set the initial particle tilt to 60◦ around a slightly eccen- +tric binary with 푒b = 0.1, as we will consider in the disc simulations. +We model various initial test particle eccentricities ranging from 0.0 +to 0.5. Figure 2 shows the eccentricity and inclination of eccentric +circumprimary particles as a function of time in binary orbital pe- +riods. An inclined circular test particle within an eccentric binary +has an increased frequency in KL oscillations when compared to a +particle orbiting one component of a circular binary, as expected by +equation (3). From Figure 2, when the particle eccentricity is in- +creased, the maximum eccentricity reached during a KL oscillation +also increases. However, the difference between the initial eccentric- +ity to the maximum eccentricity of the particle decreases as the initial +particle eccentricity increases. +Lastly, we examine the KL mechanism for a nearly polar particle. +From Fig. 1, an initially circular orbit particle with an initial orbital +tilt of 85◦ is unstable to KL oscillations but is otherwise stable. +We consider a nearly polar orbit particle with an initial orbital tilt +푖0 = 85◦ around a binary with eccentricity 푒b = 0.1. In Fig. 3 we +show the particle eccentricity and inclination as a function of time +in binary orbital periods. The various lines correspond to different +initial particle eccentricities ranging from 0.0 to 0.5. For all values +of the initial particle eccentricity we consider, the particle proceeds +through KL cycles in a periodic fashion. Unlike the particle beginning +at a tilt of 60◦, a nearly polar particle exhibits similar maximum +eccentricity close to unity during a KL oscillation regardless of initial +particle eccentricity. The minimum inclination reached during each +KL oscillation is roughly independent of particle initial eccentricity. +MNRAS 000, 1–13 (2023) + +100.8 +0.6 +0.4 +0.2 +0 +80 +60 +40 +0 +100 +200 +300 +400 +50 +t/Porb4 +Smallwood et al. +Figure 2. Eccentricity (upper panel) and inclination (lower panel) evolution +of circumprimary test particles under the influence of binary with eccentricity +푒b = 0.1. The initial tilt of the particle orbit is set to 60◦. We vary the initial +particle eccentricity 푒0 beginning with 푒0 = 0 (black), 0.1 (blue), 0.2 (red), +0.3 (green), 0.4 (yellow), 0.5 (purple). The initial orbital radius of the particle +is set at 푟0 = 0.06푎, where 푎 is the separation of the binary. The time is in +units of binary orbital period 푃orb. +3 HYDRODYNAMICAL-SIMULATION SETUP +We use the smoothed particle hydrodynamics (SPH) code phantom +(Price et al. 2018a) to model gaseous circumbinary and circumstellar +discs. phantom has been tested extensively for modeling misaligned +circumbinary discs (Nixon 2012; Nixon et al. 2013; Nixon & Lubow +2015; Facchini et al. 2018; Smallwood et al. 2019; Poblete et al. +2019; Smallwood et al. 2020; Aly & Lodato 2020; Hirsh et al. 2020; +Smallwood et al. 2021b), as well as misaligned circumstellar discs +around individual binary components (e.g. Martin et al. 2014b; +Doğan et al. 2015; Franchini et al. 2020). The suite of simulations +is summarised in Table 1. In this section we describe the setup for +the binary star, circumprimary disc, and circumbinary disc in further +detail. +3.1 Binary star setup +We model the binary star system as a pair of sink particles, with an +initial binary separation 푎. The binary is not static but rather evolves +freely in time. Each sink particle is given an initial mass with 푀1 +being the primary mass and 푀2 being the secondary mass. The total +binary mass is thereby 푀 = 푀1 + 푀2. All of our simulations assume +an equal-mass binary (푀1 = 푀2). In Cartesian coordinates, the orbit +of the binary lies in the 푥-푦 plane initially. The binary begins ini- +tially at apastron along the 푥-axis. The massive sink particles have +a hard accretion boundary, meaning that when particles penetrate +the sink accretion radius, the particle’s mass and angular momentum +are deposited onto the star (e.g., Bate et al. 1995). A large accretion +radius is often used to reduce the computation time significantly by +neglecting to resolve close-in particle orbits. In this work, however, +we are interested in resolving the formation and evolution of the cir- +Figure 3. Same as Fig. 2 but for nearly polar test particles with an initial +orbital tilt 푖0 = 85◦. +cumstellar material. Therefore, we adopt a relatively small accretion +radius of 0.05푎 for simulations that begin with a circumbinary disc +and an accretion radius of 0.025푎 for simulations that begin with +a circumprimary disc. Using a smaller accretion radius for the cir- +cumprimary disc simulations ensures that the disc lifetime is longer, +along with higher disc resolution. The more eccentric the binary, +the smaller the outer truncation radius for the circumstellar discs +(Artymowicz & Lubow 1994). Having a small binary eccentricity +helps with the resolution of the circumstellar discs. On the other +hand, to have a stable polar circumbinary disc, the binary eccentric- +ity needs to be a non-zero value. The initial binary eccentricity is set +to 푒b = 0.1, with the binary eccentricity vector along the positive +푥–axis. With this value of binary eccentricity, the critical tilt of the +circumbinary disc to remain nearly polar is ∼ 77◦ (see eq. 33 in +Martin & Lubow 2019). +3.2 Circumprimary disc setup +To model a circumprimary disc, we follow the methods of +Martin et al. (2014b). Runs 1-5 in Table 1 simulate initially a circum- +primary disc. The inner and outer disc radii are set at 푟in = 0.025푎 +and 푟out = 0.25푎, respectively, with a initial total disc mass +푀CPD = 10−3푀. The circumprimary disc consists of 750, 000 equal- +mass Lagrangian particles. We neglect any effects of self-gravity. The +disc surface density profile is initially a power law distribution given +by +Σ(푟) = Σ0 +� 푟 +푟in +�−푝 +, +(4) +where we set 푝 = 3/2. We adopt a locally isothermal disc with +sound speed 푐s ∝ 푅−3/4, 퐻/푟 = 0.035 at 푟 = 푟in, and 퐻/푟 = 0.02 +at 푟 = 푟out. With this prescription, the viscosity parameter 훼 and +⟨ℎ⟩/퐻 are effectively constant over the radial extend of the disc +(Lodato & Pringle 2007). For the circumprimary disc simulations, +we take the Shakura & Sunyaev (1973) 훼 parameter to be 0.01. To +MNRAS 000, 1–13 (2023) + +0.8 +0.6 +0.4 +0.2 +0 +90 +80 +70 +60 +50 +40 +30 +0 +20 +40 +60 +80 +10 +t/Porb10.8 +0.6 +0.4 +0.2 +0 +60 +50 +40 +30 +0 +20 +40 +60 +80 +10 +t/Porb0Formation of polar circumstellar discs +5 +Table 1. The setup of the SPH simulations that includes an initial circumprimary disc (CPD) or circumbinary disc (CBD). The table lists the initial parameters +beginning with the disc tilt 푖0, inner disc radius 푟in, outer disc radius 푟out, 훼 viscosity parameter, disc aspect ratio at inner disc radius 퐻/푟in, disc aspect ratio at +outer disc radius 퐻/푟out, the number of particles, and whether or not the circumstellar discs undergo the Kozai-Lidov (KL) instability. +Model +Disc Setup +푖0/◦ +푟in/푎 +푟out/푎 +훼 +퐻/푟in +퐻/푟out +# Particles +KL unstable? +run1 +CPD +60 +0.025 +0.25 +0.01 +0.035 +0.02 +750, 000 +Yes +run2 +CPD +70 +0.025 +0.25 +0.01 +0.035 +0.02 +750, 000 +Yes +run3 +CPD +80 +0.025 +0.25 +0.01 +0.035 +0.02 +750, 000 +Yes +run4 +CPD +90 +0.025 +0.25 +0.01 +0.035 +0.02 +750, 000 +Yes +run5 +CPD +100 +0.025 +0.25 +0.01 +0.035 +0.02 +750, 000 +Yes +run6∗ +CBD +60 +1.6 +2.6 +0.1 +0.1 +0.088 +1.5 × 106 +Yes +run7 +CBD +60 +1.6 +2.6 +0.1 +0.1 +0.088 +750, 000 +Yes +run8 +CBD +90 +1.6 +2.6 +0.1 +0.1 +0.088 +1.5 × 106 +Yes +∗ Simulation from Smallwood et al. (2021b) +accomplish this, the SPH artificial viscosity coefficients are set as +훼AV = 0.18 and 훽AV = 2.0. The disc is resolved with shell-averaged +smoothing length per scale height ⟨ℎ⟩/퐻 ≈ 0.55. +3.3 Circumbinary disc setup +To model an initially flat but tilted gaseous circumbinary disc, we +follow the methods of Smallwood et al. (2021b). Runs 6, 7, and 8 +in Table 1 describe the simulations of a circumbinary disc. The disc +initially consists of 1.5 × 106 equal-mass Lagrangian SPH particles. +We also model a 750, 000 particle simulation for a resolution study. +The simulations run for 45 푃orb, where 푃orb is the orbital period of +the binary. This is sufficient time for the forming circumstellar discs +to reach a quasi-steady state. We simulate initially highly misaligned +disc inclinations of 푖0 = 60◦, 90◦. A disc with 푖0 = 90◦ is in a polar +configuration, where the angular momentum vector of the disc is +aligned to the eccentricity vector of the binary. At the beginning +of our simulations, we select an initial inner disc radius, 푟in, and +outer disc radius, 푟out, where the initial total disc mass, 푀CBD, is +confined. All of the simulations model a low-mass circumbinary disc +such that 푀CBD = 10−3푀. We choose the circumbinary disc to be +radially very narrow and close to the binary orbit. This is done to +maximise the accretion rate onto the binary and hence the resolution +of the circumstellar discs (e.g., Smallwood et al. 2021b). For our +simulations, we take 푟in = 1.6푎 and 푟out = 2.6푎. The tidal torque +is weaker at a given radius for a more highly misaligned disc which +allows the inner disc radius to lie closer to the binary than a coplanar +disc (e.g., Lubow et al. 2015; Miranda & Lai 2015; Lubow & Martin +2018). The inner truncation radius of a polar circumbinary disc is +around 1.6 푎 (Franchini et al. 2019b), much smaller than the 2 − 3 푎 +expected for coplanar discs (Artymowicz & Lubow 1994). +The disc surface density profile follows from Equation (4). The +physical disc viscosity is incorporated by using artificial viscosity +훼av, which is detailed in Lodato & Price (2010). By using our sur- +face density profile and a disc aspect ratio 퐻/푟 = 0.1 at 푟in, the +shell-averaged smoothing length per scale height ⟨ℎ⟩/퐻 and the disc +viscosity parameter 훼 are constant over the radial extent of the disc +(Lodato & Pringle 2007). The circumbinary disc is initially resolved +with ⟨ℎ⟩/퐻 ≈ 0.11. The parameters for the simulations require a +high viscosity in order to maximize the accretion rate on to the +circumstellar discs and provide better resolution. We consider a +relatively high value for the Shakura & Sunyaev (1973) 훼SS of 0.1. +In a more realistic system, the disc viscosity may be lower. +In order to more accurately simulate the formation and develop- +ment of circumstellar discs, we adopt the locally isothermal equation +of state of Farris et al. (2014) and set the sound speed 푐s to be +푐s = F 푐s0 +� +푎 +푀1 + 푀2 +�푞 � 푀1 +푟1 ++ 푀2 +푟2 +�푞 +, +(5) +where 푟1 and 푟2 are the radial distances from the primary and sec- +ondary stars, respectively, and 푐s0 is a constant with dimensions of +velocity. 푞 is set to 3/4. F is a dimensionless function of position +that we define below. This sound speed prescription guarantees that +the temperature profiles in the circumprimary and circumsecondary +discs are set by the primary and secondary stars, respectively. For +푟1, 푟2 ≫ 푎, 푐s is set by the distance from the binary centre of mass. +To increase the resolution of the circumstellar discs, we include a +function F in Equation (5) as detailed in Smallwood et al. (2021b). +The purpose of F is to modify the sound speed around each binary +component so that the viscous timescale is longer. This increases +the mass (and hence the resolution) in the steady-state circumstellar +discs. We take +F = +�√ +0.001, +if 푟1 or 푟2 < 푟c, +1, +otherwise, +(6) +where 푟c is the cutoff radius. We set a cutoff radius of 푟c = 0.35푎 +from each binary component (e.g., Smallwood et al. 2021b). Using +the prescription mentioned above ensures that the disc aspect ratio +of the circumstellar discs at radius 푟 = 0.1푎 is 퐻/푟 ∼ 0.01, which +is one-tenth of the disc aspect ratio at the initial inner circumbinary +disc radius. +3.4 Analysis routine +We analyse the disc and binary parameters as a function of time. The +parameters include tilt, eccentricity, the longitude of the ascending +node, mass, and mass accretion rate. To probe the circumprimary +disc simulations, we average over particles in the radial range from +0.025푎 to a distance of 0.30푎. For the circumbinary disc simulations, +we average over particles in the radial range from 1.4푎 to a distance +of 10푎. For the forming circumstellar discs, we average over all +particles bound to each binary component (i.e., the specific energies, +kinetic plus potential, of the particles are negative, neglecting the +thermal energy). The tilt, 푖, is defined as the angle between the initial +angular momentum vector of the binary (the 푧-axis) and the angular +momentum vector of the disc. The longitude of the ascending node, +휙, is measured relative to the 푥-axis (the initial binary eccentricity +vector). +MNRAS 000, 1–13 (2023) + +6 +Smallwood et al. +20 +40 +60 +80 +100 +120 +0 +0.5 +1 +-100 +0 +100 +0 +0.5 +1 +0 +5 +10 +15 +10-5 +2 +3 +4 +5 +Panel 1 +Figure 4. Evolution of a KL unstable circumprimary disc as a function of time +in units of the binary orbital period 푃orb. We simulate five different initial disc +inclinations, which are 60◦ (run1 from Table 1, black), 70◦ (run2, blue), 80◦ +(run3, red), 90◦ (run4, green), and 100◦ (run5, yellow). The disc parameters +are tilt 푖 (panel 1), eccentricity 푒 (panel 2), longitude of the ascending node +휙 (panel 3), and disc mass 푀d (panel 4). The mass accretion rate �푀 onto the +primary star is shown in panel 5. +4 HYDRODYNAMICAL RESULTS WITH A +CIRCUMPRIMARY DISC +This section considers the evolution of a circumprimary disc in the +absence of accretion from a circumbinary disc. This enables us to +disentangle the effect of accretion onto the circumstellar discs. We +focus on large circumprimary disc misalignments in an eccentric +binary star system. We consider five different initial disc tilts, 60◦ +(run1 from Table 1), 70◦ (run2), 80◦ (run3), 90◦ (run4), and 100◦ +(run5). Figure 4 shows the disc tilt, eccentricity, the longitude of +the ascending node, the mass of the circumprimary disc, and the +accretion rate onto the primary star as a function of time in binary +orbital periods. The disc exhibits KL cycles for each initial tilt, where +the disc eccentricity and inclination are exchanged. For a disc with +an initial tilt of 60◦, Martin et al. (2014a) found that the first KL +oscillation occurred around 10 Porb for a circular binary. In our case, +the disc with the same initial tilt undergoes the first KL oscillation +much sooner due to the binary having a slightly eccentric orbit (see +Fig.12 in Fu et al. 2015a). Due to viscous dissipation and the lack +of circumbinary material, the KL oscillations damp quickly in time. +For higher initial inclinations, 70◦, 88◦, 90◦ and 100◦, the discs do +not survive after one KL oscillation for our given sink size. The +x-z plane t = 0 Porb +0.2a +y-z plane t = 0 Porb +0.2a +x-z plane t = 10 Porb +0.2a +y-z plane t = 10 Porb +0.2a +x-z plane t = 15 Porb +0.2a +y-z plane t = 15 Porb +0.2a +Figure 5. The evolution of polar circumprimary disc (run4 from Table 1). +The white circles denote the eccentric orbit binary components with an initial +binary separation of 푎. The top row shows the initial disc setup. The middle +and bottom rows show the disc evolution at 푡 = 10 푃orb and 푡 = 15 푃orb, +respectively, where 푃orb is the binary orbital period. The color denotes the +gas surface density, with the orange regions being about three orders of mag- +nitude larger than the purple regions. The left column shows the 푥–푧 plane, +and the right column shows the 푦–푧 plane. At 푡 = 10 푃orb, the circumpri- +mary disc is highly eccentric due to the Kozai-Lidov instability. Also, at this +time, a circumsecondary disc is being formed from material flowing close to +the secondary binary component from the eccentric circumprimary disc. At +푡 = 15 푃orb, the circumprimary disc has completely dissipated from being +accreted onto the primary star and transferring material to the secondary star. +At this time, there is more material in the newly formed circumsecondary +disc. +discs become very eccentric, which leads to the majority of the disc +material being accreted by the primary star. Increasing the resolution +of these simulations does not lengthen the disc lifetime. However, +if we were to use a smaller sink size, then the disc could survive +through the KL oscillations. A smaller sink size would ensure that +a larger portion of the disc could survive. An accretion radius of +∼ 0.01 au is comparable to the size of the star, but we simulate +a larger sink size for computational reasons and to compare with +the circumbinary disc simulations detailed in the next Section. The +initially polar disc’s tilt does not change much from polar before the +majority of the disc is accreted. This is likely a consequence of the +high disc eccentricities that are developed which is consistent with +the results for test particle orbits (see Fig. 1). In the retrograde case, +MNRAS 000, 1–13 (2023) + +Formation of polar circumstellar discs +7 +60 +70 +80 +0 +0.2 +0.4 +0.6 +-100 +0 +100 +10-6 +10-4 +10 +20 +30 +40 +10-6 +10-4 +2 +3 +4 +5 +Panel 1 +Figure 6. Resolution study for a circumbinary disc that is initially misaligned +by 60◦. The blue curves represent the simulation with initially 1.5 × 106 +particles in the circumbinary disc, while the red curves denotes the simulation +with initially 750, 000 particles. The first four panels show the disc parameters +for the newly forming circumprimary disc as a function of time in units of the +binary orbital period, 푃orb. The disc parameters are tilt 푖 (panel 1), eccentricity +푒 (panel 2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d +(panel 4). The black dotted curve in the third panel denotes the circumbinary +disc. The lower panel shows the mass accretion rate onto the primary star +�푀pri (panel 5). +푖0 = 100◦, as the disc eccentricity increases, the inclination also +increases, opposite to the prograde cases. +Highly inclined particle orbits experience a large (nearly 180◦) +shift in 휙 within a small time interval centered about the eccentricity +maximum (see the plot for Ω(푡) in Figure 1 of Lubow (2021)). This +large shift does not appear in Figure 4 or in any of our other phase +results. We are not sure why this is the case. Perhaps the disc is +unable to respond to such a large shift within a short time. +We further examine the evolution of the polar (푖0 = 90◦) circumpri- +mary disc. In Fig. 5, we show the polar circumprimary disc structure +at three different times, 푡 = 0 푃orb, 10 푃orb, and 15 푃orb. Initially, the +polar disc around the primary star (left white dot) is edge-on in the +푥-푧 plane and face-on 푦-푧 plane. At 푡 = 10 푃orb, the disc is at peak ec- +centricity growth from the KL instability. Also, at this time, streams +of material from the circumprimary disc flow around the secondary +star (right white dot) and begin forming a circumsecondary disc. At +푡 = 15 푃orb, the circumprimary disc has dissipated due to accretion +onto the primary star and transporting material to the circumsec- +ondary disc. The newly formed circumsecondary disc is at a lower +tilt, below the threshold, to induce the KL cycles. +5 HYDRODYNAMICAL RESULTS WITH A +CIRCUMBINARY DISC +In this section we examine how misaligned and polar circumbinary +material flows through the binary cavity and forms circumstellar +discs around each binary component. We first conduct a resolution +study of our earlier work from Smallwood et al. (2021b), modeling +an initially 60◦ misaligned circumbinary disc. We then focus on the +polar circumbinary disc case. +5.1 Resolution Study +We examine a circumbinary disc with an initial misalignment of +푖0 = 60◦ with two different initial numbers of particles, 1.5 × 106 +(run6) and 750, 000 (run7). The upper four panels in Figure 6 show +the circumprimary disc parameters as a function of time. The bot- +tom panel shows the mass accretion rate onto the primary star. The +blue curves represent the 1.5 × 106 particle simulation, while the red +curves represent the 750, 000 particle simulation. Panels 1 and 2 show +the evolution of disc eccentricity and inclination where the forming +circumprimary disc undergoes KL oscillations from the continuous +accretion of material from the circumbinary disc. The oscillations +damp in time at both resolutions, with the lower resolution simu- +lation damping more quickly. Therefore, the oscillations are likely +limited by resolution. If the accretion timescale is long compared +to the KL timescale, we expect the KL oscillations to damp over +time, similar to the circumprimary disc simulations without accre- +tion shown in the previous Section. If the accretion timescale is short +compared to the KL timescale, there should be no KL oscillations +present. In this case, the material moves through the disc faster than +it becomes unstable to KL oscillations. We expect the optimal oscil- +lations when the timescales are comparable because the disc refills +mass on the timescale that the oscillations take place. For the simula- +tion with a 60◦ tilted circumbinary disc, the accretion timescales for +the primary and secondary are ∼ 1.5 Porb, whereas the KL timescale +for this simulation is ∼ 5 Porb. The simulation is in the regime where +the accretion timescale is shorter than the KL oscillation timescale +because when the disc becomes eccentric during the KL oscillations, +a large amount of disc material is accreted, reducing the accretion +timescale. However, the accretion timescale is dependent on the disc +viscosity. In our hydrodynamical simulations, we use an artificial +viscosity to model an expected Shakura & Sunyaev (1973) viscos- +ity coefficient. The number of Lagrangian particles determines how +close the artificial viscosity is to the actual value. Thus, the 훼 is +artificially higher at lower resolutions, leading to a shorter accre- +tion timescale. For our higher-resolution simulation, the 훼 is lower, +leading to a longer accretion timescale. +Panel 3 in Fig. 6 shows the longitude of the ascending node as +a function of time. The precession rate of the circumprimary disc +is only slightly faster than the circumbinary disc on average. In the +absence of the effects of KL oscillations, the nodal precession rate +of the primary disc, assuming constant surface density Σ out to disc +radius 푟 from the primary, is given by +휔pr = − 15푀2푟3 +32푀1푎3 +b +cos (푖) Ω(푟), +(7) +where 푖 is inclination angle of the primary disc relative to the binary +orbital plane and Ω = +� +퐺푀1/푟3 is the angular velocity in the disc +MNRAS 000, 1–13 (2023) + +8 +Smallwood et al. +x-z plane t = 0 Porb +a +y-z plane t = 0 Porb +a +x-z plane t = 25 Porb +a +y-z plane t = 25 Porb +a +Figure 7. The formation of polar circumstellar discs from an initially low- +mass polar circumbinary disc (run8). The white circles denote the eccentric +orbit binary components with an initial binary separation of 푎. The upper +panels denote the initial disc setup, while the bottom panels show the disc +evolution at 푡 = 25 푃orb, where 푃orb is the binary orbital period. At this time, +nearly polar circumstellar discs are forming around each binary component. +The color denotes the gas density using a weighted density interpolation, +which gives a mass-weighted line of sight average. The yellow regions are +about three orders of magnitude larger than the purple. The left column shows +the 푥–푧 plane, and the right column shows the 푦–푧 plane. +(Larwood et al. 1996). With 푟 = 0.35 푎b, we find 휔pr = 6◦/푃orb +with a revolution period of ∼ 56 Porb. Therefore, the circumstellar +discs should have nodally precessed 75 per cent of a revolution in +45 Porb. In panel 3 we see that the circumstellar discs have only +completed roughly 30 per cent of a nodal revolution. It is possible +that the circumprimary phase is affected by the phase of accreted +gas from the circumbinary disc that undergoes relatively slow nodal +precession. As discussed in Section 4, KL oscillations modify the +nodal precession rate of a test particle in a way that we do not +see in the disc simulations. Lastly, the mass in the circumprimary +discs oscillates in time, with the troughs corresponding with each +high eccentricity period. During each high eccentricity phase, the +accretion rate peaks as seen in panel 5. +5.2 Polar discs +In this section, we present a hydrodynamical simulation of the flow of +material from a polar circumbinary disc onto the binary components +(run8). The top row of Fig. 7 shows the initial configuration of +the polar circumbinary disc around an eccentric binary. The bottom +row shows the disc structure at 푡 = 25 푃orb. The circumbinary disc +remains nearly polar (∼ 90◦) as shown in the 푥-푧 plane. Material flows +from the polar circumbinary disc and forms nearly polar circumstellar +discs around each binary component. The cavity size is smaller in +the polar disc compared to a coplanar disc simulation as expected +(Lubow et al. 2015; Miranda & Lai 2015). +The upper four panels in Fig. 8 show the inclination, eccentricity, +the longitude of the ascending node, and disc mass for the three +60 +80 +100 +120 +0 +0.2 +0.4 +0.6 +-100 +0 +100 +10-6 +10-4 +10-2 +10 +20 +30 +40 +10-4 +2 +3 +4 +5 +Panel 1 +Figure 8. Simulation results for run8 for an initially polar circumbinary disc. +The disc parameters are shown for the circumprimary, circumsecondary, and +circumbinary discs as a function of time in units of the binary orbital period, +푃orb. The upper four panels show the disc tilt 푖 (panel 1), eccentricity 푒 (panel +2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d (panel 4) +for the three discs. The lower panel shows the mass accretion rate onto the +sinks �푀 (panel 5). +discs as a function of time in binary orbital periods. The lower panel +shows the mass accretion rate onto the sinks. The circumstellar discs +form at a time of ∼ 10 Porb, later than in the simulation with a lower +level of circumbinary disc misalignment. The circumbinary disc tilt +evolves in time. Since we model a disc with a non-zero mass, it +will align to a generalised polar state with an inclination that is < +90◦ (e.g., Martin & Lubow 2019; Chen et al. 2019). In this case, the +circumstellar discs form slightly retrograde, with a tilt just above 90◦. +The primary and secondary discs form with an eccentricity of ∼ 0.25. +However, the polar circumstellar discs undergo the KL instability, +which forces the disc eccentricity and tilt to oscillate in time. Looking +at panels 1 and 2, we see that as the disc eccentricity increases, the +disc tilt also increases, the opposite of the conventional KL case +involving prograde orbits. However, this result is consistent with the +KL mechanism for retrograde orbits. Panel 3 shows the evolution of +the longitude of the ascending node in time. Since the circumprimary +and circumsecondary discs are nearly polar, they exhibit very little +precession (see equation 7 and discussion below it). The mass of the +polar circumstellar discs oscillates in time (panel 4), likely due to +the oscillating disc eccentricity. The polar circumbinary disc has lost +∼ 25 per cent of its initial mass. +MNRAS 000, 1–13 (2023) + +0Formation of polar circumstellar discs +9 +a +Figure 9. Edge-on view (푥–푧 plane) of a polar circumbinary disc (run8) at +a time 푡 = 5 푃orb. We ignore the main portions of the disc confined within +푟 < 0.45푎b, where 푎b is the separation of the binary. The binary components +are shown as the green dots. The colours denote the disc surface density, +with the orange regions being about three orders of magnitude larger than the +purple regions. We overlay the velocity vectors shown by the black arrows. +The length of the arrow is proportional to the velocities of the particles. We +see two asymmetric lobes of material that are produced by the binary. Several +of the velocity vectors are directed away from the plane of the circumbinary +disc; however, the material then falls back onto the disc gap. +The KL oscillations from Fig. 8 damp in time. However, from our +resolution study, the damping is primarily due to the initial number of +particles. The accretion timescale for this simulation is ∼ 15 Porb, and +the KL timescale in this case is ∼ 10 Porb. The accretion timescale +is longer in the polar simulation than in the 60◦ simulation because +the polar circumstellar discs become less eccentric during each KL +cycle, accreting less disc material. For a higher resolution, we expect +the KL oscillations to be long-lived even for polar circumstellar discs. +On the bottom-left panel in Fig. 7, we see that some material is +flung out of the disc plane on both sides of the polar circumbinary +disc. This material forms two lobes on both sides of the disc. Figure 9 +shows the edge-on view of the disc surface density, along with the +velocity vectors. The material is being flung outwards but remains +bound to the binary. Therefore, the material then falls back into the +gap region of the circumbinary disc. Throughout the simulation, the +material is periodically flung out every 0.5 푃orb when the binary +components pass through the polar circumbinary disc plane. +We further examine the flow of polar circumbinary material onto +the forming circumstellar discs. First, we investigate the tilt of the +gaseous streams that accrete onto the circumstellar discs as a function +of time. Figure 10 shows the circumbinary disc tilt as a function of +disc radius. The inner edge of the disc lies roughly at 1.6푎. The +curves that are shown at radii < 1.6푎 map the tilt of the streams. +We show the disc tilt for a full binary orbital period from 20 Porb +to 21 Porb in increments of 0.1 Porb. At every 0.5 Porb, the tilt of the +streams are low at ∼ 80◦. When the binary orbital period is not at +half increments, the tilt of the streams increases beyond 90◦. For +example, at times 20.2 − 20.3 Porb and 20.6 − 20.7 Porb, the streams +are highly tilted. Recall that the forming circumstellar discs initially +1 +1.5 +2 +2.5 +75 +80 +85 +90 +95 +100 +20.0 +20.1 +20.2 +20.3 +20.4 +20.5 +20.6 +20.7 +20.8 +20.9 +21.0 +Figure 10. Circumbinary disc tilt, 푖, as a function of radius, 푟, for the polar +circumbinary disc. The color corresponds to the time in binary orbital periods, +Porb. +form at a high disc tilt, > 90◦. Therefore, whenever the gaseous +streams are highly tilted, there is an increased accretion of material +onto the circumstellar discs from the circumbinary disc. When the +streams are less inclined, every 0.5 Porb, there will be less material +accreted onto the polar circumstellar discs. This phenomenon is also +consistent with Fig. 9, where material is flung out of the plane of the +circumbinary disc every 0.5 Porb. We test this by further visualizing +the inflow of material. Figure 11 shows snapshots of zoomed-in views +in the 푥–푧 and 푦–푧 planes of the disc surface density, showing the +gaseous streams accreting onto the nearly polar circumstellar discs. +The snapshots show the flow of material over 20 Porb to 20.9 Porb +in increments of 0.1 Porb. Higher density streams occur at times +20.3 Porb and 20.7 Porb. The flow of material decreases every 0.5 Porb +during the orbit. At these times, the steams are less dense, leading to +less material accreting onto the circumstellar discs. +We relate the flow of material from Fig. 11 to the mass of the +circumstellar discs. Figure 12 shows the mass of the circumprimary +disc from 20 Porb to 25 Porb folded on top of one another for each +orbital period. The vertical dashed-lines denote the times when the +binary is aligned with the circumbinary disc plane, which is assumed +when the stars are both aligned with 푥–푧 plane. Each time the binary +aligns to the plane of the disc, the masses of the circumstellar discs +increase. The mass of the disc decreases every 0.5 Porb. This be- +haviour repeats every orbital period. Overall, the disc mass deceases +in time due to the KL mechanism. +6 SUMMARY +In this work, we investigated the flow of material from a circumbi- +nary disc that results in the formation circumstellar discs around each +binary component. We simulated an initially highly misaligned and +polar circumbinary disc using three-dimensional SPH. We consid- +ered cases of low initial binary eccentricity (typically 푒b = 0.1) and +binary mass ratio of unity. We also simulated cases of test particles +MNRAS 000, 1–13 (2023) + +10 +Smallwood et al. +x-z plane t = 20.0 Porb +0.25a +y-z plane t = 20.0 Porb +0.25a +x-z plane t = 20.5 Porb +0.25a +y-z plane t = 20.5 Porb +0.25a +x-z plane t = 20.1 Porb +0.25a +y-z plane t = 20.1 Porb +0.25a +x-z plane t = 20.6 Porb +0.25a +y-z plane t = 20.6 Porb +0.25a +x-z plane t = 20.2 Porb +0.25a +y-z plane t = 20.2 Porb +0.25a +x-z plane t = 20.7 Porb +0.25a +y-z plane t = 20.7 Porb +0.25a +x-z plane t = 20.3 Porb +0.25a +y-z plane t = 20.3 Porb +0.25a +x-z plane t = 20.8 Porb +0.25a +y-z plane t = 20.8 Porb +0.25a +x-z plane t = 20.4 Porb +0.25a +y-z plane t = 20.4 Porb +0.25a +x-z plane t = 20.9 Porb +0.25a +y-z plane t = 20.9 Porb +0.25a +Figure 11. Zoomed-in snapshots of the disc surface density showing the flow of material from a polar circumbinary disc onto the nearly polar circumstellar +discs. The white circles denote the eccentric orbit binary components with an initial binary separation of 푎. The color denotes the gas density using a weighted +density interpolation, which gives a mass-weighted line of sight average. The yellow regions are about three orders of magnitude larger than the purple. We view +the orbit of the binary in the 푥–푧 and 푦–푧 planes. The snapshots show a period from 20 Porb to 20.9 Porb in increments of 0.1 Porb, where 푃orb is time in binary +orbital periods. +MNRAS 000, 1–13 (2023) + +00001100Formation of polar circumstellar discs +11 +0.2 +0.4 +0.6 +0.8 +2.5 +3 +3.5 +4 +4.5 +5 +10-6 +20-21 Porb +21-22 Porb +22-23 Porb +23-24 Porb +24-25 Porb +Figure 12. The circumprimary disc mass evolution during one binary orbital +period, Porb, at times 20 − 21 Porb (blue), 21 − 22 Porb (red), 22 − 23 Porb +(yellow), 23−24 Porb (purple), and 24−25 Porb (green). The mass of the disc +decreases every 0.5 Porb. The vertical dashed-lines denote the times when +the binary is aligned with the circumbinary disc plane during 20 − 21 Porb. +An increased flow of material onto the circumstellar discs occurs when the +binary is aligned with the circumbinary disc plane. +around the primary star and cases of circumprimary discs only (i.e., +no circumbinary or circumsecondary discs) for comparison. +In order to carry out these simulations in a reasonable amount of +time, we made some compromises on our choice of parameters. In +particular, we introduced a higher viscosity parameter for the cir- +cumbinary disc than is likely to occur and a lower temperature of +the gas in the gap region. These choices were made to improve the +resolution of the simulations. Even with these parameters, the reso- +lution is still playing a role in our results (see Fig. 6). While we have +chosen the disc parameters (훼 and 퐻/푅) in our simulations to max- +imise the accretion rate on to the binary components and therefore +the simulation resolution, we expect the general behaviour to persist +for more realistic parameters applicable to protoplanetary discs. The +mass of the circumstellar disc scales with the infall accretion rate. +If the resolution of the circumstellar disc is too poor, then the disc +artificially accretes rapidly due to the artificially enhanced effects of +viscosity at low density in the SPH code. +We first examined the behavior of initially highly inclined circum- +stellar discs that are not supplied with material from a circumbinary +disc. A polar test particle in orbit around a primary star reaches an +eccentricity of nearly unity during the first KL cycle, forcing the +particle to become unbound or hit the central star. Similarly, initially +highly inclined circumstellar discs around individual binary compo- +nents can experience very strong KL oscillations. For an equal mass +binary containing only a single circumstellar disc at high inclina- +tion between 70◦ and 100◦, the disc undergoes only a single KL +oscillation before losing nearly all its mass for our given sink size. +Some of the disc mass is transferred to the companion star to form +a low inclination disc that does not undergo KL oscillations. These +results suggests that such high inclinations of discs are short-lived +due to enhanced dissipation from shocks that leads to tilt evolution +on short timescales. In contrast, discs that are highly inclined but are +not subject to KL oscillations would undergo much slower evolution. +In particular, a polar disc would not precess (see e.g., equation (7)) +and therefore not warp. The disc would then not be subject to torques +that act to change its inclination. +In this work, and from Smallwood et al. (2021b), we showed that +the continuous accretion of material from the circumbinary disc +allows the effects of KL oscillations on circumstellar discs to be +much longer-lived. In this process, the circumbinary material is con- +tinuously delivered with a high inclination to the lower inclination +circumstellar discs. We found that the simulation resolution is impor- +tant for modeling the longevity of the KL oscillations. We find longer +lived KL oscillations that show signs of mild weakening in time, pos- +sibly due to the resolution (e.g., Figure 6). The balance between the +accretion timescale and the KL timescale determines whether the +oscillations are sustained or damp in time. If the circumstellar disc +material were to accrete on a much shorter timescale than the KL os- +cillation period, we would not expect the KL oscillations to operate. +We found that with increasing resolution, the accretion timescale +becomes comparable to the KL timescale, favoring sustained KL +oscillations. +Planet formation is thought to still occur in non-zero eccentric- +ity discs (Silsbee & Rafikov 2021). In the case of S-type planets +(planets orbiting one of the stellar companions in a binary), gravita- +tional perturbations from an eccentric orbit stellar companion and an +eccentric disc increase planetesimal eccentricities, leading to colli- +sional fragmentation, rather than growth, of planetesimals. However, +Rafikov & Silsbee (2015) analyzed the planetesimal motion in ec- +centric protoplanetary discs when the planetesimals were affected by +gas drag and disc gravity. They found that the planetesimals could +withstand collisional fragmentation and erosion, thereby providing +a pathway to forming planetary cores by coagulation in a binary. It +is not clear how those results carry over to the case of highly ec- +centric discs undergoing KL oscillations. However, the formation of +nearly polar circumstellar discs from this work may give rise to the +formation of nearly polar planets that become Kozai-unstable. Planet +formation in a polar circumstellar disc requires the disc to last for a +sufficiently long time. We speculate that this is possible provided that +the disc is continuously accreting material in a polar configuration. +Observations of misaligned planetary systems show a prefer- +ence for nearly polar orbits with true obliquities 휓 in the range +휓 = 80◦ − 125◦ (Albrecht et al. 2021; Dawson & Albrecht 2021). +For example, two observed ultra-short-period hot Jupiters in po- +lar orbits around an A-type star are Kelt-9b (Ahlers et al. 2020a) +and MASCARA-4b (Ahlers et al. 2020b). The majority of planets +studied by Albrecht et al. (2021) were hot Jupiters, since the mea- +surements for these types of planets are more precise. However, +a few warm-Neptunes with polar orbits were observed, including +HAT-P-11b (Sanchis-Ojeda & Winn 2011), GJ 436b (Bourrier et al. +2018, 2022), HD 3167c (Dalal et al. 2019; Bourrier et al. 2021), and +WASP-107b (Dai & Winn 2017; Rubenzahl et al. 2021). A more re- +cent warm Neptune, GJ 3470b, is also observed to be on a polar orbit +(Stefànsson et al. 2022). +ACKNOWLEDGEMENTS +We thank the anonymous reviewer for helpful suggestions that pos- +itively impacted the work. We thank Daniel Price for providing the +phantom code for SPH simulations and acknowledge the use of +SPLASH (Price 2007) for the rendering of the figures. Computer +support was provided by UNLV’s National Supercomputing Center. +MNRAS 000, 1–13 (2023) + +12 +Smallwood et al. +We acknowledge support from NASA XRP grants 80NSSC19K0443 +and 80NSSC21K0395. This research was supported in part by the +National Science Foundation under Grant No. NSF PHY-1748958. +SHL thanks the Simons Foundation for support during a visit to the +Flatiron Institute. +DATA AVAILABILITY +The data supporting the plots within this article are available +on reasonable request to the corresponding author. 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J., Lai D., 2018, MNRAS, 473, 603 +Ziegler C., et al., 2018, AJ, 155, 161 +Zrake J., Tiede C., MacFadyen A., Haiman Z., 2021, ApJ, 909, L13 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2023) + diff --git a/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/load_file.txt b/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73c22d71f328451e8823c4e3fff270674efa9f65 --- /dev/null +++ b/3tFKT4oBgHgl3EQfRC0N/content/tmp_files/load_file.txt @@ -0,0 +1,1468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf,len=1467 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='11769v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='EP] 27 Jan 2023 MNRAS 000, 1–13 (2023) Preprint 30 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 Formation of polar circumstellar discs in binary star systems Jeremy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood,1,2★ Rebecca G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Martin2 and Stephen H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lubow3 1Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan 2Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA 3Space Telescope Science Institute, Baltimore, MD 21218, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We investigate the flow of material from highly misaligned and polar circumbinary discs that feed the formation of circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' With three-dimensional hydrodynamic simulations we consider equal mass binaries with low eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We also simulate inclined test particles and highly-misaligned circumstellar discs around one binary component for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='During Kozai-Lidov (KL) cycles, the circumstellar disc structure is altered through exchangesof disc eccentricity with disc tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Highly inclined circumstellar discs and test particles around individual binary components can experience very strong KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The continuous accretion of highly misaligned material from the circumbinary disc allows the KL oscillations of circumstellar discs to be long-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this process, the circumbinary material is continuously delivered with a high inclination to the lower inclination circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We find that the simulation resolution is important for modeling the longevity of the KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' An initially polar circumbinary disc forms nearly polar, circumstellar discs that undergo KL cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The gas steams accreting onto the polar circumstellar discs vary in tilt during each binary orbital period, which determines how much material is accreted onto the discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The long-lived KL cycles in polar circumstellar discs may lead to the formation of polar S-type planets in binary star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Key words: binaries: general – circumstellar matter– accretion, accretion discs 1 INTRODUCTION The majority of stars born in dense stellar clusters are part of binary star systems (Duquennoy & Mayor 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Duchêne & Kraus 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The observed orbital eccentrici- ties of binaries vary with orbital separation (Raghavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Tokovinin & Kiyaeva 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For tight binaries, the eccentricities are small, which implies that there has been circularization of the bi- nary orbit caused by stellar tidal dissipation (Zahn 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' More widely-separated binaries have observed eccentricities ranging from 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='39 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='59, with a considerable number of highly eccentric systems with 푒b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The interactions of the binary with sur- rounding gas may be responsible for the present-day observed binary eccentricities (Goldreich & Tremaine 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Artymowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Artymowicz 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Armitage & Natarajan 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Cuadra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Roedig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Muñoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Zrake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Circumbi- nary discs of gas and dust are sometimes observed to be responsible to be providing accreting material onto the binary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Alves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The gas flow dynamics from the circumbinary disc onto the binary components has significant implications for planet formation scenarios in binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Circumbinary discs are commonly observed to be moderately to highly misaligned to the binary orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, the pre- main sequence binary KH 15D has a circumbinary disc inclined by 5 − 16◦ (Chiang & Murray-Clay 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Poon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The radial extent of the disc is narrow and pre- ★ E-mail: jlsmallwood@asiaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='sinica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='tw sumed to be rigidly precessing to explain the unique periodic light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A ∼ 60◦ inclined circumbinary disc is found around the main-sequence binary IRS 43 (Brinch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2016), along with mis- aligned circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' There is an observed misalignment of about 70◦ between the circumbinary disc and the circumprimary disc in HD 142527 (Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Owen & Lai 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Another young binary, HD 98800 BaBb, has the only observed polar (inclined by ∼ 90◦) gaseous circumbinary disc (Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The 6–10 Gyr old binary system, 99 Herculis, has a nearly polar (about 87◦) debris ring (Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Apart from binaries, stars may also form in higher-order systems (Tokovinin 2014a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumtriple disc around the hierarchical triple star system, GW Ori, is tilted by about 38◦ (Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The observations of inclined circumbinary discs have implications on planet formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Observations from space and ground- based telescopes reveal that ∼ 50 per cent of the confirmed exoplan- ets reside in binary systems (Horch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Deacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, the binary system 훾 Cep AB hosts a giant planet around the primary star, 훾 Cep Ab (Hatzes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' It is crucial to study the structure and evolution of protoplanetary discs since these are the sites for planet formation (D’Angelo & Lissauer 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A forming planet’s orbital properties are directly related to the orientation of the protoplanetary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, the observed young binary system XZ Tau shows both the circumprimary and circumsecondary discs are misaligned to the binary orbital plane (Ichikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The binary system HD 142527 shows the presence of a misaligned inner disc around one of the stellar com- © 2023 The Authors 2 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' ponents, presumably fed from the circumbinary disc (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Furthermore, IRAS 04158+2805 is a binary system where the two circumstellar discs and the circumbinary discs have been observed to be misaligned (Ragusa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, highly- inclined circumstellar discs may give birth to planets on highly-tilted orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Due to viscous dissipation, a misaligned circumbinary disc un- dergoes nodal precession and evolves towards either a coplanar or polar alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For an initially low-inclination circumbinary disc, the disc precesses about the angular momentum vector of the bi- nary and eventually evolves to be coplanar to the binary orbital plane (Facchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Foucart & Lai 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Slightly misaligned discs around an eccentric binary undergo tilt oscillations as they align, due to the nonaxisymmetric potential produced by the ec- centric binary (Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For highly inclined discs around eccentric orbit binaries, the angular momentum vec- tor of the disc precesses about the eccentricity vector of the bi- nary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Aly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015), which leads the disc to align perpen- dicular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', polar) to the binary orbital plane (Martin & Lubow 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lubow & Martin 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Zanazzi & Lai 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Martin & Lubow 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Cuello & Giuppone 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A massive circumbinary disc that is undergoing polar alignment aligns to a generalized polar state which is less than 90◦ (Zanazzi & Lai 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Martin & Lubow 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Circumbinary gas discs contain a central cavity around the bi- nary where little material is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The cavity size is determined by where the tidal torque is balanced with the viscous torque (Artymowicz & Lubow 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lubow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Miranda & Lai 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Franchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Hirsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Ragusa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The strength of the binary torque on the disc is dependent on the tilt of the circumbinary disc and binary eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The tidal torque at a given radius is zero when the circumbinary disc is polar and the binary eccentricity approaches 푒b = 1 (Lubow & Martin 2018) or if the disc is retrograde (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In the simplest models, the production of an outward forcing torque by the binary can prevent circumbinary material from flowing through the cavity (Lynden-Bell & Pringle 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Pringle 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, material from the circumbinary disc flows through the binary cavity in the form of gaseous streams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Artymowicz & Lubow 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Günther & Kley 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Nixon & King 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' D’Orazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Farris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Muñoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Alves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' These streams are respon- sible for forming and replenishing circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The accretion of material onto the circumstellar discs may aid in the formation of 푆–type planets, those that orbit one component of a binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Accretion of material onto the central binary may be suppressed for small disc aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The structure of a circumstellar disc around one star is strongly affected by the tidal field of the binary compan- ion (Papaloizou & Pringle 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Artymowicz & Lubow 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Pichardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Jang-Condell 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Circumstellar discs around each binary component undergo tidal truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A circumstellar disc in a circular orbit binary is typically truncated to about one-third to one-half of the binary orbital separation The tidal truncation radius is expected to decrease with increasing binary eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Kozai-Lidov (KL) oscillations (Kozai 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lidov 1962) have been studied extensively to analyze several astronomical pro- cesses involving bodies that orbit a member of a binary sys- tem that begin on highly misaligned orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' During KL os- cillations, the object’s inclination is exchanged for eccentricity, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' These processes include asteroids and irregular satellites (Kozai 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Nesvorný et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2003), artificial satellites (Lidov 1962), tidal disruption events (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2011), forma- tion of Type Ia supernovae (Kushnir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013), triple star systems (Eggleton & Kiseleva-Eggleton 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Fabrycky & Tremaine 2007), planet formation with inclined stellar companions (Wu & Murray 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Takeda & Rasio 2005), giant outbursts in Be/X-ray bi- naries (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Martin & Franchini 2019), inclined planetary companions (Nagasawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2008), mergers of bina- ries in galactic nuclei (Blaes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Antonini & Perets 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Hamers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Fragione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019a,b), stel- lar compact objects (Thompson 2011), and blue straggler stars (Perets & Fabrycky 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A highly misaligned initially circular disc around one compo- nent of a binary undergoes KL cycles in which its inclination is exchanged for eccentricity, and vice versa (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Due to disc dissipation by viscosity and shocks, these oscillations are typically significantly damped after a few oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' KL oscilla- tions can occur in a fluid disc with a wide variety of disc and binary parameters (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' When the disc becomes eccentric, it overflows its Roche lobe and transfers material to the companion star (Franchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Self-gravity of a disc can suppress disc KL oscillations if the disc is close to being gravitationally unstable (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' KL oscillations in a circumstellar disc may have significant consequences for planet formation since strong shocks in the gas are produced during high eccentricity phases (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A misaligned circumbinary disc may form misaligned circumstel- lar discs around the individual binary components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A highly misaligned disc around one component of a binary may be unstable to the Kozai-Lidov (KL) mechanism (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b) simu- lated the flow of gas originating from an initially misaligned cir- cumbinary disc by 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The misaligned gas streams that flow into the binary cavity result in formation of highly tilted circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The inclined circumstellar discs in turn undergo KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, the KL oscillations are long- lived, due to the continuous accretion of inclined material from the circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Long-lived KL cycles have important implica- tions for planet formation in binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this work, we extend the previous study Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b) and consider more highly inclined circumbinary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We first revisit the dynamics of highly inclined test particle orbits around one component of a binary in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Section 3, we describe the setup for our hydrodynamical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Section 4, we dis- cuss the results of our circumprimary disc simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We simulate a highly inclined circumprimary disc in a binary to explore the dy- namics of the KL cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Previous studies have only dealt with cir- cumprimary disc inclinations ≲ 60◦, while we consider higher tilts, including a polar circumprimary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Section 5, we show the re- sults of our hydrodynamical simulations with an initial circumbinary disc, where we consider the flow of material from discs with various initial misalignments, including a polar circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Finally, a summary is given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2 KOZAI-LIDOV OSCILLATIONS OF TEST PARTICLES Before considering discs, we consider the properties of test parti- cle orbits that undergo KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' As a consequence of the conservation of the component of the angular momentum that is perpendicular to the binary orbital plane, the test particle’s inclina- tion is recurrently exchanged for eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This conservation is MNRAS 000, 1–13 (2023) Formation of polar circumstellar discs 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Eccentricity (upper panel) and inclination (lower panel) evolution of circumprimary test particles under the influence of a circular binary for initially circular orbit particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We vary the initial particle orbital tilt, 푖0, beginning with 30◦ (black), 45◦ (blue), 60◦ (red), 75◦ (green), 80◦ (yellow), 85◦ (purple), and 90◦ (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The initial orbital radius of the particle is set at 푟0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='06푎, where 푎 is the separation of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The time is in units of binary orbital period 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' expressed as � 1 − 푒2p cos 푖p ≈ const, (1) where 푖p is the particle inclination with respect to the binary orbital plane and 푒p is the eccentricity of the test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A initially circular orbit particle initially gains eccentricity while reducing its orbital tilt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' going towards alignment which means higher values of | cos 푖p|) and then circularizes while gaining orbital tilt back to its original inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For an initially circular orbit particle, KL oscillations only occur if the initial tilt of the test particle 푖p0 satisfies cos2 푖p0 < cos2 푖cr = 3/5 (Innanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1997), which requires that 39◦ ≲ 푖p0 ≲ 141◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (1), an initially circular particle orbit can achieve a maximum eccentricity given by 푒max = � 1 − 5 3 cos2 푖p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2) The increase in a circular particle’s eccentricity can be quite signif- icant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, if the particle’s initial orbit is tilted by 60◦, the maximum eccentricity reached during a KL cycle is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For eccentric binaries, stronger effects from KL oscillations have been found to exist (Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lithwick & Naoz 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Naoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2011, 2013a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Teyssandier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The KL oscillation period for a particle in the po- tential of an eccentric binary is approximately given by 휏KL 푃b ≈ 푀1 + 푀2 푀2 푃b 푃 (1 − 푒2 b)3/2 (3) (Holman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Innanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Kiseleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1998), where 푀1 and 푀2 are the masses of the primary and secondary components of the binary, respectively, 푃 = 2휋/ � 퐺푀1/푎3p is the orbital period of the particle with semimajor axis 푎p, 푃b = 2휋/Ωb is the orbital period of the binary, 푒b is the binary eccentricity, and Ωb = � 퐺(푀1 + 푀2)/푎3 b is the binary orbital frequency for binary semimajor axis 푎b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' To simulate an inclined circumprimary test particle in a binary, we use the 푁–body integrator, MERCURY (Chambers 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The test particle is orbiting the primary companion with an initial tilt 푖0 relative to the binary orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The binary components have equal mass so that 푀1 = 푀2 = 푀/2, where 푀 is the total mass of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2015b) ran numerous test particle orbits showing the effects the particle and binary parameters have on the induced KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Following their work, we model an eccentric inclined particle around one component of an eccentric binary, more applicable to binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We first simulate an inclined particle in a circular binary to match previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1 shows the eccentricity and inclination of a circumprimary particle as a function of time that begins on a cir- cular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The analytic solution for these test particle orbits in the quadrupole approximation is given in Lubow (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We consider various initial tilts of the test particle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The critical inclination that the test particle orbit must have to induce KL cycles is ∼ 39◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Thus, a particle tilt of 30◦ (black line) does not undergo KL oscil- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' As the initial inclination of the particle increases, the KL oscillations become more frequent, and the growth in the eccentric- ity becomes more prominent (in agreement with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1 in Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2015b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The trough in the inclination profile of a test particle be- comes narrower with initial inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' An initial particle orbit tilt of 90◦ becomes unstable and collides with the primary star during the first KL oscillation because the particles eccentricity exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The eccentricity of the polar particle increases almost up to its maximum eccentricity before the tilt begins to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Next, we set the initial particle tilt to 60◦ around a slightly eccen- tric binary with 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1, as we will consider in the disc simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We model various initial test particle eccentricities ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 2 shows the eccentricity and inclination of eccentric circumprimary particles as a function of time in binary orbital pe- riods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' An inclined circular test particle within an eccentric binary has an increased frequency in KL oscillations when compared to a particle orbiting one component of a circular binary, as expected by equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' From Figure 2, when the particle eccentricity is in- creased, the maximum eccentricity reached during a KL oscillation also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, the difference between the initial eccentric- ity to the maximum eccentricity of the particle decreases as the initial particle eccentricity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lastly, we examine the KL mechanism for a nearly polar particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1, an initially circular orbit particle with an initial orbital tilt of 85◦ is unstable to KL oscillations but is otherwise stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We consider a nearly polar orbit particle with an initial orbital tilt 푖0 = 85◦ around a binary with eccentricity 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3 we show the particle eccentricity and inclination as a function of time in binary orbital periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The various lines correspond to different initial particle eccentricities ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For all values of the initial particle eccentricity we consider, the particle proceeds through KL cycles in a periodic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Unlike the particle beginning at a tilt of 60◦, a nearly polar particle exhibits similar maximum eccentricity close to unity during a KL oscillation regardless of initial particle eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The minimum inclination reached during each KL oscillation is roughly independent of particle initial eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0 80 60 40 0 100 200 300 400 50 t/Porb4 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Eccentricity (upper panel) and inclination (lower panel) evolution of circumprimary test particles under the influence of binary with eccentricity 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The initial tilt of the particle orbit is set to 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We vary the initial particle eccentricity 푒0 beginning with 푒0 = 0 (black), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 (red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 (green), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 (yellow), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The initial orbital radius of the particle is set at 푟0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='06푎, where 푎 is the separation of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The time is in units of binary orbital period 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3 HYDRODYNAMICAL-SIMULATION SETUP We use the smoothed particle hydrodynamics (SPH) code phantom (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018a) to model gaseous circumbinary and circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' phantom has been tested extensively for modeling misaligned circumbinary discs (Nixon 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Nixon & Lubow 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Facchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Poblete et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Aly & Lodato 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Hirsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021b), as well as misaligned circumstellar discs around individual binary components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Doğan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Franchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The suite of simulations is summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this section we describe the setup for the binary star, circumprimary disc, and circumbinary disc in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Binary star setup We model the binary star system as a pair of sink particles, with an initial binary separation 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The binary is not static but rather evolves freely in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Each sink particle is given an initial mass with 푀1 being the primary mass and 푀2 being the secondary mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The total binary mass is thereby 푀 = 푀1 + 푀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' All of our simulations assume an equal-mass binary (푀1 = 푀2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Cartesian coordinates, the orbit of the binary lies in the 푥-푦 plane initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The binary begins ini- tially at apastron along the 푥-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The massive sink particles have a hard accretion boundary, meaning that when particles penetrate the sink accretion radius, the particle’s mass and angular momentum are deposited onto the star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Bate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A large accretion radius is often used to reduce the computation time significantly by neglecting to resolve close-in particle orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this work, however, we are interested in resolving the formation and evolution of the cir- Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2 but for nearly polar test particles with an initial orbital tilt 푖0 = 85◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' cumstellar material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, we adopt a relatively small accretion radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='05푎 for simulations that begin with a circumbinary disc and an accretion radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025푎 for simulations that begin with a circumprimary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Using a smaller accretion radius for the cir- cumprimary disc simulations ensures that the disc lifetime is longer, along with higher disc resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The more eccentric the binary, the smaller the outer truncation radius for the circumstellar discs (Artymowicz & Lubow 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Having a small binary eccentricity helps with the resolution of the circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' On the other hand, to have a stable polar circumbinary disc, the binary eccentric- ity needs to be a non-zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The initial binary eccentricity is set to 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1, with the binary eccentricity vector along the positive 푥–axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' With this value of binary eccentricity, the critical tilt of the circumbinary disc to remain nearly polar is ∼ 77◦ (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 33 in Martin & Lubow 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 Circumprimary disc setup To model a circumprimary disc, we follow the methods of Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2014b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Runs 1-5 in Table 1 simulate initially a circum- primary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The inner and outer disc radii are set at 푟in = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025푎 and 푟out = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25푎, respectively, with a initial total disc mass 푀CPD = 10−3푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumprimary disc consists of 750, 000 equal- mass Lagrangian particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We neglect any effects of self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc surface density profile is initially a power law distribution given by Σ(푟) = Σ0 � 푟 푟in �−푝 , (4) where we set 푝 = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We adopt a locally isothermal disc with sound speed 푐s ∝ 푅−3/4, 퐻/푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 at 푟 = 푟in, and 퐻/푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 at 푟 = 푟out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' With this prescription, the viscosity parameter 훼 and ⟨ℎ⟩/퐻 are effectively constant over the radial extend of the disc (Lodato & Pringle 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For the circumprimary disc simulations, we take the Shakura & Sunyaev (1973) 훼 parameter to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' To MNRAS 000, 1–13 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0 90 80 70 60 50 40 30 0 20 40 60 80 10 t/Porb10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0 60 50 40 30 0 20 40 60 80 10 t/Porb0Formation of polar circumstellar discs 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The setup of the SPH simulations that includes an initial circumprimary disc (CPD) or circumbinary disc (CBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The table lists the initial parameters beginning with the disc tilt 푖0, inner disc radius 푟in, outer disc radius 푟out, 훼 viscosity parameter, disc aspect ratio at inner disc radius 퐻/푟in, disc aspect ratio at outer disc radius 퐻/푟out, the number of particles, and whether or not the circumstellar discs undergo the Kozai-Lidov (KL) instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Model Disc Setup 푖0/◦ 푟in/푎 푟out/푎 훼 퐻/푟in 퐻/푟out # Particles KL unstable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' run1 CPD 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 750, 000 Yes run2 CPD 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 750, 000 Yes run3 CPD 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 750, 000 Yes run4 CPD 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 750, 000 Yes run5 CPD 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='02 750, 000 Yes run6∗ CBD 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='088 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 Yes run7 CBD 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='088 750, 000 Yes run8 CBD 90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='088 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 Yes ∗ Simulation from Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b) accomplish this, the SPH artificial viscosity coefficients are set as 훼AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='18 and 훽AV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc is resolved with shell-averaged smoothing length per scale height ⟨ℎ⟩/퐻 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 Circumbinary disc setup To model an initially flat but tilted gaseous circumbinary disc, we follow the methods of Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Runs 6, 7, and 8 in Table 1 describe the simulations of a circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc initially consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 equal-mass Lagrangian SPH particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We also model a 750, 000 particle simulation for a resolution study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The simulations run for 45 푃orb, where 푃orb is the orbital period of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This is sufficient time for the forming circumstellar discs to reach a quasi-steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We simulate initially highly misaligned disc inclinations of 푖0 = 60◦, 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A disc with 푖0 = 90◦ is in a polar configuration, where the angular momentum vector of the disc is aligned to the eccentricity vector of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At the beginning of our simulations, we select an initial inner disc radius, 푟in, and outer disc radius, 푟out, where the initial total disc mass, 푀CBD, is confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' All of the simulations model a low-mass circumbinary disc such that 푀CBD = 10−3푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We choose the circumbinary disc to be radially very narrow and close to the binary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This is done to maximise the accretion rate onto the binary and hence the resolution of the circumstellar discs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For our simulations, we take 푟in = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6푎 and 푟out = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The tidal torque is weaker at a given radius for a more highly misaligned disc which allows the inner disc radius to lie closer to the binary than a coplanar disc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Lubow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Miranda & Lai 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lubow & Martin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The inner truncation radius of a polar circumbinary disc is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 푎 (Franchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019b), much smaller than the 2 − 3 푎 expected for coplanar discs (Artymowicz & Lubow 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc surface density profile follows from Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The physical disc viscosity is incorporated by using artificial viscosity 훼av, which is detailed in Lodato & Price (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' By using our sur- face density profile and a disc aspect ratio 퐻/푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 at 푟in, the shell-averaged smoothing length per scale height ⟨ℎ⟩/퐻 and the disc viscosity parameter 훼 are constant over the radial extent of the disc (Lodato & Pringle 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumbinary disc is initially resolved with ⟨ℎ⟩/퐻 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The parameters for the simulations require a high viscosity in order to maximize the accretion rate on to the circumstellar discs and provide better resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We consider a relatively high value for the Shakura & Sunyaev (1973) 훼SS of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In a more realistic system, the disc viscosity may be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In order to more accurately simulate the formation and develop- ment of circumstellar discs, we adopt the locally isothermal equation of state of Farris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2014) and set the sound speed 푐s to be 푐s = F 푐s0 � 푎 푀1 + 푀2 �푞 � 푀1 푟1 + 푀2 푟2 �푞 , (5) where 푟1 and 푟2 are the radial distances from the primary and sec- ondary stars, respectively, and 푐s0 is a constant with dimensions of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 푞 is set to 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' F is a dimensionless function of position that we define below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This sound speed prescription guarantees that the temperature profiles in the circumprimary and circumsecondary discs are set by the primary and secondary stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For 푟1, 푟2 ≫ 푎, 푐s is set by the distance from the binary centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' To increase the resolution of the circumstellar discs, we include a function F in Equation (5) as detailed in Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The purpose of F is to modify the sound speed around each binary component so that the viscous timescale is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This increases the mass (and hence the resolution) in the steady-state circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We take F = �√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='001, if 푟1 or 푟2 < 푟c, 1, otherwise, (6) where 푟c is the cutoff radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We set a cutoff radius of 푟c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='35푎 from each binary component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Using the prescription mentioned above ensures that the disc aspect ratio of the circumstellar discs at radius 푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1푎 is 퐻/푟 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01, which is one-tenth of the disc aspect ratio at the initial inner circumbinary disc radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 Analysis routine We analyse the disc and binary parameters as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The parameters include tilt, eccentricity, the longitude of the ascending node, mass, and mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' To probe the circumprimary disc simulations, we average over particles in the radial range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='025푎 to a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='30푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For the circumbinary disc simulations, we average over particles in the radial range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4푎 to a distance of 10푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For the forming circumstellar discs, we average over all particles bound to each binary component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', the specific energies, kinetic plus potential, of the particles are negative, neglecting the thermal energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The tilt, 푖, is defined as the angle between the initial angular momentum vector of the binary (the 푧-axis) and the angular momentum vector of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The longitude of the ascending node, 휙, is measured relative to the 푥-axis (the initial binary eccentricity vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 6 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 20 40 60 80 100 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 1 100 0 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 1 0 5 10 15 10-5 2 3 4 5 Panel 1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Evolution of a KL unstable circumprimary disc as a function of time in units of the binary orbital period 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We simulate five different initial disc inclinations, which are 60◦ (run1 from Table 1, black), 70◦ (run2, blue), 80◦ (run3, red), 90◦ (run4, green), and 100◦ (run5, yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc parameters are tilt 푖 (panel 1), eccentricity 푒 (panel 2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d (panel 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The mass accretion rate �푀 onto the primary star is shown in panel 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 4 HYDRODYNAMICAL RESULTS WITH A CIRCUMPRIMARY DISC This section considers the evolution of a circumprimary disc in the absence of accretion from a circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This enables us to disentangle the effect of accretion onto the circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We focus on large circumprimary disc misalignments in an eccentric binary star system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We consider five different initial disc tilts, 60◦ (run1 from Table 1), 70◦ (run2), 80◦ (run3), 90◦ (run4), and 100◦ (run5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 4 shows the disc tilt, eccentricity, the longitude of the ascending node, the mass of the circumprimary disc, and the accretion rate onto the primary star as a function of time in binary orbital periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc exhibits KL cycles for each initial tilt, where the disc eccentricity and inclination are exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For a disc with an initial tilt of 60◦, Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2014a) found that the first KL oscillation occurred around 10 Porb for a circular binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In our case, the disc with the same initial tilt undergoes the first KL oscillation much sooner due to the binary having a slightly eccentric orbit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='12 in Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Due to viscous dissipation and the lack of circumbinary material, the KL oscillations damp quickly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For higher initial inclinations, 70◦, 88◦, 90◦ and 100◦, the discs do not survive after one KL oscillation for our given sink size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The x-z plane t = 0 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a y-z plane t = 0 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a x-z plane t = 10 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a y-z plane t = 10 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a x-z plane t = 15 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a y-z plane t = 15 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2a Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The evolution of polar circumprimary disc (run4 from Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The white circles denote the eccentric orbit binary components with an initial binary separation of 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The top row shows the initial disc setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The middle and bottom rows show the disc evolution at 푡 = 10 푃orb and 푡 = 15 푃orb, respectively, where 푃orb is the binary orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The color denotes the gas surface density, with the orange regions being about three orders of mag- nitude larger than the purple regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The left column shows the 푥–푧 plane, and the right column shows the 푦–푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At 푡 = 10 푃orb, the circumpri- mary disc is highly eccentric due to the Kozai-Lidov instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Also, at this time, a circumsecondary disc is being formed from material flowing close to the secondary binary component from the eccentric circumprimary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At 푡 = 15 푃orb, the circumprimary disc has completely dissipated from being accreted onto the primary star and transferring material to the secondary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At this time, there is more material in the newly formed circumsecondary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' discs become very eccentric, which leads to the majority of the disc material being accreted by the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Increasing the resolution of these simulations does not lengthen the disc lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, if we were to use a smaller sink size, then the disc could survive through the KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A smaller sink size would ensure that a larger portion of the disc could survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' An accretion radius of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='01 au is comparable to the size of the star, but we simulate a larger sink size for computational reasons and to compare with the circumbinary disc simulations detailed in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The initially polar disc’s tilt does not change much from polar before the majority of the disc is accreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This is likely a consequence of the high disc eccentricities that are developed which is consistent with the results for test particle orbits (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In the retrograde case, MNRAS 000, 1–13 (2023) Formation of polar circumstellar discs 7 60 70 80 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 100 0 100 10-6 10-4 10 20 30 40 10-6 10-4 2 3 4 5 Panel 1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Resolution study for a circumbinary disc that is initially misaligned by 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The blue curves represent the simulation with initially 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 particles in the circumbinary disc, while the red curves denotes the simulation with initially 750, 000 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The first four panels show the disc parameters for the newly forming circumprimary disc as a function of time in units of the binary orbital period, 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc parameters are tilt 푖 (panel 1), eccentricity 푒 (panel 2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d (panel 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The black dotted curve in the third panel denotes the circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The lower panel shows the mass accretion rate onto the primary star �푀pri (panel 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 푖0 = 100◦, as the disc eccentricity increases, the inclination also increases, opposite to the prograde cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Highly inclined particle orbits experience a large (nearly 180◦) shift in 휙 within a small time interval centered about the eccentricity maximum (see the plot for Ω(푡) in Figure 1 of Lubow (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This large shift does not appear in Figure 4 or in any of our other phase results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We are not sure why this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Perhaps the disc is unable to respond to such a large shift within a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We further examine the evolution of the polar (푖0 = 90◦) circumpri- mary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 5, we show the polar circumprimary disc structure at three different times, 푡 = 0 푃orb, 10 푃orb, and 15 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Initially, the polar disc around the primary star (left white dot) is edge-on in the 푥-푧 plane and face-on 푦-푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At 푡 = 10 푃orb, the disc is at peak ec- centricity growth from the KL instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Also, at this time, streams of material from the circumprimary disc flow around the secondary star (right white dot) and begin forming a circumsecondary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At 푡 = 15 푃orb, the circumprimary disc has dissipated due to accretion onto the primary star and transporting material to the circumsec- ondary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The newly formed circumsecondary disc is at a lower tilt, below the threshold, to induce the KL cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 5 HYDRODYNAMICAL RESULTS WITH A CIRCUMBINARY DISC In this section we examine how misaligned and polar circumbinary material flows through the binary cavity and forms circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We first conduct a resolution study of our earlier work from Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b), modeling an initially 60◦ misaligned circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We then focus on the polar circumbinary disc case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Resolution Study We examine a circumbinary disc with an initial misalignment of 푖0 = 60◦ with two different initial numbers of particles, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 (run6) and 750, 000 (run7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The upper four panels in Figure 6 show the circumprimary disc parameters as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The bot- tom panel shows the mass accretion rate onto the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The blue curves represent the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 × 106 particle simulation, while the red curves represent the 750, 000 particle simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Panels 1 and 2 show the evolution of disc eccentricity and inclination where the forming circumprimary disc undergoes KL oscillations from the continuous accretion of material from the circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The oscillations damp in time at both resolutions, with the lower resolution simu- lation damping more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, the oscillations are likely limited by resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' If the accretion timescale is long compared to the KL timescale, we expect the KL oscillations to damp over time, similar to the circumprimary disc simulations without accre- tion shown in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' If the accretion timescale is short compared to the KL timescale, there should be no KL oscillations present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this case, the material moves through the disc faster than it becomes unstable to KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We expect the optimal oscil- lations when the timescales are comparable because the disc refills mass on the timescale that the oscillations take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For the simula- tion with a 60◦ tilted circumbinary disc, the accretion timescales for the primary and secondary are ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb, whereas the KL timescale for this simulation is ∼ 5 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The simulation is in the regime where the accretion timescale is shorter than the KL oscillation timescale because when the disc becomes eccentric during the KL oscillations, a large amount of disc material is accreted, reducing the accretion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, the accretion timescale is dependent on the disc viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In our hydrodynamical simulations, we use an artificial viscosity to model an expected Shakura & Sunyaev (1973) viscos- ity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The number of Lagrangian particles determines how close the artificial viscosity is to the actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Thus, the 훼 is artificially higher at lower resolutions, leading to a shorter accre- tion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For our higher-resolution simulation, the 훼 is lower, leading to a longer accretion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Panel 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 6 shows the longitude of the ascending node as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The precession rate of the circumprimary disc is only slightly faster than the circumbinary disc on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In the absence of the effects of KL oscillations, the nodal precession rate of the primary disc, assuming constant surface density Σ out to disc radius 푟 from the primary, is given by 휔pr = − 15푀2푟3 32푀1푎3 b cos (푖) Ω(푟), (7) where 푖 is inclination angle of the primary disc relative to the binary orbital plane and Ω = � 퐺푀1/푟3 is the angular velocity in the disc MNRAS 000, 1–13 (2023) 8 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' x-z plane t = 0 Porb a y-z plane t = 0 Porb a x-z plane t = 25 Porb a y-z plane t = 25 Porb a Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The formation of polar circumstellar discs from an initially low- mass polar circumbinary disc (run8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The white circles denote the eccentric orbit binary components with an initial binary separation of 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The upper panels denote the initial disc setup, while the bottom panels show the disc evolution at 푡 = 25 푃orb, where 푃orb is the binary orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At this time, nearly polar circumstellar discs are forming around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The color denotes the gas density using a weighted density interpolation, which gives a mass-weighted line of sight average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The yellow regions are about three orders of magnitude larger than the purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The left column shows the 푥–푧 plane, and the right column shows the 푦–푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (Larwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' With 푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='35 푎b, we find 휔pr = 6◦/푃orb with a revolution period of ∼ 56 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, the circumstellar discs should have nodally precessed 75 per cent of a revolution in 45 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In panel 3 we see that the circumstellar discs have only completed roughly 30 per cent of a nodal revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' It is possible that the circumprimary phase is affected by the phase of accreted gas from the circumbinary disc that undergoes relatively slow nodal precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' As discussed in Section 4, KL oscillations modify the nodal precession rate of a test particle in a way that we do not see in the disc simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Lastly, the mass in the circumprimary discs oscillates in time, with the troughs corresponding with each high eccentricity period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' During each high eccentricity phase, the accretion rate peaks as seen in panel 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 Polar discs In this section, we present a hydrodynamical simulation of the flow of material from a polar circumbinary disc onto the binary components (run8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 7 shows the initial configuration of the polar circumbinary disc around an eccentric binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The bottom row shows the disc structure at 푡 = 25 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumbinary disc remains nearly polar (∼ 90◦) as shown in the 푥-푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Material flows from the polar circumbinary disc and forms nearly polar circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The cavity size is smaller in the polar disc compared to a coplanar disc simulation as expected (Lubow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Miranda & Lai 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The upper four panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 8 show the inclination, eccentricity, the longitude of the ascending node, and disc mass for the three 60 80 100 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 100 0 100 10-6 10-4 10-2 10 20 30 40 10-4 2 3 4 5 Panel 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Simulation results for run8 for an initially polar circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc parameters are shown for the circumprimary, circumsecondary, and circumbinary discs as a function of time in units of the binary orbital period, 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The upper four panels show the disc tilt 푖 (panel 1), eccentricity 푒 (panel 2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d (panel 4) for the three discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The lower panel shows the mass accretion rate onto the sinks �푀 (panel 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' discs as a function of time in binary orbital periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The lower panel shows the mass accretion rate onto the sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumstellar discs form at a time of ∼ 10 Porb, later than in the simulation with a lower level of circumbinary disc misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumbinary disc tilt evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Since we model a disc with a non-zero mass, it will align to a generalised polar state with an inclination that is < 90◦ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Martin & Lubow 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this case, the circumstellar discs form slightly retrograde, with a tilt just above 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The primary and secondary discs form with an eccentricity of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, the polar circumstellar discs undergo the KL instability, which forces the disc eccentricity and tilt to oscillate in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Looking at panels 1 and 2, we see that as the disc eccentricity increases, the disc tilt also increases, the opposite of the conventional KL case involving prograde orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, this result is consistent with the KL mechanism for retrograde orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Panel 3 shows the evolution of the longitude of the ascending node in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Since the circumprimary and circumsecondary discs are nearly polar, they exhibit very little precession (see equation 7 and discussion below it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The mass of the polar circumstellar discs oscillates in time (panel 4), likely due to the oscillating disc eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The polar circumbinary disc has lost ∼ 25 per cent of its initial mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 0Formation of polar circumstellar discs 9 a Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Edge-on view (푥–푧 plane) of a polar circumbinary disc (run8) at a time 푡 = 5 푃orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We ignore the main portions of the disc confined within 푟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='45푎b, where 푎b is the separation of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The binary components are shown as the green dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The colours denote the disc surface density, with the orange regions being about three orders of magnitude larger than the purple regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We overlay the velocity vectors shown by the black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The length of the arrow is proportional to the velocities of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We see two asymmetric lobes of material that are produced by the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Several of the velocity vectors are directed away from the plane of the circumbinary disc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' however, the material then falls back onto the disc gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The KL oscillations from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 8 damp in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, from our resolution study, the damping is primarily due to the initial number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The accretion timescale for this simulation is ∼ 15 Porb, and the KL timescale in this case is ∼ 10 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The accretion timescale is longer in the polar simulation than in the 60◦ simulation because the polar circumstellar discs become less eccentric during each KL cycle, accreting less disc material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For a higher resolution, we expect the KL oscillations to be long-lived even for polar circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' On the bottom-left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 7, we see that some material is flung out of the disc plane on both sides of the polar circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This material forms two lobes on both sides of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 9 shows the edge-on view of the disc surface density, along with the velocity vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The material is being flung outwards but remains bound to the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, the material then falls back into the gap region of the circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Throughout the simulation, the material is periodically flung out every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 푃orb when the binary components pass through the polar circumbinary disc plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We further examine the flow of polar circumbinary material onto the forming circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' First, we investigate the tilt of the gaseous streams that accrete onto the circumstellar discs as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 10 shows the circumbinary disc tilt as a function of disc radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The inner edge of the disc lies roughly at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The curves that are shown at radii < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6푎 map the tilt of the streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We show the disc tilt for a full binary orbital period from 20 Porb to 21 Porb in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb, the tilt of the streams are low at ∼ 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' When the binary orbital period is not at half increments, the tilt of the streams increases beyond 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, at times 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 Porb and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='7 Porb, the streams are highly tilted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Recall that the forming circumstellar discs initially 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 75 80 85 90 95 100 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Circumbinary disc tilt, 푖, as a function of radius, 푟, for the polar circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The color corresponds to the time in binary orbital periods, Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' form at a high disc tilt, > 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Therefore, whenever the gaseous streams are highly tilted, there is an increased accretion of material onto the circumstellar discs from the circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' When the streams are less inclined, every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb, there will be less material accreted onto the polar circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This phenomenon is also consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 9, where material is flung out of the plane of the circumbinary disc every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We test this by further visualizing the inflow of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 11 shows snapshots of zoomed-in views in the 푥–푧 and 푦–푧 planes of the disc surface density, showing the gaseous streams accreting onto the nearly polar circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The snapshots show the flow of material over 20 Porb to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='9 Porb in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Higher density streams occur at times 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 Porb and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='7 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The flow of material decreases every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb during the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' At these times, the steams are less dense, leading to less material accreting onto the circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We relate the flow of material from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 11 to the mass of the circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Figure 12 shows the mass of the circumprimary disc from 20 Porb to 25 Porb folded on top of one another for each orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The vertical dashed-lines denote the times when the binary is aligned with the circumbinary disc plane, which is assumed when the stars are both aligned with 푥–푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Each time the binary aligns to the plane of the disc, the masses of the circumstellar discs increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The mass of the disc decreases every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This be- haviour repeats every orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Overall, the disc mass deceases in time due to the KL mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 6 SUMMARY In this work, we investigated the flow of material from a circumbi- nary disc that results in the formation circumstellar discs around each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We simulated an initially highly misaligned and polar circumbinary disc using three-dimensional SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We consid- ered cases of low initial binary eccentricity (typically 푒b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1) and binary mass ratio of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We also simulated cases of test particles MNRAS 000, 1–13 (2023) 10 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='0 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='7 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='7 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='3 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a x-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='9 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a y-z plane t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='9 Porb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='25a Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Zoomed-in snapshots of the disc surface density showing the flow of material from a polar circumbinary disc onto the nearly polar circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The white circles denote the eccentric orbit binary components with an initial binary separation of 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The color denotes the gas density using a weighted density interpolation, which gives a mass-weighted line of sight average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The yellow regions are about three orders of magnitude larger than the purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We view the orbit of the binary in the 푥–푧 and 푦–푧 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The snapshots show a period from 20 Porb to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='9 Porb in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='1 Porb, where 푃orb is time in binary orbital periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 00001100Formation of polar circumstellar discs 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 5 10-6 20-21 Porb 21-22 Porb 22-23 Porb 23-24 Porb 24-25 Porb Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The circumprimary disc mass evolution during one binary orbital period, Porb, at times 20 − 21 Porb (blue), 21 − 22 Porb (red), 22 − 23 Porb (yellow), 23−24 Porb (purple), and 24−25 Porb (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The mass of the disc decreases every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='5 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The vertical dashed-lines denote the times when the binary is aligned with the circumbinary disc plane during 20 − 21 Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' An increased flow of material onto the circumstellar discs occurs when the binary is aligned with the circumbinary disc plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' around the primary star and cases of circumprimary discs only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', no circumbinary or circumsecondary discs) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In order to carry out these simulations in a reasonable amount of time, we made some compromises on our choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In particular, we introduced a higher viscosity parameter for the cir- cumbinary disc than is likely to occur and a lower temperature of the gas in the gap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' These choices were made to improve the resolution of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Even with these parameters, the reso- lution is still playing a role in our results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' While we have chosen the disc parameters (훼 and 퐻/푅) in our simulations to max- imise the accretion rate on to the binary components and therefore the simulation resolution, we expect the general behaviour to persist for more realistic parameters applicable to protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The mass of the circumstellar disc scales with the infall accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' If the resolution of the circumstellar disc is too poor, then the disc artificially accretes rapidly due to the artificially enhanced effects of viscosity at low density in the SPH code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We first examined the behavior of initially highly inclined circum- stellar discs that are not supplied with material from a circumbinary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A polar test particle in orbit around a primary star reaches an eccentricity of nearly unity during the first KL cycle, forcing the particle to become unbound or hit the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Similarly, initially highly inclined circumstellar discs around individual binary compo- nents can experience very strong KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For an equal mass binary containing only a single circumstellar disc at high inclina- tion between 70◦ and 100◦, the disc undergoes only a single KL oscillation before losing nearly all its mass for our given sink size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Some of the disc mass is transferred to the companion star to form a low inclination disc that does not undergo KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' These results suggests that such high inclinations of discs are short-lived due to enhanced dissipation from shocks that leads to tilt evolution on short timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In contrast, discs that are highly inclined but are not subject to KL oscillations would undergo much slower evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In particular, a polar disc would not precess (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', equation (7)) and therefore not warp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The disc would then not be subject to torques that act to change its inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this work, and from Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021b), we showed that the continuous accretion of material from the circumbinary disc allows the effects of KL oscillations on circumstellar discs to be much longer-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In this process, the circumbinary material is con- tinuously delivered with a high inclination to the lower inclination circumstellar discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We found that the simulation resolution is impor- tant for modeling the longevity of the KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We find longer lived KL oscillations that show signs of mild weakening in time, pos- sibly due to the resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=', Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The balance between the accretion timescale and the KL timescale determines whether the oscillations are sustained or damp in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' If the circumstellar disc material were to accrete on a much shorter timescale than the KL os- cillation period, we would not expect the KL oscillations to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We found that with increasing resolution, the accretion timescale becomes comparable to the KL timescale, favoring sustained KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Planet formation is thought to still occur in non-zero eccentric- ity discs (Silsbee & Rafikov 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' In the case of S-type planets (planets orbiting one of the stellar companions in a binary), gravita- tional perturbations from an eccentric orbit stellar companion and an eccentric disc increase planetesimal eccentricities, leading to colli- sional fragmentation, rather than growth, of planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, Rafikov & Silsbee (2015) analyzed the planetesimal motion in ec- centric protoplanetary discs when the planetesimals were affected by gas drag and disc gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' They found that the planetesimals could withstand collisional fragmentation and erosion, thereby providing a pathway to forming planetary cores by coagulation in a binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' It is not clear how those results carry over to the case of highly ec- centric discs undergoing KL oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, the formation of nearly polar circumstellar discs from this work may give rise to the formation of nearly polar planets that become Kozai-unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Planet formation in a polar circumstellar disc requires the disc to last for a sufficiently long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We speculate that this is possible provided that the disc is continuously accreting material in a polar configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Observations of misaligned planetary systems show a prefer- ence for nearly polar orbits with true obliquities 휓 in the range 휓 = 80◦ − 125◦ (Albrecht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Dawson & Albrecht 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' For example, two observed ultra-short-period hot Jupiters in po- lar orbits around an A-type star are Kelt-9b (Ahlers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020a) and MASCARA-4b (Ahlers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' The majority of planets studied by Albrecht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' (2021) were hot Jupiters, since the mea- surements for these types of planets are more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' However, a few warm-Neptunes with polar orbits were observed, including HAT-P-11b (Sanchis-Ojeda & Winn 2011), GJ 436b (Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2018, 2022), HD 3167c (Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021), and WASP-107b (Dai & Winn 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Rubenzahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A more re- cent warm Neptune, GJ 3470b, is also observed to be on a polar orbit (Stefànsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous reviewer for helpful suggestions that pos- itively impacted the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We thank Daniel Price for providing the phantom code for SPH simulations and acknowledge the use of SPLASH (Price 2007) for the rendering of the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' Computer support was provided by UNLV’s National Supercomputing Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 12 Smallwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' We acknowledge support from NASA XRP grants 80NSSC19K0443 and 80NSSC21K0395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' This research was supported in part by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' NSF PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' SHL thanks the Simons Foundation for support during a visit to the Flatiron Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' DATA AVAILABILITY The data supporting the plots within this article are available on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' A pub- lic version of the phantom, splash, and mercury codes are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='com/danieljprice/phantom, http://users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='au/~dprice/splash/download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='html, and https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content='com/4xxi/mercury, 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prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFKT4oBgHgl3EQfRC0N/content/2301.11769v1.pdf'} diff --git a/49E1T4oBgHgl3EQfAwK8/content/tmp_files/2301.02844v1.pdf.txt b/49E1T4oBgHgl3EQfAwK8/content/tmp_files/2301.02844v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c57088b1d7c6eea20cc46ff7f55430afc83efa1b --- /dev/null +++ b/49E1T4oBgHgl3EQfAwK8/content/tmp_files/2301.02844v1.pdf.txt @@ -0,0 +1,1215 @@ +Topological superconductor candidates PdBi2Te4 and PdBi2Te5 from a generic ab +initio strategy +Aiyun Luo,1, ∗ Ying Li,1, ∗ Yi Qin,1, ∗ Jingnan Hu,1 Biao Lian,2 and Gang Xu1, 3, 4, † +1Wuhan National High Magnetic Field Center & School of Physics, +Huazhong University of Science and Technology, Wuhan 430074, China +2Department of Physics, Princeton University, Princeton, NJ 08544, United States of America +3Institute for Quantum Science and Engineering, +Huazhong University of Science and Technology, Wuhan 430074, China +4Wuhan Institute of Quantum Technology, Wuhan 430074, China +Superconducting topological metals (SCTMs) have recently emerged as a promising platform of +topological superconductivity (TSC) and Majorana zero modes(MZMs) for quantum computation. +Despite their importance in both fundamental research and applications, SCTMs are very rare in +nature. In addition, some superconductors with topological electronic structures have been reported +recently, but a feasible program to determine their TSC properties is still lacking. Here, we propose +a new strategy to design SCTMs by intercalating the superconducting units into the topological +insulators. +A program that characterizes the superconducting BdG Chern number of 2D BdG +Hamiltonian from ab initio calculations is also developed. Following this strategy, PdBi2Te5 and +PdBi2Te4 are found to be experimentally synthesizable and ideal SCTMs. Chiral TSC could be +realized in such SCTMs by incorporating topological surface states with Zeeman effect, which can +be realized by an external magnetic field or in proximity to ferromagnetic (FM) insulator. Our +strategy provides a new method for identifying the SCTMs and TSC candidates, and the program +makes it possible to design and modulate the TSC candidates from ab initio calculations. +MAIN +As one of the most important systems in both fundamental physics and topological quantum computation, topo- +logical superconductors (TSCs) have attracted increasing interest for their ability to support Majorana fermions +and anyons with non-Abelian statistics [1–17]. Currently, the search for TSCs candidates has been focused on two +∗ These authors made equal contributions to this work. +† e-mail address: gangxu@hust.edu.cn +arXiv:2301.02844v1 [cond-mat.mtrl-sci] 7 Jan 2023 + +2 +experimental schemes. One is the architectures by the combination of conventional superconductors with topological +insulators (TIs) [18–20] or 1D nanowires [21, 22], but this approach brings high requirements for sample fabrication +and interface engineering. The other route is to achieve TSCs in superconducting topological metals (SCTMs) that +host both topological electronic structures at the Fermi level and superconductivity in one compound [23–36], in which +the topological surface states are gapped by the “self-proximity effect” of bulk superconductivity, thus avoiding the +complications of interface engineering. This approach has successfully predicted the SCTM FeTe0.55Se0.45 [24–26] and +similar compounds of iron-based superconductors [27–30], owing to the favorable SC gap and non-trivial band topology. +Beside the MZMs, 1D helical/chiral Majorana states have also been reported in domain walls of FeTe0.55Se0.45 [37] +and the magnetism-superconductor heterostructures [38–45]. It is also proposed that the propagating chiral Majorana +states can be applied to realize non-Abelian quantum gate operations, which could be 103 faster than the currently +existing quantum computation schemes [46]. +Encouraged by the success of Fe(Se,Te) [24–26], many topological materials that host both superconductivity and +topological electronic structures are proposed [47–52]. However, very rare experimental progress of TSC has been +made in such SCTMs. This is because, on the one hand, all of them are not the ideal SCTMs, whose band structures +are too complicated, the topological surface states are usually buried in the bulk states and difficult to form the +pairing required by TSC. On the other hand, lacking a direct characterization of the TSC properties from ab initio +calculations also hinders the effective experimental search in such materials. Therefore, a general program that could +calculate the TSC invariant from first-principles calculations is highly desirable. +In this work, we develop a program to characterize the superconducting topological invariant of 2D system from ab +initio calculations. Besides, we also propose a new strategy to design ideal SCTMs by intercalating superconducting +units into topological insulators. +Following this strategy, PdBi2Te5 and PdBi2Te4 are found to be ideal SCTMs +that host topological surface states at the Fermi level and superconductivity at 0.57 K and 3.11 K respectively. By +performing the superconducting energy spectrum and topological invariant calculations, we identify that chiral TSC +could be realized in the slab of such SCTMs by introducing considerable Zeeman splitting on the topological surface +states, which can be realized by an external magnetic field or in proximity to FM insulators. Our strategy provides +a new framework to enrich SCTMs and TSC candidates, and the program makes it possible to design and modulate +the TSC system from ab initio calculations, which can also be extended to study the TSC properties in other system, +such as magnetic TI/SC heterostructure, SC/FM heterostructure and SC/TI/SC heterostructure. +Inspired by the construction of magnetic TI MnBi2Te4 [53, 54], we propose that the SCTMs can be designed by + +3 +intercalating the SC units into the TIs, as illustrated by the schematic of Fig. 1(a). As an ideal SCTM, the target +crystal should be relatively stable in both energy and structure. More importantly, it must inherit the topological +electronic structures of the parent TI near the Fermi level, and also the superconductivity of the parent SC as shown in +Fig. 1(b). However, the combination of topological electronic structures and SC does not result in TSC eventually. The +realization of TSC generally requires a delicate modulation of many parameters, such as SC pairing, Zeeman splitting +and chemical potential, et al [18–26, 38–45]. Thus, the ability to characterize the TSC invariant and determine the +required parameters in real materials from the ab initio calculations is not only of theoretical significance, but also +highly desirable in experiment. +Here we develop a program to simulate the superconducting properties and characterize its topological invariant in +2D slab system from ab initio calculations, in which the necessary ingredients to realize chiral TSC based on SCTM +are included, such as bulk band structures, SC pairing, Zeeman splitting, Rashba spin-orbit coupling and chemical +potential. The workflow of this program is shown in Fig. 2. First, one should calculate the electronic structures of +SCTM materials, and construct the localized Wannier functions that capture all electronic features from the first- +principles calculations, referred as ˆHbulk. The next step is to construct the slab Hamiltonian ˆHslab with open boundary +condition along a certain direction [55]. In general, the spin-orbit coupling (SOC) and surface effect can be included +automatically in ˆHslab through the first-principles calculations with SOC. So that the topological properties, such as +the surface states and spin-texture, can be directly studied by using ˆHslab. On the other hand, one can also construct +a slab Hamiltonian ˆHnsoc +slab that excluded SOC from the non-SOC first-principles calculations, and add ˆHSOC and ˆHsurf +manually to simulate the variable SOC and surface effect in the topological electronic states and TSC. In this work, +we will adopt the former type of ˆHslab, in which only the intrinsic SOC of the real material is included. With adopting +particle-hole transformation, the ˆHslab can be extended to BdG Hamiltonian ˆHBdG +slab by adding SC pairing ˆHsc and +Zeeman splitting ˆHz. In the Nambu basis Φk = (ck,j,α,↑, ck,j,α,↓, c† +−k,j,α,↑, c† +−k,j,α,↓), where the cj,α,σ is the fermion +operator denotes an electron at j layer with orbital α and spin σ(↑, ↓), the BdG Hamiltonian is formulated as: +HBdG +slab (k) = +� +� +� +Hslab(k) − µ +∆(k) +∆†(k) +−H∗ +slab(−k) + µ +� +� +� + Mzτz. +(1) +In Eq. 1, µ is the chemical potential, which can be used to simulate the carriers doping. ∆(k) denotes the SC pairing +matrices, which could be both singlet and triplet pairing form. For the conventional s-wave SC, ∆(k) is expressed as: +∆(k) = ∆s × Islab ⊗ (iσy ⊗ Iorb), +(2) + +4 +where ∆s is the magnitude of intrinsic bulk s-wave pairing, σy is the Pauli matrix in spin space, Islab (Iorb) is an +Nslab × Nslab (Norb × Norb) identity matrix that represents the number of slab layers (Wannier orbitals). τz is the +Pauli matrix in particle-hole space, Mz is the Zeeman splitting energy, and Hz = Mzτz is used to simulate the +influence of the external magnetic field or the proximity effect of the FM insulator. Thus, Hz can be chosen to be +applied for the whole slab or just few surface layers, depending on the slab thickness, strength of magnetic field, the +type of the SC et al. In principle, chiral TSC can be achieved by modulating the SC pairing, Zeeman splitting and +chemical potential [38–45], which can be further revealed by calculating the superconducting energy spectrum and +the superconducting topological invariant. +In the gaped 2D superconducting system, the topological superconductors are classified by BdG Chern number in +the absence of time-reversal symmetry [3]. Such superconducting topological invariants can be characterized by the +evolution of Wilson loop [56–58]. For the occupied quasiparticle states |uBdG +n,k1,k2⟩, where k1 and k2 are momenta along +two primitive vectors of the Brillouin zone (BZ), the Berry phase of the Wilson loop along k2 at a fixed k1 can be +expressed as: +W(k1) = −Im ln +� +i +det M (i) +k1 , +(3) +with the overlap matrix M (i) +k1,mn = ⟨uBdG +m,k1,k(i) +2 |uBdG +n,k1,k(i+1) +2 +⟩, where k(i) +2 +is the i-th discretized momenta along k2 direction. +The winding number of W(k1) with respect to k1 is equal to the superconducting BdG Chern number CBdG. +Next, we take TI Bi2Te3 [59, 60], SC PdTe [61, 62] and SC PdTe2 [50–52] as parent compounds to demonstrate that +our SCTMs strategy is feasible. Experimentally, Bi2Te3 (space group R¯3m, a = 4.35 ˚A, c = 30.36 ˚A), PdTe (space +group space group P63/mmc, a = 4.152 ˚A, c = 5.671 ˚A, Tc = 2.3 K) and PdTe2 (space group P¯3m1, a = 4.03 ˚A, +c = 5.12 ˚A, Tc = 1.64 K) all adopt the triangle lattice and have very similar in-plane lattice constants, which makes +it much easier to integrate them together to form a new compound. According to our calculations, the stable unit of +PdBi2Te5 and PdBi2Te4 adopt octuple-layer (OL) structure and septuple-layer (SL) structure respectively, as shown +in Fig. 3(a)(also Fig. S1) and Fig. S2 of Supplementary Material(SM) [63]. They both favor the ABC stacking along +c-direction, and form the rhombohedral unit cell as shown in Fig. 3(a), which is 73 meV/f.u. +(73 meV/f.u. +for +PdBi2Te4) and 46 meV/f.u. (12 meV/f.u. PdBi2Te4) lower than the AA and AB stacking structures. The detailed +crystal parameters and total energy of different stacking PbBi2Te5 and PbBi2Te4 are tabulated in the Table. S1 and +Table. S2, respectively [63]. +The formation energy of PdBi2Te5 and PdBi2Te4 are calculated to study their thermodynamic stability by using +EPdmBinTel +f += EPdmBinTel − mEPd − nEBi − lETe, with Ei(i=PdmBinTel, Pd, Bi and Te) means the calculated + +5 +total energy per formula in the ground state. The calculated EP dBi2T e5 +f +and EP dBi2T e4 +f +are −3.184 eV/f.u. and +−2.476 eV/f.u., which means that 3.184 eV and 2.476 eV can be released during their synthesis processes from the +constituent elements. To further manifest their thermodynamic stability, we construct the convex hull diagram in +Fig. 3(b) with all of the synthesized Pd-Bi-Te compounds, whose crystal parameters and the calculated formation +energy have been tabulated in Table. S3 and Table. S4, respectively [63]. Fig. 3(b) shows that PdBi2Te5 and PdBi2Te4 +are 13 meV/atom and 61 meV/atom above the convex hull respectively. +Moreover, considering that metastable +PdBi2Te3, 52 meV and 3 meV higher than PdBi2Te5 and PdBi2Te4 as shown in Fig. 3(b), has been synthesized +in experiments [64, 65], we thus conclude that PdBi2Te5 and PdBi2Te4 could be synthesized in experiments. For +PdBi2Te5, we propose a synthetic route through the growth of Bi2Te3 and PdTe2 layer by layer. Our calculated +results reveal that bulk PdBi2Te5 is 59 meV/f.u. lower than the total energy of free standing Bi2Te3 and PdTe2 layers, +which strongly suggest that PdTe2 layer tends to deposit on Bi2Te3 to form new PdBi2Te5 crystal. To investigate +their dynamical stability, we calculate the phonon dispersion of PdBi2Te5 and PdBi2Te4, and plot them in Fig. 3(c) +and Fig. S3(a) [63]. There are 24 (21) phonon modes with fully real positive frequencies for PdBi2Te5 (PdBi2Te4), +which indicates that the rhombohedral unit cells are dynamically stable. Based on these results, we conclude that +PdBi2Te5 and PdBi2Te4 are relatively thermodynamically and dynamically stability in the rhombohedral structure, +and further experimental investigation is called for. +Then we study the electronic structures and topological properties of PdBi2Te5 and PdBi2Te4. Since PdBi2Te5 +and PbBi2Te4 exhibit similar electronic structures and non-trivial band topology, we only show the detailed density +of states (DOS), band structures, and topological surface states of PdBi2Te5 as an example in the main text, one +can check the results of PdBi2Te4 in Section III and Figs. S3 of the SM [63]. In Fig. 3(d), we plot the total and +projected DOS of PdBi2Te5, which gives rise to DOS(0 eV) = 1.91 states/eV at Fermi level, indicating its metallic +nature and the possibility of superconductivity. The projected DOS demonstrates that the states between −1 eV +and 1 eV are dominated by the p-orbitals of Te hybridized with d-orbitals from Pd and p-orbitals from Bi. The +hybridization is also manifested by the projected band structures shown in Fig. 3(e), which shows that two bands +with p-orbital components from Te or Bi cross the Fermi level and form several Fermi surfaces. Further detailed +orbital components analysis demonstrates that a continuous band gap (yellow region in Fig. 3(e)) and band inversion +exists between the nominal valence band and conduction band around the Fermi level, which implies that PdBi2Te5 +inherits the topological electronic nature of Bi2Te3 successfully. The nontrivial band topology can be confirmed by +calculating the Z2 topological invariant of time-reversal invariant insulators [66]. Given that rhombohedral PdBi2Te5 + +6 +possesses inversion symmetry and a continuous band gap, the Z2 topological invariant νTI = (1 − P)/2 is determined +by the product P of the parity of the wave function at the TRIM points [66]. Our calculated results give Z2 index +νTI = 1, confirming PdBi2Te5 is a Z2 topological metal. To visualize the bulk–boundary correspondence, we calculate +and plot the topological surface states on the (001) surface in Fig. 3(f). The surface states are similar to that of +Bi2Te3 [59, 60], the Dirac cone at the Γ point manifest approximately −6.3 meV below the Fermi level (the dashed +line in Fig. 3(f)). +To investigate the superconducting property of PdBi2Te5, we perform the electron-phonon calculations based on +density functional perturbation theory [67]. The calculated electron-phonon coupling constant λ = 0.43 and loga- +rithmic average phonon frequency ωlog = 97 cm−1, as tabulated in Table. S5 [63]. Furthermore, the superconducting +transition temperature (Tc) is estimated by using the reduced Allen-Dynes formula [68, 69]: +Tc = ωlog +1.20 exp +� +− +1.04(1 + λ) +λ − µ∗(1 + 0.62λ) +� +, +(4) +where µ∗ is the effective Coulomb potential. By adopting a typical µ∗ = 0.1, the Tc of PdBi2Te5 is estimated as +0.57 K. As comparison, the calculated λ and ωlog in PdTe2 is 0.52 and 112 cm−1, respectively. Accordingly, the +estimated Tc in PdTe2 is 1.59 K, which agrees well with the experimental Tc of 1.64 K [50–52]. These results clearly +demonstrate that the SC in PdTe2 is well inherited into the PdBi2Te5. +We now study the TSC property of the PdBi2Te5 slab by introducing the SC pairing and Zeeman splitting into the +topological surface states. Usually, the Zeeman splitting is applied by external magnetic field or in proximity to a FM +insulator, as illustrated in Fig. 4(a). As a concrete example, we use a 2D slab consisting of 10-OL PdBi2Te5, which is +thick enough to avoid the hybridization between top layer and bottom layer (Fig. 3(f)). Since PdBi2Te5 is an intrinsic +SC, the estimated s-wave superconducting gap ∆s = 1.0 meV is introduced globally for all 10-OLs. The out-of-plane +Zeeman splitting is applied only in the bottom layer consisting of one Bi2Te3 and one PdTe2, by assuming PdBi2Te5 +is the conventional SC from the parent type-I SC PdTe2 [50]. The chemical potential µ is set at the energy of surface +Dirac cone at the Γ point (about −6.3 meV below the Fermi level). In Fig. 4(b), we show the low energy spectrum +of HBdG at Γ point as a function of Zeeman splitting energy Mz, which manifest that the superconducting spectrum +is fully gaped with an energy gap of ∆ at Mz = 0. As Mz increases, the superconducting gap at the Γ point closes +and reopens. This behavior indicates that a topological phase transition happens at critical point Mz/∆ = 3.1, and +this 2D slab enters chiral TSC phase characterized by a nonzero BdG Chern number and chiral Majorana edge states +according to previous model simulations [38–40]. +To firmly verify its topological property and visualize the low energy physics in the TSC phase, we calculate the + +7 +superconducting energy spectrum at Mz=5 meV and ∆=1 meV in Fig. 4(c). The corresponding Wilson loop evolutions +for the occupied states are ploted in Fig. 4(d). The zoom-in image of Fig. 4(c) reveals that a full superconducting gap +is opened in the whole BZ, indicating that the system is a well defined chiral TSC. The Wilson loop evolution exhibits +a nontrivial chiral winding number 1, which directly confirms the superconducting BdG Chern number CBdG = 1. +Given that the experimental accessible magnetization energy usually reaches a few tens of meV, our results provide +a feasible guideline for discovery the chiral TSC phase in PdBi2Te5. +Finally, we would like to point out that the chiral TSC phase could also be realized in PdBi2Te4 as shown in +Fig. S4 [63], which exhibits a similar superconducting spectrum gap closing behavior with respect to Mz/∆ as in +PdBi2Te5. +In addition, we emphasize that our material design strategy can also be applied to search for other +SCTM candidates. For example, our calculated results demonstrate that AuBi2Te5 formed by SC AuTe2 interacting +into Bi2Te3 is also an ideal SCTM, whose detailed crystal structures, dynamic stability, electronic structures, and +topological surface states are discussed in Section V and Fig. S5 of SM [63]. +Therefore, we expect that SCTM +AuBi2Te5 could also be a TSC candidate. Last, we would like to point out that the program can be extended to study +many 2D topological superconducting heterostructure systems, such as magnetic TI/SC heterostructure, SC/FM +heterostructure and SC/TI/SC heterostructure. This will make it possible to determine the accurate parameters of +the TSC phase and simulate their TSC property in such systems from first-principles calculations. We expect our +program to be also useful for optimizing the experimental setup, stimulating the field of TSC study. +ACKNOWLEDGMENTS +This work is supported by the National Key Research and Development Program of China (2018YFA0307000), +and the National Natural Science Foundation of China (12274154, 11874022). B.L. is supported by the Alfred P. +Sloan Foundation, the National Science Foundation through Princeton University’s Materials Research Science and +Engineering Center DMR-2011750, and the National Science Foundation under award DMR-2141966. +METHOD +The first-principles calculations based on density functional theory are performed by the Vienna ab initio simulation +package [70, 71] with treating Perdew–Burke–Ernzerhof type of generalized gradient approximation as the exchange- +correlation potential [72]. The cutoff energy for wave function expansion is set as 450 eV, k-points grid 13×13×13 + +8 +is used for sampling the first BZ. All crystal structures are fully optimized until the force on each atom is less than +0.01 eV/˚A, and the SOC is included self-consistently. The electron-phonon coupling calculations with van der Waals +correction [73] are carried out in Quantum Espresso [74] based on the perturbation theory. A Hermite-Gaussian +smearing of 0.0025 Ryd is used for the electronic integration. The 8×8×8 k-mesh is used for the electron-phonon +coupling strength λ calculations, and the dynamical matrices are calculated on a 4×4×4 phonon-momentum grid. +Besides, a 2×2×2 supercell is built to calculate the phonon dispersion by using PHONOPY [75]. For the surface +calculation, the Wannier functions of Pd-d, Bi-p and Te-p orbitals are constructed by using WANNIER90 [76]. 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First principles phonon calculations in materials science. Scr. Mater. 108, 1–5 (2015). +[76] Mostofi, A. A. et al. +wannier90: +A tool for obtaining maximally-localised wannier functions. +Computer Physics +Communications 178, 685–699 (2008). + +13 +FIG. 1. The strategy to design idea SCTMs by intercalating the SC units into the TI. +FIG. 2. The generic flow chart to characterize the TSC properties from ab initio calculations. + +(a) +TI +SC +SCTM +(b)SCTM structure +Stability +Band Topology +Superconductivity +DFT + Wannier90 Hbulk +V +Slab Hamiltonian Hslab +Zeeman coupling H +SC pairing Hsc +BdG Hamiltonian HBdG +Energy dispersion +Wilson loop +TSC candidate14 +FIG. 3. (a) The side view of the crystal structures of PdBi2Te5, in which the octuple-layer (OL) unit of PdBi2Te5 formed by +the Bi2Te3 quintuple-layer and PdTe2 triple-layer is marked by a grey dashed rectangle. (b) Convex hull diagram for Pd-Bi-Te +system, the energy above convex hull is displayed by color-bar. (c) The phonon dispersion of PdBi2Te5. (d) The total DOS +and projected DOS of the Pd, Te, Bi atoms in PdBi2Te5, the zoom-in image shows the projected DOS near the Fermi level. +(e) The orbital-projected band structures of PdBi2Te5, where a continuous band gap around the Fermi level is marked by the +yellow shade. (f) The topological surface states on (001) surface of PdBi2Te5. + +(a) +b +(e) +Bi +Pd +(meV/atom) +Bi-p +Pd +60 +. +Te +e +0.5 +Bi +40 +PdBi2 +RdBi +Bi4Te3 +PdBi2Te3 +Energy +20 +e +Te +BiTe +Energy +PdBi2Te4 +BizTe3 +PdBi2Tes +OL +PdTe2 PdTe +c +-0.5 +F +04 +IZ +F +d) +20 +0.2 +Total +4 +- Pd d +5 +Te_p +0 +Bi p +0.0 +5 +-0.18 +4 +2 +0 +2 +4 +k +Energy (eV) +M15 +FIG. 4. (a) The schematic to realize chiral TSC in PdBi2T5 slab. (b) The low energy spectrum at the Γ point as the function of +Zeeman splitting energy Mz. (c) The superconducting spectrum along high symmetry paths with Mz = 5 meV and ∆ = 1 meV, +the zoom-in image shows the full gap in the whole BZ. (d) The Wilson loop spectrum for all occupied states of 4(c), which +manifest the superconducting BdG Chern number CBdG = 1 clearly. + +(b) 3 +(a) +CBdG=0 +CBdG=1 +2 +chiral Majorana states +(n-OL PdBizTes) +V +SCTM +0 +M +-1 +FM insulator +-2 +-3 +0 +2 +M/△ +8 +10 +(c) +(d) +40 +0.5 +20 +(meV) +0 +0(2元) +20 +Energy +40 +V +0 +-0.5 +K +M +-元 +元Supplementary Material: Topological superconductor candidates PdBi2Te4 and +PdBi2Te5 from a generic ab initio strategy +Aiyun Luo,1, ∗ Ying Li,1, ∗ Yi Qin,1, ∗ Jingnan Hu,1 Biao Lian,2 and Gang Xu1, 3, 4, † +1Wuhan National High Magnetic Field Center & School of Physics, +Huazhong University of Science and Technology, Wuhan 430074, China +2Department of Physics, Princeton University, Princeton, NJ 08544, United States of America +3Institute for Quantum Science and Engineering, +Huazhong University of Science and Technology, Wuhan 430074, China +4Wuhan Institute of Quantum Technology, Wuhan 430074, China +I. +The crystal structures of PdBi2Te5 and PdBi2Te4 +The crystal structures of PdBi2Te5 are shown in Fig. S1. Since both Bi2Te3(Fig. S1(a)) and PdTe2(Fig. S1(b)) +are Van der Waals materials, a naturally stable unit of PdBi2Te5 would be an octuple-layer (OL) block as shown in +Fig. S1(e). With different stacking sequences along c-direction, the crystal structures of AA-stacking, AB-stacking +and ABC-stacking are shown in Fig. S1(c), (d) and (e), respectively. As a result, the relaxed lattice parameters and +total energies of these structures are tabulated in Table. S1. Our calculated results demonstrate that the most stable +structure of PdBi2Te5 is the ABC-stacking rhombohedral unit cell as shown in Fig. S1(e), which is 73 meV/f.u. and +46 meV/f.u. lower than the AA and AB stacking structures. +The crystal structures of PdBi2Te4 are shown in Fig. S2, which is formed by the integration of Bi2Te3(Fig. S2(a)) +and PdTe(Fig. S2(b)). Like the construction of MnBi2Te4 [1, 2], a naturally stable unit of PdBi2Te4 would be a +septuple-layer (SL) block as shown by the grey dash rectangle in Fig. S2(c)-(e). Similar to the case of PdBi2Te5, the +relaxed lattice parameters and total energies of AA-stacking, AB-stacking, and ABC-stacking sequences of PdBi2Te4 +are tabulated in Table. S2. Our calculations demonstrate that SL-PdBi2Te4 favor the ABC stacking rhombohedral unit +cell as shown in Fig. S2(e), which is 73 meV/f.u. and 12 meV/f.u. lower than the AA(Fig. S2(c)) and AB(Fig. S2(d)) +stacking structures. +∗ These authors made equal contributions to this work. +† e-mail address: gangxu@hust.edu.cn +arXiv:2301.02844v1 [cond-mat.mtrl-sci] 7 Jan 2023 + +2 +FIG. S1. The crystal structures of different stacking sequences of PdBi2Te5. (a) A side view of the crystal structures of Bi2Te3, +the quintuple-layer block is marked by the green dashed rectangle. (b) PdTe2, the triple-layer block is marked by the purple +dashed rectangle. PdBi2Te5 for (c) AA stacking, (d) AB stacking, and (e) ABC stacking, respectively. A octuple-layer(OL) +block of PdBi2Te5 is marked by the grey dashed rectangle. + +(c) +(e) +a +OL +(d) +(b) +Te +Bi +Pd3 +FIG. S2. The crystal structures of different stacking sequences of PdBi2Te4. (a) Bi2Te3. (b) PdTe. PdBi2Te4 for (c) AA +stacking, (d) AB stacking, and (e) ABC stacking, respectively. A SL block of PdBi2Te4 is marked by the grey dashed rectangle. + +(c) +a +(e) +SL +(d) +(b) +Te +Bi +Pd4 +TABLE S1. The crystal parameters and total energies of different stacking sequences of PdBi2Te5 from our relaxed results. +Compound +Stacking +Space group +Lattice (˚A) +Atom +Wyckoff +Coordinate +Energy(eV/f.u.) +PdBi2Te5 +AA +P3m1 +a = b = 4.287 +Pd +1c +(0.667, 0.333, 0.013) +-30.385 +c = 16.325 +Bi1 +1a +0, 0, 0.363 +Bi2 +1a +0, 0, 0.623 +Te1 +1b +(0.333, 0.667, 0.090) +Te2 +1b +(0.333, 0.667, 0.492) +Te3 +1c +(0.667, 0.333, 0.251) +Te4 +1c +(0.667, 0.333, 0.936) +Te5 +1a +(0, 0, 0.733) +PdBi2Te5 +AB +P3m1 +a = b = 4.315 +Pd1 +1c +(0.667, 0.333, 0.483) +-30.412 +c = 30.987 +Pd2 +1a +(0, 0, 0) +Bi1 +1b +(0.333, 0.667, 0.792) +Bi2 +1c +(0.667, 0.333, 0.658) +Bi3 +1c +(0.667, 0.333, 0.308) +Bi4 +1a +(0, 0, 0.175) +Te1 +1b +(0.333, 0.667, 0.046) +Te2 +1b +(0.333, 0.667, 0.437) +Te3 +1b +(0.333, 0.667, 0.600) +Te4 +1b +(0.333, 0.667, 0.241) +Te5 +1c +(0.667, 0.333, 0.956) +Te6 +1c +(0.667, 0.333, 0.115) +Te7 +1c +(0.667, 0.333, 0.851) +Te8 +1a +(0, 0, 0.528) +Te9 +1a +(0, 0, 0.726) +Te10 +1a +(0, 0, 0.367) +PdBi2Te5 +ABC +R¯3m +a = b = 4.304 +Pd +3b +(0, 0, 0.5) +-30.458 +c = 46.391 +Bi +6c +(0, 0, 0.3788) +Te1 +3a +(0, 0, 0) +Te2 +6c +(0, 0, 0.1396) +Te3 +6c +(0, 0, 0.2488) + +5 +TABLE S2. The crystal parameters and total energies of different stacking sequences of PdBi2Te4 from our relaxed results. +Compound +Stacking +Space group +Lattice (˚A) +Atom +Wyckoff +Coordinate +Energy(eV/f.u.) +PdBi2Te4 +AA +P¯3m1 +a = b = 4.275 +Pd +1b +(0, 0, 0.5) +-26.776 +c = 13.035 +Bi +2d +(0.667, 0.333, 0.2325) +Te1 +2d +(0.333, 0.667, 0.397) +Te2 +2c +(0, 0, 0.09) +PdBi2Te4 +AB +P¯3m1 +a = b = 4.275 +Pd +2d +(0.333, 0.667, 0.75025) +-26.837 +c = 26.070 +Bi1 +2c +(0, 0, 0.61675) +Bi2 +2d +(0.667, 0.333, 0.88375) +Te1 +2d +(0.333, 0.667, 0.95375) +Te2 +2d +(0.333, 0.667, 0.54675) +Te3 +2c +(0, 0, 0.80175) +Te4 +2d +(0.667, 0.333, 0.88375) +PdBi2Te4 +ABC +R¯3m +a = b =4.296 +Pd +3a +(0, 0, 0) +-26.849 +c = 40.546 +Bi +6c +(0, 0, 0.4215) +Te1 +6c +(0, 0, 0.3000) +Te2 +6c +(0, 0, 0.8668) + +6 +II. +The detailed crystal parameters and formation energies of Pd-Bi-Te compounds for convex hull +The fundamental thermodynamic nature of Pd-Bi-Te compounds can be fully characterized in terms of the convex +hull diagram [3, 4], which is defined as formation energy versus composition diagram with the ground state at special +compositions. For all of the known Pd-Bi-Te compounds as tabulated in Table. S3, we calculate their formation +energy and tabular the results in Table. S4. Based on these results, we construct the convex hull diagram of Pd-Bi-Te +compounds in Fig.3(b) of the main text. The ternary compounds PdBi2Te3, PdBi2Te4 and Pd2Te5 are 65 meV/atom, +59 meV/atom and 13 meV/atom above the convex hull respectively, indicating that they are metastable phases with +respect to decomposition into the energetically favorable binary phases. Significantly, PdBi2Te3 above the convex +hull has already been synthesized in experiments, which is higher 52 meV/atom and 7 meV/atom than PdBi2Te5 and +PdBi2Te4. Therefore, we expect that PdBi2Te5 and PdBi2Te4 could be synthesized in the future. + +7 +TABLE S3. The space group, lattice constants and atomic coordinates of Pd-Bi-Te compounds used in the convex hull. +Compound +Space group +Lattice (˚A) +Atom +Wyckoff +Coordinate +Pd +Fm¯3m +a = b = c = 3.889 +Pd +4a +(0, 0, 0) +Bi +R¯3m +a = 4.523, c = 11.800 +Bi +3b +(0, 0, 0.227) +Te +P3121 +a = 4.445, c = 5.91 +Te +3a +(0.225, 0.0, 0.0) +PdBi +Cmc21 +a = 8.707 +Pd1 +8b +(0.274, 0.125, 0.053) +b = 7.203 +Pd2 +4a +(0, 0.108, 0.225) +c = 10.662 +Pd3 +4a +(0, 0.65, 0.225) +Bi1 +8b +(0.226, 0.375, 0.278) +Bi2 +4a +(0, 0.108, 0.5) +Bi3 +4a +(0, 0.35, 0) +PdBi2 +I4/mmm +a = b = 3.362 +Pd +2a +(0, 0, 0) +c = 12.983 +Bi +4e +(0, 0, 0.363) +PdTe +P63/mmc +a = b = 4.152 +Pd +2a +(0, 0, 0) +c = 5.671 +Te +2c +(0.333, 0.667, 0.25) +PdTe2 +P¯3m1 +a = b = 4.028 +Pd +1a +(0, 0, 0) +c = 5.118 +Te +2d +(0.333, 0.667, 0.25) +BiTe +P¯3m1 +a = b = 4.400 +Bi1 +2d +(0.333, 0.667, 0.291) +c = 24.000 +Bi2 +2d +(0.333, 0.667, 0.541) +Bi3 +2c +(0, 0, 0.126) +Te1 +2d +(0.333, 0.667, 0.056) +Te2 +2d +(0.333, 0.667, 0.789) +Te3 +2c +(0, 0, 0.362) +Bi2Te3 +R¯3m +a = b = 4.35 +Bi +6c +(0, 0, 0.600) +c = 30.36 +Te1 +6c +(0, 0, 0.790) +Te2 +3a +(0, 0, 0) +Bi4Te3 +R¯3m +a = b = 4.451 +Bi1 +6c +(0, 0, 0.146) +c = 41.888 +Bi2 +6c +(0, 0, 0.283) +Te1 +6c +(0, 0, 0.426) +Te2 +3a +(0, 0, 0) +PdBi2Te3 +R¯3m +a = b = 4.421 +Pd +3b +(0, 0, 0.5) +c = 30.337 +Bi +6c +(0, 0, 0.561) +Te1 +3a +(0, 0, 0) +Te2 +6c +(0, 0, 0.796) + +8 +TABLE S4. The total energy and formation energy of Pd-Bi-Te compounds used in the convex hull. +Compound +Energy(eV/f.u.) +Formation energy(eV/atom) +Pd +-5.161 +– +Bi +-3.804 +– +Te +-2.901 +– +PdTe +-9.022 +-0.480 +PdTe2 +-1.349 +-0.450 +PdBi +-9.565 +-0.3 +PdBi2 +-13.568 +-0.266 +BiTe +-7.346 +-0.320 +Bi2Te3 +-18.258 +-0.389 +Bi4Te3 +-25.979 +-0.294 +PdBi2Te3 +-23.038 +-0.261 +PdBi2Te4 +-26.849 +-0.354 +PdBi2Te5 +-30.458 +-0.398 + +9 +III. +The phonon spectrum, electronic structures and superconducting properties of PdBi2Te4 +In Fig. S3(a), we calculate and plot the phonon dispersion of PdBi2Te4. It clearly shows that there are 21 phonon +modes with fully real positive frequencies, indicating that the rhombohedral PdBi2Te4 are dynamically stable. Based +on the stable rhombohedral structure, we carry out the calculations with SOC of electronic properties of PdBi2Te4. +The total and projected density of states (DOS) are shown in Fig. S3(b), which shows that the total DOS at the Fermi +level is about of 3.95 states/eV, indicating its metallic nature and the probability of superconductivity. Fig. S3(c) +shows the projected band structures of PdBi2Te4, from which we can see that two bands with p-orbital components +from Te or Bi cross the Fermi level and a continuous direct gap (the yellow region in Fig. S3(c)) exists around the +Fermi level. Further detailed orbital components analysis demonstrates that band inversion is present between the +nominal valence band and conduction band, indicating the non-trivial band topology of bulk PdBi2Te4. In Fig. S3(d), +we calculate and plot the topological surface states on the (001) surface of PdBi2Te4, in which the Dirac surface states +at the Γ point manifest approximately −62 meV below the Fermi level, confirming PdBi2Te4 is a Z2 topological metal. +To investigate the superconducting properties of PdBi2Te4 and PdBi2Te5, we calculate their electron-phonon +coupling constant λ and logarithmic average phonon frequency ωlog as tabulated in Table. S5. As a comparison, the +λ and ωlog of parent SC PdTe and PdTe2 are also calculated. Moreover, the superconducting transition temperature +(Tc) is estimated by using the reduced Allen-Dynes formula as Eq.(4) in the main text. By adopting a typical µ∗ = +0.1, the Tc of PdBi2Te4 and PdBi2Te5 is estimated as 3.11 K and 0.57 K, respectively. Furthermore, the Tc of PdTe +and PdTe2 is estimated as 2.55 K and 1.59 K, which are consistent with the experimental results very well [5–9]. +These results clearly demonstrate that the SC in PdTe and PdTe2 is well inherited into the bulk of PdBi2Te4 and +PdBi2Te5. +TABLE S5. The calculated electron-phonon coupling strength λ, logarithmic average phonon frequency ωlog and the estimated +Tc (µ∗ = 0.1) of PdTe, PdTe2, PdBi2Te4 and PdBi2Te5. +Compound +λ +ωlog(cm−1) +Tc(K) +Texp +c +(K) +PdTe +0.61 +106 +2.55 +2.3 [5], 4.5 [6] +PdTe2 +0.52 +112 +1.59 +1.64 [7–9] +PdBi2Te4 +0.70 +89 +3.11 +– +PdBi2Te5 +0.43 +97 +0.57 +– + +10 +FIG. S3. The phonon dispersion and electronic properties of PdBi2Te4. (a) The phonon dispersion of PdBi2Te4. (b) The total +DOS and projected DOS of the Pd, Te, Bi atoms in PdBi2Te4, the zoom-in image shows the projected DOS near the Fermi +level. (c) The orbital-projected band structures of PdBi2Te4, a continuous band gap around the Fermi level is marked by the +yellow shade. (d) The topological surface states on (001) surface of PdBi2Te4. +IV. +The chiral TSC phase in PdBi2Te4 +In this section, we study the chiral TSC phase in 20-SL PdBi2Te4 by incorporating its topological and supercon- +ducting properties with Zeeman splitting. In this case, the chemical potential µ is set at the energy of surface Dirac +cone at the Γ point (about −62 meV below the Fermi level), the estimated s-wave superconducting gap ∆s = 1.0 meV +is introduced globally for all 20-SLs, and the out-of-plane Zeeman splitting is applied only in the bottom layer. In +Fig. S4(a), we show the low energy spectrum of HBdG at Γ point as a function of Zeeman splitting energy Mz, which +manifests that two superconducting bands cross each other at critical point Mz/∆ = 5. This behavior indicates that + +a +b +20 +Total +4 +(states/eV +-Pd d +Frequency (THz) +15 +-Te p +-Bi p +10 +Density +0 +F +-2 +0 +2 +4 +Energy (eV) +C +Bi-p +Te-p +0.5 +0.0 +Energy (eV) +-0.1 +-0.5 +-0.2 +F +M11 +a topological phase transition happens at the critical point. Fig. S4(b) shows the superconducting energy dispersion +with Mz=10 meV and ∆=1 meV. Its zoomed-in image, calculated in the whole BZ, directly reveals the full gap +characters of superconducting spectrum, indicating that the system is a well defined chiral TSC. To further verify +its nontrivial topological properties, the superconducting BdG Chern number for all occupied states is calculated by +the Wilson loop method as shown in Fig. S4(c). The spectrum of Wilson loop exhibits a non-trivial chiral winding +number 1, which directly confirms the superconducting BdG Chern number CBdG = 1. +FIG. S4. The chiral TSC phase in slab PdBi2T4. (a) The low energy spectrum at the Γ point as the function of Zeeman +splitting energy Mz, where ∆ = 1 meV. (b) The superconducting spectrum along high symmetry paths with Mz = 10 meV +and ∆ = 1 meV, the zoom-in image shows the full gap in the whole BZ. (c) The Wilson loop spectrum of all occupied states +in (b), which manifest the superconducting BdG Chern number CBdG = 1 clearly. +V. +The electronic structures and topological properties of AuBi2Te5 +The crystal structure of rhombohedral AuBi2Te5 is shown in Fig. S5(a), which is formed by the layered intercalation +of Bi2Te3 and AuTe2. In Fig. S5(b), we show the calculated phonon dispersion along the high symmetry lines of +AuBi2Te5, in which the most important observation is that all phonon modes have positive frequency throughout the +BZ, indicating the dynamical stability of the rhombohedral structure. In Fig. S5(c), we calculate and plot the total +and projected DOS of AuBi2Te5, which gives rise to DOS(0 eV)=5.41 states/eV at Fermi level, indicating its metallic +nature and the possibility of superconductivity. Fig. S5(d) shows the orbital-projected band structures of AuBi2Te5, +in which a continuous band gap around the Fermi level is marked by the yellow shades. Further detailed orbital +components analysis demonstrates that a band inversion exists between the nominal valence band and conduction + +(a) +(b) +(c) +40 +0.5 +CBdG=0 +CBdG=1 +2 +20 +(meV) +2元） +0 +A +-1 +-2 +0 +-3 +群 +V +-0.5 +0 +5 Mz/△ 10 +15 +K +M +Ky +-元 +元12 +band, which implies that AuBi2Te5 is a topological metal that inherits from parent Bi2Te3. In Fig. S5(e), we show +the calculated surface states of AuBi2Te5 in the (001) surface. It clearly shows that the topological surface states are +presented in the bulk gap, confirming AuBi2Te5 is a Z2 topological metal. +FIG. S5. The crystal structure, phonon dispersion, and electronic properties of AuBi2Te5. (a) The side view of the crystal +structure. (b) The phonon dispersion along high symmetry path. (d) The total DOS and projected DOS of the Au, Te, Bi +atoms. (e) The orbital-projected band structures, a continuous band gap around the Fermi level is marked by the yellow shade. +(f) The topological surface states on (001) surface. +[1] Lee, D. S. et al. Crystal structure, properties and nanostructuring of a new layered chalcogenide semiconductor, Bi2MnTe4. +CrystEngComm 15, 5532–5538 (2013). +[2] Zhang, D. et al. Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect. +Phys. Rev. Lett. 122, 206401 (2019). + +(a) +(b) +(c) +10 +- Total +Au +Density (states/eV) +Au d +8 +Te p +Bi +Bi p +6 +Te +OL +F +0 +(e) +Energy (ev) +(p) +0.2 +Bi-p +te +0.1 +(eV) +0.0 +Energy ( +Energy +0.1 +-0.2 +-0.3 +F +7 +K +M13 +[3] Sun, Y., Lv, J., Xie, Y., Liu, H. & Ma, Y. Route to a superconducting phase above room temperature in electron-doped +hydride compounds under high pressure. Phys. Rev. Lett. 123, 097001 (2019). +[4] Sharan, A. & Lany, S. Computational discovery of stable and metastable ternary oxynitrides. The Journal of Chemical +Physics 154, 234706 (2021). +[5] Matthias, B. T. Superconducting compounds of nonsuperconducting elements. Phys. Rev. 90, 487–487 (1953). +[6] Karki, A. B., Browne, D. A., Stadler, S., Li, J. & Jin, R. PdTe: a strongly coupled superconductor. Journal of Physics: +Condensed Matter 24, 055701 (2012). +[7] Kudo, K., Ishii, H. & Nohara, M. Composition-induced structural instability and strong-coupling superconductivity in +Au1−xPdxTe2. Phys. Rev. B 93, 140505 (2016). +[8] Leng, H., Paulsen, C., Huang, Y. K. & de Visser, A. Type-I superconductivity in the Dirac semimetal PdTe2. Phys. Rev. +B 96, 220506 (2017). +[9] Das, S. et al. Conventional superconductivity in the type-II Dirac semimetal PdTe2. Phys. Rev. B 97, 014523 (2018). + diff --git a/49E1T4oBgHgl3EQfAwK8/content/tmp_files/load_file.txt b/49E1T4oBgHgl3EQfAwK8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..90192d64bcc20e8ff8f45fe9dc64d2f184ed2b13 --- /dev/null +++ b/49E1T4oBgHgl3EQfAwK8/content/tmp_files/load_file.txt @@ -0,0 +1,1218 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf,len=1217 +page_content='Topological superconductor candidates PdBi2Te4 and PdBi2Te5 from a generic ab initio strategy Aiyun Luo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Ying Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Yi Qin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Jingnan Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1 Biao Lian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 and Gang Xu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' † 1Wuhan National High Magnetic Field Center & School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' NJ 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' United States of America 3Institute for Quantum Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China 4Wuhan Institute of Quantum Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China Superconducting topological metals (SCTMs) have recently emerged as a promising platform of topological superconductivity (TSC) and Majorana zero modes(MZMs) for quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Despite their importance in both fundamental research and applications, SCTMs are very rare in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In addition, some superconductors with topological electronic structures have been reported recently, but a feasible program to determine their TSC properties is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Here, we propose a new strategy to design SCTMs by intercalating the superconducting units into the topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A program that characterizes the superconducting BdG Chern number of 2D BdG Hamiltonian from ab initio calculations is also developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Following this strategy, PdBi2Te5 and PdBi2Te4 are found to be experimentally synthesizable and ideal SCTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Chiral TSC could be realized in such SCTMs by incorporating topological surface states with Zeeman effect, which can be realized by an external magnetic field or in proximity to ferromagnetic (FM) insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our strategy provides a new method for identifying the SCTMs and TSC candidates, and the program makes it possible to design and modulate the TSC candidates from ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' MAIN As one of the most important systems in both fundamental physics and topological quantum computation, topo- logical superconductors (TSCs) have attracted increasing interest for their ability to support Majorana fermions and anyons with non-Abelian statistics [1–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Currently, the search for TSCs candidates has been focused on two ∗ These authors made equal contributions to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' † e-mail address: gangxu@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='02844v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='mtrl-sci] 7 Jan 2023 2 experimental schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' One is the architectures by the combination of conventional superconductors with topological insulators (TIs) [18–20] or 1D nanowires [21, 22], but this approach brings high requirements for sample fabrication and interface engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The other route is to achieve TSCs in superconducting topological metals (SCTMs) that host both topological electronic structures at the Fermi level and superconductivity in one compound [23–36], in which the topological surface states are gapped by the “self-proximity effect” of bulk superconductivity, thus avoiding the complications of interface engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' This approach has successfully predicted the SCTM FeTe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='55Se0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='45 [24–26] and similar compounds of iron-based superconductors [27–30], owing to the favorable SC gap and non-trivial band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Beside the MZMs, 1D helical/chiral Majorana states have also been reported in domain walls of FeTe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='55Se0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='45 [37] and the magnetism-superconductor heterostructures [38–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' It is also proposed that the propagating chiral Majorana states can be applied to realize non-Abelian quantum gate operations, which could be 103 faster than the currently existing quantum computation schemes [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Encouraged by the success of Fe(Se,Te) [24–26], many topological materials that host both superconductivity and topological electronic structures are proposed [47–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' However, very rare experimental progress of TSC has been made in such SCTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' This is because, on the one hand, all of them are not the ideal SCTMs, whose band structures are too complicated, the topological surface states are usually buried in the bulk states and difficult to form the pairing required by TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' On the other hand, lacking a direct characterization of the TSC properties from ab initio calculations also hinders the effective experimental search in such materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Therefore, a general program that could calculate the TSC invariant from first-principles calculations is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In this work, we develop a program to characterize the superconducting topological invariant of 2D system from ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Besides, we also propose a new strategy to design ideal SCTMs by intercalating superconducting units into topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Following this strategy, PdBi2Te5 and PdBi2Te4 are found to be ideal SCTMs that host topological surface states at the Fermi level and superconductivity at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='57 K and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='11 K respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' By performing the superconducting energy spectrum and topological invariant calculations, we identify that chiral TSC could be realized in the slab of such SCTMs by introducing considerable Zeeman splitting on the topological surface states, which can be realized by an external magnetic field or in proximity to FM insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our strategy provides a new framework to enrich SCTMs and TSC candidates, and the program makes it possible to design and modulate the TSC system from ab initio calculations, which can also be extended to study the TSC properties in other system, such as magnetic TI/SC heterostructure, SC/FM heterostructure and SC/TI/SC heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Inspired by the construction of magnetic TI MnBi2Te4 [53, 54], we propose that the SCTMs can be designed by 3 intercalating the SC units into the TIs, as illustrated by the schematic of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As an ideal SCTM, the target crystal should be relatively stable in both energy and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' More importantly, it must inherit the topological electronic structures of the parent TI near the Fermi level, and also the superconductivity of the parent SC as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' However, the combination of topological electronic structures and SC does not result in TSC eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The realization of TSC generally requires a delicate modulation of many parameters, such as SC pairing, Zeeman splitting and chemical potential, et al [18–26, 38–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Thus, the ability to characterize the TSC invariant and determine the required parameters in real materials from the ab initio calculations is not only of theoretical significance, but also highly desirable in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Here we develop a program to simulate the superconducting properties and characterize its topological invariant in 2D slab system from ab initio calculations, in which the necessary ingredients to realize chiral TSC based on SCTM are included, such as bulk band structures, SC pairing, Zeeman splitting, Rashba spin-orbit coupling and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The workflow of this program is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' First, one should calculate the electronic structures of SCTM materials, and construct the localized Wannier functions that capture all electronic features from the first- principles calculations, referred as ˆHbulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The next step is to construct the slab Hamiltonian ˆHslab with open boundary condition along a certain direction [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In general, the spin-orbit coupling (SOC) and surface effect can be included automatically in ˆHslab through the first-principles calculations with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' So that the topological properties, such as the surface states and spin-texture, can be directly studied by using ˆHslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' On the other hand, one can also construct a slab Hamiltonian ˆHnsoc slab that excluded SOC from the non-SOC first-principles calculations, and add ˆHSOC and ˆHsurf manually to simulate the variable SOC and surface effect in the topological electronic states and TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In this work, we will adopt the former type of ˆHslab, in which only the intrinsic SOC of the real material is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' With adopting particle-hole transformation, the ˆHslab can be extended to BdG Hamiltonian ˆHBdG slab by adding SC pairing ˆHsc and Zeeman splitting ˆHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In the Nambu basis Φk = (ck,j,α,↑, ck,j,α,↓, c† −k,j,α,↑, c† −k,j,α,↓), where the cj,α,σ is the fermion operator denotes an electron at j layer with orbital α and spin σ(↑, ↓), the BdG Hamiltonian is formulated as: HBdG slab (k) = � � � Hslab(k) − µ ∆(k) ∆†(k) −H∗ slab(−k) + µ � � � + Mzτz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (1) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 1, µ is the chemical potential, which can be used to simulate the carriers doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∆(k) denotes the SC pairing matrices, which could be both singlet and triplet pairing form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For the conventional s-wave SC, ∆(k) is expressed as: ∆(k) = ∆s × Islab ⊗ (iσy ⊗ Iorb), (2) 4 where ∆s is the magnitude of intrinsic bulk s-wave pairing, σy is the Pauli matrix in spin space, Islab (Iorb) is an Nslab × Nslab (Norb × Norb) identity matrix that represents the number of slab layers (Wannier orbitals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' τz is the Pauli matrix in particle-hole space, Mz is the Zeeman splitting energy, and Hz = Mzτz is used to simulate the influence of the external magnetic field or the proximity effect of the FM insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Thus, Hz can be chosen to be applied for the whole slab or just few surface layers, depending on the slab thickness, strength of magnetic field, the type of the SC et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In principle, chiral TSC can be achieved by modulating the SC pairing, Zeeman splitting and chemical potential [38–45], which can be further revealed by calculating the superconducting energy spectrum and the superconducting topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In the gaped 2D superconducting system, the topological superconductors are classified by BdG Chern number in the absence of time-reversal symmetry [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Such superconducting topological invariants can be characterized by the evolution of Wilson loop [56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For the occupied quasiparticle states |uBdG n,k1,k2⟩, where k1 and k2 are momenta along two primitive vectors of the Brillouin zone (BZ), the Berry phase of the Wilson loop along k2 at a fixed k1 can be expressed as: W(k1) = −Im ln � i det M (i) k1 , (3) with the overlap matrix M (i) k1,mn = ⟨uBdG m,k1,k(i) 2 |uBdG n,k1,k(i+1) 2 ⟩, where k(i) 2 is the i-th discretized momenta along k2 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The winding number of W(k1) with respect to k1 is equal to the superconducting BdG Chern number CBdG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Next, we take TI Bi2Te3 [59, 60], SC PdTe [61, 62] and SC PdTe2 [50–52] as parent compounds to demonstrate that our SCTMs strategy is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Experimentally, Bi2Te3 (space group R¯3m, a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='35 ˚A, c = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='36 ˚A), PdTe (space group space group P63/mmc, a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='152 ˚A, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='671 ˚A, Tc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 K) and PdTe2 (space group P¯3m1, a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='03 ˚A, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='12 ˚A, Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='64 K) all adopt the triangle lattice and have very similar in-plane lattice constants, which makes it much easier to integrate them together to form a new compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' According to our calculations, the stable unit of PdBi2Te5 and PdBi2Te4 adopt octuple-layer (OL) structure and septuple-layer (SL) structure respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(a)(also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2 of Supplementary Material(SM) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' They both favor the ABC stacking along c-direction, and form the rhombohedral unit cell as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(a), which is 73 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (73 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' for PdBi2Te4) and 46 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (12 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' PdBi2Te4) lower than the AA and AB stacking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The detailed crystal parameters and total energy of different stacking PbBi2Te5 and PbBi2Te4 are tabulated in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1 and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2, respectively [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The formation energy of PdBi2Te5 and PdBi2Te4 are calculated to study their thermodynamic stability by using EPdmBinTel f = EPdmBinTel − mEPd − nEBi − lETe, with Ei(i=PdmBinTel, Pd, Bi and Te) means the calculated 5 total energy per formula in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The calculated EP dBi2T e5 f and EP dBi2T e4 f are −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='184 eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='476 eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', which means that 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='184 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='476 eV can be released during their synthesis processes from the constituent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To further manifest their thermodynamic stability, we construct the convex hull diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(b) with all of the synthesized Pd-Bi-Te compounds, whose crystal parameters and the calculated formation energy have been tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3 and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4, respectively [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(b) shows that PdBi2Te5 and PdBi2Te4 are 13 meV/atom and 61 meV/atom above the convex hull respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Moreover, considering that metastable PdBi2Te3, 52 meV and 3 meV higher than PdBi2Te5 and PdBi2Te4 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(b), has been synthesized in experiments [64, 65], we thus conclude that PdBi2Te5 and PdBi2Te4 could be synthesized in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For PdBi2Te5, we propose a synthetic route through the growth of Bi2Te3 and PdTe2 layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our calculated results reveal that bulk PdBi2Te5 is 59 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' lower than the total energy of free standing Bi2Te3 and PdTe2 layers, which strongly suggest that PdTe2 layer tends to deposit on Bi2Te3 to form new PdBi2Te5 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To investigate their dynamical stability, we calculate the phonon dispersion of PdBi2Te5 and PdBi2Te4, and plot them in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(a) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' There are 24 (21) phonon modes with fully real positive frequencies for PdBi2Te5 (PdBi2Te4), which indicates that the rhombohedral unit cells are dynamically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Based on these results, we conclude that PdBi2Te5 and PdBi2Te4 are relatively thermodynamically and dynamically stability in the rhombohedral structure, and further experimental investigation is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Then we study the electronic structures and topological properties of PdBi2Te5 and PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Since PdBi2Te5 and PbBi2Te4 exhibit similar electronic structures and non-trivial band topology, we only show the detailed density of states (DOS), band structures, and topological surface states of PdBi2Te5 as an example in the main text, one can check the results of PdBi2Te4 in Section III and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3 of the SM [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(d), we plot the total and projected DOS of PdBi2Te5, which gives rise to DOS(0 eV) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='91 states/eV at Fermi level, indicating its metallic nature and the possibility of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The projected DOS demonstrates that the states between −1 eV and 1 eV are dominated by the p-orbitals of Te hybridized with d-orbitals from Pd and p-orbitals from Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The hybridization is also manifested by the projected band structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(e), which shows that two bands with p-orbital components from Te or Bi cross the Fermi level and form several Fermi surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Further detailed orbital components analysis demonstrates that a continuous band gap (yellow region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(e)) and band inversion exists between the nominal valence band and conduction band around the Fermi level, which implies that PdBi2Te5 inherits the topological electronic nature of Bi2Te3 successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The nontrivial band topology can be confirmed by calculating the Z2 topological invariant of time-reversal invariant insulators [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Given that rhombohedral PdBi2Te5 6 possesses inversion symmetry and a continuous band gap, the Z2 topological invariant νTI = (1 − P)/2 is determined by the product P of the parity of the wave function at the TRIM points [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our calculated results give Z2 index νTI = 1, confirming PdBi2Te5 is a Z2 topological metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To visualize the bulk–boundary correspondence, we calculate and plot the topological surface states on the (001) surface in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The surface states are similar to that of Bi2Te3 [59, 60], the Dirac cone at the Γ point manifest approximately −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 meV below the Fermi level (the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To investigate the superconducting property of PdBi2Te5, we perform the electron-phonon calculations based on density functional perturbation theory [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The calculated electron-phonon coupling constant λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='43 and loga- rithmic average phonon frequency ωlog = 97 cm−1, as tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Furthermore, the superconducting transition temperature (Tc) is estimated by using the reduced Allen-Dynes formula [68, 69]: Tc = ωlog 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='20 exp � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='04(1 + λ) λ − µ∗(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='62λ) � , (4) where µ∗ is the effective Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' By adopting a typical µ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1, the Tc of PdBi2Te5 is estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='57 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As comparison, the calculated λ and ωlog in PdTe2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='52 and 112 cm−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Accordingly, the estimated Tc in PdTe2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='59 K, which agrees well with the experimental Tc of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='64 K [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' These results clearly demonstrate that the SC in PdTe2 is well inherited into the PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' We now study the TSC property of the PdBi2Te5 slab by introducing the SC pairing and Zeeman splitting into the topological surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Usually, the Zeeman splitting is applied by external magnetic field or in proximity to a FM insulator, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As a concrete example, we use a 2D slab consisting of 10-OL PdBi2Te5, which is thick enough to avoid the hybridization between top layer and bottom layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Since PdBi2Te5 is an intrinsic SC, the estimated s-wave superconducting gap ∆s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0 meV is introduced globally for all 10-OLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The out-of-plane Zeeman splitting is applied only in the bottom layer consisting of one Bi2Te3 and one PdTe2, by assuming PdBi2Te5 is the conventional SC from the parent type-I SC PdTe2 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The chemical potential µ is set at the energy of surface Dirac cone at the Γ point (about −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 meV below the Fermi level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4(b), we show the low energy spectrum of HBdG at Γ point as a function of Zeeman splitting energy Mz, which manifest that the superconducting spectrum is fully gaped with an energy gap of ∆ at Mz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As Mz increases, the superconducting gap at the Γ point closes and reopens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' This behavior indicates that a topological phase transition happens at critical point Mz/∆ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1, and this 2D slab enters chiral TSC phase characterized by a nonzero BdG Chern number and chiral Majorana edge states according to previous model simulations [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To firmly verify its topological property and visualize the low energy physics in the TSC phase, we calculate the 7 superconducting energy spectrum at Mz=5 meV and ∆=1 meV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The corresponding Wilson loop evolutions for the occupied states are ploted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The zoom-in image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4(c) reveals that a full superconducting gap is opened in the whole BZ, indicating that the system is a well defined chiral TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The Wilson loop evolution exhibits a nontrivial chiral winding number 1, which directly confirms the superconducting BdG Chern number CBdG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Given that the experimental accessible magnetization energy usually reaches a few tens of meV, our results provide a feasible guideline for discovery the chiral TSC phase in PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Finally, we would like to point out that the chiral TSC phase could also be realized in PdBi2Te4 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4 [63], which exhibits a similar superconducting spectrum gap closing behavior with respect to Mz/∆ as in PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In addition, we emphasize that our material design strategy can also be applied to search for other SCTM candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For example, our calculated results demonstrate that AuBi2Te5 formed by SC AuTe2 interacting into Bi2Te3 is also an ideal SCTM, whose detailed crystal structures, dynamic stability, electronic structures, and topological surface states are discussed in Section V and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5 of SM [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Therefore, we expect that SCTM AuBi2Te5 could also be a TSC candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Last, we would like to point out that the program can be extended to study many 2D topological superconducting heterostructure systems, such as magnetic TI/SC heterostructure, SC/FM heterostructure and SC/TI/SC heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' This will make it possible to determine the accurate parameters of the TSC phase and simulate their TSC property in such systems from first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' We expect our program to be also useful for optimizing the experimental setup, stimulating the field of TSC study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported by the National Key Research and Development Program of China (2018YFA0307000), and the National Natural Science Foundation of China (12274154, 11874022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' is supported by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Sloan Foundation, the National Science Foundation through Princeton University’s Materials Research Science and Engineering Center DMR-2011750, and the National Science Foundation under award DMR-2141966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' METHOD The first-principles calculations based on density functional theory are performed by the Vienna ab initio simulation package [70, 71] with treating Perdew–Burke–Ernzerhof type of generalized gradient approximation as the exchange- correlation potential [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The cutoff energy for wave function expansion is set as 450 eV, k-points grid 13×13×13 8 is used for sampling the first BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' All crystal structures are fully optimized until the force on each atom is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='01 eV/˚A, and the SOC is included self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The electron-phonon coupling calculations with van der Waals correction [73] are carried out in Quantum Espresso [74] based on the perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A Hermite-Gaussian smearing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0025 Ryd is used for the electronic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The 8×8×8 k-mesh is used for the electron-phonon coupling strength λ calculations, and the dynamical matrices are calculated on a 4×4×4 phonon-momentum grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Besides, a 2×2×2 supercell is built to calculate the phonon dispersion by using PHONOPY [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For the surface calculation, the Wannier functions of Pd-d, Bi-p and Te-p orbitals are constructed by using WANNIER90 [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A slab consisting of 10-OL PdBi2Te5 layers with a bottom surface terminated as the Bi2Te3 layer is implemented in WannierTools [55], which is further used to calculate the electronic surface states, the superconducting spectrum, and the superconducting BdG Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [1] Nayak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Simon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Stern, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Freedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Das Sarma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Non-abelian anyons and topological quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 80, 1083–1159 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [2] Sarma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Freedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Nayak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Majorana zero modes and topological quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' npj Quantum Information 1, 1–13 (2015).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Tanaka, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' First principles phonon calculations in materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 108, 1–5 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [76] Mostofi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' wannier90: A tool for obtaining maximally-localised wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Computer Physics Communications 178, 685–699 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The strategy to design idea SCTMs by intercalating the SC units into the TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The generic flow chart to characterize the TSC properties from ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) TI SC SCTM (b)SCTM structure Stability Band Topology Superconductivity DFT + Wannier90 Hbulk V Slab Hamiltonian Hslab Zeeman coupling H SC pairing Hsc BdG Hamiltonian HBdG Energy dispersion Wilson loop TSC candidate14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) The side view of the crystal structures of PdBi2Te5, in which the octuple-layer (OL) unit of PdBi2Te5 formed by the Bi2Te3 quintuple-layer and PdTe2 triple-layer is marked by a grey dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) Convex hull diagram for Pd-Bi-Te system, the energy above convex hull is displayed by color-bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) The phonon dispersion of PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (d) The total DOS and projected DOS of the Pd, Te, Bi atoms in PdBi2Te5, the zoom-in image shows the projected DOS near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (e) The orbital-projected band structures of PdBi2Te5, where a continuous band gap around the Fermi level is marked by the yellow shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (f) The topological surface states on (001) surface of PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) b (e) Bi Pd (meV/atom) Bi-p Pd 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Te e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 Bi 40 PdBi2 RdBi Bi4Te3 PdBi2Te3 Energy 20 e Te BiTe Energy PdBi2Te4 BizTe3 PdBi2Tes OL PdTe2 PdTe c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 F 04 IZ F d) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 Total 4 Pd d 5 Te_p 0 Bi p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='18 4 2 0 2 4 k Energy (eV) M15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) The schematic to realize chiral TSC in PdBi2T5 slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) The low energy spectrum at the Γ point as the function of Zeeman splitting energy Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) The superconducting spectrum along high symmetry paths with Mz = 5 meV and ∆ = 1 meV, the zoom-in image shows the full gap in the whole BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (d) The Wilson loop spectrum for all occupied states of 4(c), which manifest the superconducting BdG Chern number CBdG = 1 clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) 3 (a) CBdG=0 CBdG=1 2 chiral Majorana states (n-OL PdBizTes) V SCTM 0 M 1 FM insulator 2 3 0 2 M/△ 8 10 (c) (d) 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 20 (meV) 0 0(2元) 20 Energy 40 V 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 K M 元 元Supplementary Material: Topological superconductor candidates PdBi2Te4 and PdBi2Te5 from a generic ab initio strategy Aiyun Luo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Ying Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Yi Qin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ Jingnan Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1 Biao Lian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 and Gang Xu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' † 1Wuhan National High Magnetic Field Center & School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' NJ 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' United States of America 3Institute for Quantum Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China 4Wuhan Institute of Quantum Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Wuhan 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' China I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal structures of PdBi2Te5 and PdBi2Te4 The crystal structures of PdBi2Te5 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Since both Bi2Te3(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1(a)) and PdTe2(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1(b)) are Van der Waals materials, a naturally stable unit of PdBi2Te5 would be an octuple-layer (OL) block as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' With different stacking sequences along c-direction, the crystal structures of AA-stacking, AB-stacking and ABC-stacking are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1(c), (d) and (e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As a result, the relaxed lattice parameters and total energies of these structures are tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our calculated results demonstrate that the most stable structure of PdBi2Te5 is the ABC-stacking rhombohedral unit cell as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1(e), which is 73 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' and 46 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' lower than the AA and AB stacking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal structures of PdBi2Te4 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2, which is formed by the integration of Bi2Te3(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(a)) and PdTe(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Like the construction of MnBi2Te4 [1, 2], a naturally stable unit of PdBi2Te4 would be a septuple-layer (SL) block as shown by the grey dash rectangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(c)-(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Similar to the case of PdBi2Te5, the relaxed lattice parameters and total energies of AA-stacking, AB-stacking, and ABC-stacking sequences of PdBi2Te4 are tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Our calculations demonstrate that SL-PdBi2Te4 favor the ABC stacking rhombohedral unit cell as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(e), which is 73 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' and 12 meV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' lower than the AA(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(c)) and AB(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2(d)) stacking structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' ∗ These authors made equal contributions to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' † e-mail address: gangxu@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='02844v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='mtrl-sci] 7 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal structures of different stacking sequences of PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) A side view of the crystal structures of Bi2Te3, the quintuple-layer block is marked by the green dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) PdTe2, the triple-layer block is marked by the purple dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' PdBi2Te5 for (c) AA stacking, (d) AB stacking, and (e) ABC stacking, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A octuple-layer(OL) block of PdBi2Te5 is marked by the grey dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) (e) a OL (d) (b) Te Bi Pd3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal structures of different stacking sequences of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) Bi2Te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' PdBi2Te4 for (c) AA stacking, (d) AB stacking, and (e) ABC stacking, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A SL block of PdBi2Te4 is marked by the grey dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) a (e) SL (d) (b) Te Bi Pd4 TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal parameters and total energies of different stacking sequences of PdBi2Te5 from our relaxed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Compound Stacking Space group Lattice (˚A) Atom Wyckoff Coordinate Energy(eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=') PdBi2Te5 AA P3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='287 Pd 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='013) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='385 c = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='325 Bi1 1a 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='363 Bi2 1a 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='623 Te1 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='090) Te2 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='492) Te3 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='251) Te4 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='936) Te5 1a (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='733) PdBi2Te5 AB P3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='315 Pd1 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='483) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='412 c = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='987 Pd2 1a (0, 0, 0) Bi1 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='792) Bi2 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='658) Bi3 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='308) Bi4 1a (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='175) Te1 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='046) Te2 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='437) Te3 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='600) Te4 1b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='241) Te5 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='956) Te6 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='115) Te7 1c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='851) Te8 1a (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='528) Te9 1a (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='726) Te10 1a (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='367) PdBi2Te5 ABC R¯3m a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='304 Pd 3b (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='458 c = 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='391 Bi 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3788) Te1 3a (0, 0, 0) Te2 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1396) Te3 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2488) 5 TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal parameters and total energies of different stacking sequences of PdBi2Te4 from our relaxed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Compound Stacking Space group Lattice (˚A) Atom Wyckoff Coordinate Energy(eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=') PdBi2Te4 AA P¯3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='275 Pd 1b (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='776 c = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='035 Bi 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2325) Te1 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='397) Te2 2c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='09) PdBi2Te4 AB P¯3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='275 Pd 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='75025) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='837 c = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='070 Bi1 2c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='61675) Bi2 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='88375) Te1 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='95375) Te2 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='54675) Te3 2c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='80175) Te4 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='88375) PdBi2Te4 ABC R¯3m a = b =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='296 Pd 3a (0, 0, 0) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='849 c = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='546 Bi 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='4215) Te1 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3000) Te2 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='8668) 6 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The detailed crystal parameters and formation energies of Pd-Bi-Te compounds for convex hull The fundamental thermodynamic nature of Pd-Bi-Te compounds can be fully characterized in terms of the convex hull diagram [3, 4], which is defined as formation energy versus composition diagram with the ground state at special compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' For all of the known Pd-Bi-Te compounds as tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3, we calculate their formation energy and tabular the results in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Based on these results, we construct the convex hull diagram of Pd-Bi-Te compounds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The ternary compounds PdBi2Te3, PdBi2Te4 and Pd2Te5 are 65 meV/atom, 59 meV/atom and 13 meV/atom above the convex hull respectively, indicating that they are metastable phases with respect to decomposition into the energetically favorable binary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Significantly, PdBi2Te3 above the convex hull has already been synthesized in experiments, which is higher 52 meV/atom and 7 meV/atom than PdBi2Te5 and PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Therefore, we expect that PdBi2Te5 and PdBi2Te4 could be synthesized in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 7 TABLE S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The space group, lattice constants and atomic coordinates of Pd-Bi-Te compounds used in the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Compound Space group Lattice (˚A) Atom Wyckoff Coordinate Pd Fm¯3m a = b = c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='889 Pd 4a (0, 0, 0) Bi R¯3m a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='523, c = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='800 Bi 3b (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='227) Te P3121 a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='445, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='91 Te 3a (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='225, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0) PdBi Cmc21 a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='707 Pd1 8b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='274, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='053) b = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='203 Pd2 4a (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='108, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='225) c = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='662 Pd3 4a (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='225) Bi1 8b (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='226, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='278) Bi2 4a (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='108, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5) Bi3 4a (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='35, 0) PdBi2 I4/mmm a = b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='362 Pd 2a (0, 0, 0) c = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='983 Bi 4e (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='363) PdTe P63/mmc a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='152 Pd 2a (0, 0, 0) c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='671 Te 2c (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='25) PdTe2 P¯3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='028 Pd 1a (0, 0, 0) c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='118 Te 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='25) BiTe P¯3m1 a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='400 Bi1 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='291) c = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='000 Bi2 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='541) Bi3 2c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='126) Te1 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='056) Te2 2d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='789) Te3 2c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='362) Bi2Te3 R¯3m a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='35 Bi 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='600) c = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='36 Te1 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='790) Te2 3a (0, 0, 0) Bi4Te3 R¯3m a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='451 Bi1 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='146) c = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='888 Bi2 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='283) Te1 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='426) Te2 3a (0, 0, 0) PdBi2Te3 R¯3m a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='421 Pd 3b (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5) c = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='337 Bi 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='561) Te1 3a (0, 0, 0) Te2 6c (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='796) 8 TABLE S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The total energy and formation energy of Pd-Bi-Te compounds used in the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Compound Energy(eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=') Formation energy(eV/atom) Pd 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='161 – Bi 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='804 – Te 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='901 – PdTe 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='480 PdTe2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='450 PdBi 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 PdBi2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='266 BiTe 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='320 Bi2Te3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='389 Bi4Te3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='294 PdBi2Te3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='261 PdBi2Te4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='354 PdBi2Te5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='398 9 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The phonon spectrum, electronic structures and superconducting properties of PdBi2Te4 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(a), we calculate and plot the phonon dispersion of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' It clearly shows that there are 21 phonon modes with fully real positive frequencies, indicating that the rhombohedral PdBi2Te4 are dynamically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Based on the stable rhombohedral structure, we carry out the calculations with SOC of electronic properties of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The total and projected density of states (DOS) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(b), which shows that the total DOS at the Fermi level is about of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='95 states/eV, indicating its metallic nature and the probability of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(c) shows the projected band structures of PdBi2Te4, from which we can see that two bands with p-orbital components from Te or Bi cross the Fermi level and a continuous direct gap (the yellow region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(c)) exists around the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Further detailed orbital components analysis demonstrates that band inversion is present between the nominal valence band and conduction band, indicating the non-trivial band topology of bulk PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3(d), we calculate and plot the topological surface states on the (001) surface of PdBi2Te4, in which the Dirac surface states at the Γ point manifest approximately −62 meV below the Fermi level, confirming PdBi2Te4 is a Z2 topological metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To investigate the superconducting properties of PdBi2Te4 and PdBi2Te5, we calculate their electron-phonon coupling constant λ and logarithmic average phonon frequency ωlog as tabulated in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' As a comparison, the λ and ωlog of parent SC PdTe and PdTe2 are also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Moreover, the superconducting transition temperature (Tc) is estimated by using the reduced Allen-Dynes formula as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (4) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' By adopting a typical µ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1, the Tc of PdBi2Te4 and PdBi2Te5 is estimated as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='11 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='57 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Furthermore, the Tc of PdTe and PdTe2 is estimated as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='55 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='59 K, which are consistent with the experimental results very well [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' These results clearly demonstrate that the SC in PdTe and PdTe2 is well inherited into the bulk of PdBi2Te4 and PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' TABLE S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The calculated electron-phonon coupling strength λ, logarithmic average phonon frequency ωlog and the estimated Tc (µ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1) of PdTe, PdTe2, PdBi2Te4 and PdBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Compound λ ωlog(cm−1) Tc(K) Texp c (K) PdTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='61 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 [5], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 [6] PdTe2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='52 112 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='64 [7–9] PdBi2Te4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='70 89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='11 – PdBi2Te5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='43 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='57 – 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The phonon dispersion and electronic properties of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) The phonon dispersion of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) The total DOS and projected DOS of the Pd, Te, Bi atoms in PdBi2Te4, the zoom-in image shows the projected DOS near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) The orbital-projected band structures of PdBi2Te4, a continuous band gap around the Fermi level is marked by the yellow shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (d) The topological surface states on (001) surface of PdBi2Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The chiral TSC phase in PdBi2Te4 In this section, we study the chiral TSC phase in 20-SL PdBi2Te4 by incorporating its topological and supercon- ducting properties with Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In this case, the chemical potential µ is set at the energy of surface Dirac cone at the Γ point (about −62 meV below the Fermi level), the estimated s-wave superconducting gap ∆s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0 meV is introduced globally for all 20-SLs, and the out-of-plane Zeeman splitting is applied only in the bottom layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4(a), we show the low energy spectrum of HBdG at Γ point as a function of Zeeman splitting energy Mz, which manifests that two superconducting bands cross each other at critical point Mz/∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' This behavior indicates that a b 20 Total 4 (states/eV Pd d Frequency (THz) 15 Te p Bi p 10 Density 0 F 2 0 2 4 Energy (eV) C Bi-p Te-p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 F M11 a topological phase transition happens at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4(b) shows the superconducting energy dispersion with Mz=10 meV and ∆=1 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Its zoomed-in image, calculated in the whole BZ, directly reveals the full gap characters of superconducting spectrum, indicating that the system is a well defined chiral TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' To further verify its nontrivial topological properties, the superconducting BdG Chern number for all occupied states is calculated by the Wilson loop method as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The spectrum of Wilson loop exhibits a non-trivial chiral winding number 1, which directly confirms the superconducting BdG Chern number CBdG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The chiral TSC phase in slab PdBi2T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) The low energy spectrum at the Γ point as the function of Zeeman splitting energy Mz, where ∆ = 1 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) The superconducting spectrum along high symmetry paths with Mz = 10 meV and ∆ = 1 meV, the zoom-in image shows the full gap in the whole BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (c) The Wilson loop spectrum of all occupied states in (b), which manifest the superconducting BdG Chern number CBdG = 1 clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The electronic structures and topological properties of AuBi2Te5 The crystal structure of rhombohedral AuBi2Te5 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5(a), which is formed by the layered intercalation of Bi2Te3 and AuTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5(b), we show the calculated phonon dispersion along the high symmetry lines of AuBi2Te5, in which the most important observation is that all phonon modes have positive frequency throughout the BZ, indicating the dynamical stability of the rhombohedral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5(c), we calculate and plot the total and projected DOS of AuBi2Te5, which gives rise to DOS(0 eV)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='41 states/eV at Fermi level, indicating its metallic nature and the possibility of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5(d) shows the orbital-projected band structures of AuBi2Te5, in which a continuous band gap around the Fermi level is marked by the yellow shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Further detailed orbital components analysis demonstrates that a band inversion exists between the nominal valence band and conduction (a) (b) (c) 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 CBdG=0 CBdG=1 2 20 (meV) 2元） 0 A 1 2 0 3 群 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='5 0 5 Mz/△ 10 15 K M Ky 元 元12 band, which implies that AuBi2Te5 is a topological metal that inherits from parent Bi2Te3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5(e), we show the calculated surface states of AuBi2Te5 in the (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' It clearly shows that the topological surface states are presented in the bulk gap, confirming AuBi2Te5 is a Z2 topological metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' The crystal structure, phonon dispersion, and electronic properties of AuBi2Te5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) The side view of the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (b) The phonon dispersion along high symmetry path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (d) The total DOS and projected DOS of the Au, Te, Bi atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (e) The orbital-projected band structures, a continuous band gap around the Fermi level is marked by the yellow shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (f) The topological surface states on (001) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [1] Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Crystal structure, properties and nanostructuring of a new layered chalcogenide semiconductor, Bi2MnTe4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' CrystEngComm 15, 5532–5538 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [2] Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 122, 206401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' (a) (b) (c) 10 Total Au Density (states/eV) Au d 8 Te p Bi Bi p 6 Te OL F 0 (e) Energy (ev) (p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 Bi-p te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='0 Energy ( Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content='3 F 7 K M13 [3] Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Lv, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Route to a superconducting phase above room temperature in electron-doped hydride compounds under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 123, 097001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [4] Sharan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Lany, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' 90, 487–487 (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [6] Karki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Browne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & Nohara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Composition-induced structural instability and strong-coupling superconductivity in Au1−xPdxTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' B 93, 140505 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' [8] Leng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Paulsen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=', Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' & de Visser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Type-I superconductivity in the Dirac semimetal PdTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} +page_content=' B 97, 014523 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfAwK8/content/2301.02844v1.pdf'} diff --git a/4dE1T4oBgHgl3EQfAwIb/vector_store/index.pkl b/4dE1T4oBgHgl3EQfAwIb/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..46d847032983e206f6e23a58a6fdde988aa068d9 --- /dev/null +++ b/4dE1T4oBgHgl3EQfAwIb/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b2dee8b78416ec9151257cc89a7348d9525523ffa032967542217d8f56469695 +size 75773 diff --git a/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/2301.02850v1.pdf.txt b/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/2301.02850v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7d50d8291040781f1d0616eb24704d94da5a7b2 --- /dev/null +++ b/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/2301.02850v1.pdf.txt @@ -0,0 +1,1565 @@ +Programmable wave-based analog computing +machine: a metastructure that designs metastructures +Dimitrios C. Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗ +1Department of Electrical and Systems Engineering, +School of Engineering and Applied Sciences, +University of Pennsylvania, Philadelphia, 19104, U.S.A. +†These authors contributed equally to this work. +‡Present address: Meta Materials Inc. (Europe), +Ap. Pavlou 10A, Marousi, 15123, Greece. +∗To whom correspondence should be addressed; e-mail: engheta@seas.upenn.edu +January 10, 2023 +Abstract: The ability to perform mathematical computations using metastruc- +tures is an emergent paradigm that carries the potential of wave-based analog +computing to the realm of near-speed-of-light, low-loss, compact devices. We +theoretically introduce and experimentally verify the concept of a reconfig- +urable metastructure that performs analog complex mathematical computa- +tions using electromagnetic waves. Reconfigurable, RF-based components en- +dow our device with the ability to perform stationary and non-stationary iter- +ative algorithms. After demonstrating matrix inversion (stationary problem), +we use the machine to tackle two major non-stationary problems: root finding +with Newton’s method and inverse design (constrained optimization) via the +Lagrange multiplier method. The platform enables possible avenues for wave- +1 +arXiv:2301.02850v1 [physics.app-ph] 22 Dec 2022 + +based, analog computations for general linear algebraic problems and beyond +in compact, ultrafast, and parallelized ways. +One-Sentence Summary: +A reconfigurable wave-based analog computing metastructure that +can inverse-design a metastructure. +Calculators of various kinds have emerged by forging numerical algorithms with corre- +sponding technological platforms. While the algorithms describe the mathematical paths on +how solutions to problems can be found, the platforms are responsible for the transliteration of +this abstract path into measurable quantities. The algorithms, the platforms, and their fusion +define such systems’ features and limitations. Following the ever-growing demand for ultrafast, +compact, low/near-zero-power, and integrable cyber-physical devices for mathematical compu- +tations, it is organic that significant research efforts focus on making these numerical systems +as optimal and efficient as possible. +This quest led to the exploration and development of unconventional analog computing sys- +tems that exploit electromagnetic waves to deliver parallelized, ultrafast, compact, low-power +computations (1–4). The two main categories in this domain involve systems that utilize free- +space (scattering) elements (e.g. lenses (3)), and waveguides (e.g. photonic systems (5, 6) +and phased arrays (7)). Sufficient free space propagation can act as dense matrix multiplica- +tion (8). Realized with traditional optics, this results in bulky devices (3,9), while metasurfaces +can be more compact (10–12). However, in both there can be major bottlenecks regarding +photonic and electronic integration. +Waveguiding systems offer more mature solutions for +integrable and reconfigurable devices, at the expense of much larger footprints compared to +their free-space counterparts. In all cases, their main challenge is reconfigurability since its +implementation requires some form of a-priori mathematical calculations. For instance, meta- +surfaces/complex media (13, 14) requires optimization, and photonic meshes require operator +2 + +decomposition (15–18). +In terms of their mathematical abilities, the above examples demonstrate wave-based analog +computing with functionalities such as integration/differentiation in space (19–22) and time +(23), matrix-vector multiplication (24), emulating equations through physical phenomena (25, +26), or acting as platforms for neural network functionalities (3,6,27,28). The intersection with +the metamaterial paradigm delivered a series of remarkable analog computing devices with +matrix multiplication (4,19,29) and ultimately equation solving (matrix inversion) capabilities +(30). In most of the cases the matrix computations (especially the matrix inversion (30)) were +performed through stationary algorithms (31), such as the Jacobi method, where the matrix +(operator/kernel) does not change with the iteration count. +The fundamental and far-reaching question we address here is whether a wave-based analog +metastructure can be reconfigurable simply and intuitively, without needing a-priori calcula- +tions. Most importantly, the resolution to this question endows one with the ability to implement +stationary and non-stationary algorithms. We propose a device based on an RF waveguide archi- +tecture with reconfigurable components. Regarding stationary problems, we use this device to +perform matrix inversion of a statistically large number of matrices. As for non-stationary prob- +lems, we demonstrate both root finding using Newton’s method and Inverse Design. All three +examples are only possible due to the reconfigurability of the device and hint at the possibility +of deeper explorations into the realm of advanced numerical algebra methods. +A conceptual representation of the main idea is pictorially summarized in Fig. 1 (A). The +main property of our proof-of-concept system distinctively different from all previous metas- +tructure approaches (i.e. (30)) is its reconfigurability; the metastructure has the ability to rapidly +take on different matrices (operators or kernels) K. To facilitate this, we employ a wave-based +direct complex matrix (DCM) architecture, which offers an intuitive and simple implementation +of any desired matrix (32). Using waves instead of currents and voltages, it is analogous to the +3 + +crossbar architecture used in electronic analog computing systems (33–35), and it can be seen as +a generalized phased array feed (7). In this device, a collection of n × n tunable phase shifting +and amplifying elements (which can also act as attenuating element) connect an input vector +of n complex amplitudes on an array of transmission lines to a similar output vector through +combiners. This architecture can be seen schematically in Fig. 1 (B) and its corresponding +experimental implementation in Fig. 1 (C). +The key component of the metadevice is the multiplier module (Fig. 1 (D)) so named be- +cause given an input signal characterized by its complex amplitude at 45MHz, Vin, it will render +a similar output Vout = zVin, where z is a complex multiplication factor. This module consists of +two basic components: (a) a voltage-controlled phase shifter with over 360 degrees of potential +phase rotation, and (b) a voltage-controlled amplifier with ≈47dB of dynamic range (-30dB to ++17dB). Through the use of an embedded microcontroller unit (MCU) in each multiplier, each +device can be controlled externally through a suitable communication network and a computer +(see supplementary material). While the design frequency is 45MHz, the module could be im- +plemented at RF (GHz) and photonic (THz) platforms platforms, following the same principle +of operation. The experimental DCM implementation consists of 25 multipliers to yield 5 × 5 +complex matrices. The ingress and egress stages (32) are implemented with five 1-to-5 power +splitters (ingress stage) and five 5-to-1 signal combiners (egress stage). The multipliers are +clustered into five groups, one for each matrix row. In Fig. 1(B) we depict the planar schematic +of the DCM suitable for photonic implementation. However, for the RF implementation we +stacked and routed the components vertically (Fig. 1 (C)), making the device compact for our +particular wavelength and platform choice. Different stacking or integrated circuitry approaches +can potentially be used to further reduce its overall footprint. +The metadevice can be operated in one of two configurations with dramatically different +results. When the DCM is set in an open-loop configuration (Fig 1 (E) inset), it can be used +4 + +for rapidly calculating parallelized matrix-vector multiplication. However, a closed-loop con- +figuration can be created by connecting the outputs and the inputs with a feedback loop using +properly designed couplers (Fig 1 (F) inset). When the DCM is in a closed-loop configuration, +the metadevice can rapidly calculate parallelized matrix inversion (equation solving). This is a +unique feature of metadevices/metastructures (30,32) that incorporate feedback loops. +First we investigate the stationary analysis capabilities of our metadevice. For the assess- +ment of the open and closed loop operation, we performed a series of randomized trials, one +instance of which is presented in Fig. 1 (E) and (F). For each trial, a random passive matrix +A ∈ C5×5 was chosen and applied to the metadevice in both its configurations. Each mea- +surement was performed by exciting each input port in turn with all other inputs appropriately +terminated and then observing the complex amplitudes on each of the output ports. For the +open-loop configuration, this corresponds to performing five matrix-vector multiplications, or +as A · I where each column of I is progressively applied (one column at a time) as separate +vectors. The closed-loop configuration was measured similarly, but in this case we are prob- +ing the steady-state of the metadevice which corresponds to (I − K)−1 · I = A−1 · I. While +this measurement technique fully characterizes both configurations, in practice dense complex +vectors will be input and read to achieve parallelized results. +The estimated relative error ||Aexact − Ameas||2/||Aexact||2 for both cases (Fig. 1 (E) and (F)) +revealed an error about 0.001 and 0.005, respectively. Despite the component imperfections, +misalignments, measurement noise, and other stochastic errors, the measured results are in +excellent agreement with the theoretical values. The calibration procedure of the metadevice +and the statistical analysis of the full trial set (100 values) are presented in the Supplementary +Material. +A single multiplier module has a rise time of approximately 80 ns to achieve its desired com- +plex value. This value is approximately 4T assuming one-period duration of T = 1/45MHz ≈ +5 + +22.2 ns. In the open loop configuration, the total response requires approximately 5T, including +signal delays in connections and splitters. The duration of the closed-loop case is affected by +the platform and the condition number of the inverted matrix (32), but in principle is in the +same order of magnitude. Possible photonic implementations may further reduce this time to +the picosecond range (28) and below (36). +We now apply the implemented metadevice to two characteristic non-stationary problems +that highlight its mathematical abilities: (i) root finding of a system of five equations with five +unknowns using Newton’s iterative technique and (ii) implementing an inverse-design problem +using the Lagrangian multiplier formalism for constrained optimization. Both cases require +that the kernel be reprogrammed in each iteration step. Note that our approach is not restricted +to these two problems; instead, we choose these to highlight the potential of the introduced +metadevice. +For the first case we construct a simple nonlinear toy problem and we apply Newton’s algo- +rithm (37) (Fig. 2 (A)) for finding one possible root. The vector problem statement reads +f(z) = [f1(z), f2(z), f3(z), f4(z), f5(z)]T = 0 +(1) +where f ∈ C5×1, z = [z1, z2, z3, z4, z5]T ∈ C5×1, and 0 is the zero vector. We construct the +vector function to have the following polynomial form +f1(z) = (z1 − r1)(z2 − 4.2i)(z3 + 2)(z4 − 5i)(z5 − 3.5) +(2) +f2(z) = (z1 − 3.9)(z2 − r2)(z3 + 2.5i)(z4 − 3.2i)(z5 − 4.2) +(3) +f3(z) = (z1 + 5.2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.1) +(4) +f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i) +(5) +f5(z) = (z1 − 5.2i)(z2 − 4)(z3 + 4.75i)(z4 − 8)(z5 − r5) +(6) +where r = [r1, r2, r3, r4, r5]T are the vertices of a regular pentagon with 1/4 radius (see Fig. +2(B)) and the other factors represent additional extraneous roots far from the starting point. +6 + +For the evaluation of Newton’s method we need to calculate the Jacobian matrix, i.e., Jij = +∂fi +∂zj or +Jf(z) = +� +� +� +� +� +∂f1 +∂z1 +∂f1 +∂z2 +· · · +∂f1 +∂z5 +∂f2 +∂z1 +∂f2 +∂z2 +· · · +∂f2 +∂z5 +... +... +... +... +∂f5 +∂z1 +∂f5 +∂z2 +· · · +∂f5 +∂z5 +� +� +� +� +� +(7) +therefore the root can be estimated by the following iterative process +zn+1 = zn − αJ−1 +f (zn)f(zn) +(8) +where α = 0.2 is a relaxation constant (32). +In Fig. 2 (A), we can see the required algorithm steps that implement the iterative scheme +described by Eq. (8). Note that the Jacobian changes value in each iteration and it is required +that its inverse is calculated anew. This is traditionally a computationally expensive operation +which is accelerated through the use of our metadevice. The results are then used to update the +z. The method converges successfully after a few iterations. +A numerical version (using MATLAB) is compared with the experimental results illustrated +in Fig. 2 (B). We observe that for both MATLAB and the experiment, the estimation vector +converges close to the exact roots. Moreover, the estimated vector reaches a stationary point as +the iteration count increases. After 15 iterations the relative error is ||z − r||2/||r||2 ≈ 0.0023. +This is similar to the accuracy achieved for the stationary trials, thus representing the accuracy +floor of our system. A similar picture is also visible by comparing three specific iterations, as +illustrated in Fig. 2 (C), where a comparison of the full Jacobian is presented. +The experimental results do not precisely follow the paths indicated by the numerical im- +plementation realized using MATLAB. This can be explained by adding random noise to the +Jacobian on each iteration step. The added noise N ∈ C5×5 is a random complex matrix that fol- +lows a normal distribution inside a disk with radius rN = 0.01λmax, where λmax is the maximum +eigenvalue of the Jacobian. The noise creates many possible paths, all of which successfully +7 + +converge and we observe that our measured results comfortably lie within these families of +curves. Note that some solution branches are more susceptible to this noise than others (e.g. r1 +(blue) and r3 (red) curves in Fig. 2(B)) and this is due to the details of the toy problem solved. +Generally, the numerical accuracy of the device has a threshold that depends on both the +implementation and the measuring apparatus (vector network analyzer (VNA)). When higher +precision computations are required, this device can be a part of a mixed-precision computing +system. In these systems, part of the calculations are done in a fast, low-precision estimation +stage and then fed and further refined at a higher precision stage, similar to the in-memory +mixed-precision approaches in electronic platforms (38). +For the second example, we chose the case of an inverse design problem (Fig. 3 (A)). We +assume that our design consists of a collection of m = 5 two-dimensional (2D) scatterers with +circular cross section at fixed known locations r = [r1, ..., r5], each with an unknown bounded +permittivity ε = [ε1, ..., ε5] ∈ C5×1. The goal is to achieve a specific user-defined scattered field +measured at a series of n = 4 detection (objective) points, o = [o1, ..., o4]. Note that in our case +we assume a collection of cylindrical circular scatterers (2D) excited with a monochromatic +incident field of λw wavelength. The x-propagating incident field (kx) is a polarized in the +z-direction (TE wave - Ez) with the ejωt convention. +The scatterers are coupled, making this a nonlinear problem modeled using the Lippmann– +Schwinger (39) scheme, solved with a standard discrete dipole approximation (DDA) method- +ology (40). Each scatterer will respond to the local (self-excluded) electric field, which consists +of the known incident electric field, einc ∈ C5×1, and the scattered field from all other scatterers, +esca ∈ C5×1. The scatterers exhibits a complex polarization vector p = A(einc + esca) ∈ C5×1 +where A is the normalized polarizabilitiy diagonal matrix, i.e., A = diag(ε − εbackground). +The field interaction between the scatterers are expressed via the Greens matrix G ∈ +C5×5 +(hollow symmetric matrix) such that esca = Gp. +We may express the polarization vector +8 + +p = A(einc + Gp), which indicates the mutual dependence of p. Therefore, the polarization +vector can be calculated as p = (A−1 − G)−1einc. Finally, we use the four objective points o to +measure the scattered field vector emeas = Gprp ∈ C4×1 where Gpr ∈ C4×5 is the propagator +Greens function. The measured field is then compared to a (user-defined) objective eobj ∈ C4×1. +A typical constrained minimization problem (primal) can be written as (37) +min +x,y +f(x, y) +s.t. +g(x, y) ≤ 0 +(9) +where f(x, y) are the objectives and g(x, y) are the constraints. For such problems the La- +grangian (dual) problem is expressed as +max +λ +min +x,y +L(x, y, λ) = f(x, y) + λg(x, y) +(10) +Note that x and y may be subject to further requirements such as domains and bounds. +For our particular example we have that x = p, y = ε, and f(p, ε) = 1/2||Gprp − eobj||2 and +g(p, ε) = 1/2||(A(ε)−1 − G)p − einc||2. In our formulation, the objective is the scattered field at +the observation points. The constraints comprise the self-consistency of the polarization vector +(physics). Also, the permittivity vector is subject to specific bounds, i.e., ε ∈ R and ε ∈ [1, 5]. +Note that g(p, ε) is nonlinear with respect to p and ε and therefore requires a non-stationary +approach. +Following an initialization, our numerical evaluation of the above is implemented by a non- +stationary algorithm that requires repeated application of the following three steps. First, we +minimize with respect to ε by examining ∇εL(p, ε, λ) = 0. At this step we project the resulting +permittivity vector to the desired domain and bounds. Second, we minimize with respect to p by +examining ∇pL(p, ε, λ) = 0. At this stage the required stationary matrix inversion is performed +with our metadevice. Finally we maximize for λ by using ∇λL(p, ε, λ) = 0. These steps are +repeated until convergence is achieved, i.e., +E = ||eobj − emeas||2/||eobj||2 < δ +(11) +9 + +(for more information see SM). +As a numerical test case, the scatterers are assumed to be lossless with permittivity of ε = +[ε1, ε2, ε3, ε4, ε5] = [3.5, 1.5, 1.5, 3.5, 1.5]. The objective scattered field at the detection points +o, as depicted in (Fig. 3(A)), is eobj = [−0.0086 − 0.0078j, 0.0089 − 0.0132j, −0.0066 − +0.0120j, 0.0043 − 0.0004j]. The values were extracted from the DDA method and verified with +a full-wave COMSOL simulation. Note that the Fig 3(A) depicts the complex (hue/saturation) +of the electric scattered field (Ez), i.e. the difference between the total field and the incident +excitation. +Figure 3 (B) depicts a set of four cases for the same algorithm. In the first case (black +line), the idealized (noiseless, no filtering) computer evaluation of the algorithm is given - we +observe that after only 20 iterations the error drops below 10−3. The experimental results are +presented in Fig. 3(B) as red dots. The measured results exhibit an optimal point (minimum +error) after 87 iterations, with an error of 0.00172. As an analog device, there is an additional +systematic/stochastic/experimental noise to the system which affects the fidelity of the matrix +inversion. We apply a simple averaging filtering scheme on the polarization estimation, i.e., +pnew = (1 − αF)p + αFpprevious, with αF = 0.25, as a way to partially mitigate this noise. +The filter affects the convergence speed by increasing the iteration count but also significantly +improves the accuracy/fidelity of the matrix inversion, hence the metadevice’s performance. +This feature is illustrated in Fig. 3 (B), where the retrieved experimental results are compared +to the idealized computer evaluation with the applied filter (blue line). We also performed a +series of 100 randomized cases of the idealized filtered computer evaluation with added noise +to the estimated/measured polarization vector (faint blue lines in Fig. 3(B)). The noise profile +is similar to the one used in the first example (Newton’s method). The measured results are +well contained within these error bounds. Note that iteration count is not equivalent of time. +For a traditional computer evaluation, each iteration (with its required matrix-inversion) could +10 + +ultimately be slower than the convergence time of an optimized hardware implementation of +the metadevice. +Due to systematic/measurement noise, the error begins to grow after the experimental accu- +racy floor is obtained - an indication that a termination criterion could be applied at this point. +This result also agrees with the maximum accuracy we obtained in the previous non-stationary +example. More sophisticated error-correcting and filtering schemes can possibly push the accu- +racy below this threshold. For instance, αF could be adaptively tuned during the non-stationary +evaluation to realize a mixed-precision computing system. +At the minimum error point (iteration 87), the extracted permittivity estimation is illustrated +in (Fig. 3(C)). Notice that the values are very close to the numerical test case objectives and +permittivities. Finally, Fig. 3 (D) illustrates the path of the scattering vector, emeas, for these 87 +iterations. Similar to the above example, the faint paths represent the added noise effects to the +numerical evaluation. +For both presented non-stationary examples, it is evident that our metadevice can act ei- +ther as an ultrafast analog computing machine and mathematics calculator with waves, or in +a broader sense as an electromagnetic emulator for inverse design (41). It can be used for a +plethora of realistic problems where the linear response of a system (i.e. matrix-vector mul- +tiplication) or the solution of a system of equations (stationary problems, matrix inversion) +is required. Moreover, the intuitive reconfigurability of this metadevice also enables the per- +formance of constrained optimization tasks, like the ones required in non-stationary problems +such as inverse design, where the desired response of complex media requires intensive opti- +mization (42). In short, this metastructure can design metastructures. Finally, an adaptation +of the above proof-of-concept metadevice in RF-IC, photonic, or hybrid platforms can make it +an excellent candidate for on-the-fly or computation-through-propagation ultrafast, parallelized +calculations. +11 + +References +1. H. J. Caulfield, S. Dolev, Nat. Photonics 4, 261 (2010). +2. D. R. Solli, B. Jalali, Nat. Photonics 9, 704 (2015). +3. G. Wetzstein, et al., Nature 588, 39 (2020). +4. F. 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Horodynski, M. K¨uhmayer, C. Ferise, S. Rotter, M. Davy, Nature 607, 281 (2022). +Acknowledgments +The authors would like to thank Mario Junior Mencagli for useful discussions and preliminary +experimental survey on the subject. D.C.T acknowledges Luiz F. O. Chamon and Juan Cervi˜no +for the useful inputs and discussions regarding the constrained optimization algorithm. +Funding +: This work is supported in part by the Air Force Office of Scientific Research +(AFOSR) Multidisciplinary University Research Initiative (MURI) grant numbers FA9550-17- +1-0002 and FA9550-21-1-0312. +Competing Interests +: N.E. is a strategic scientific advisor/consultant to Meta Materials Inc. +The authors have no competing interest. +14 + +Authors Contributions: +N.E. conceived the idea for the reconfigurable device that solve +equations, acquired the funds, and supervised the project. D.C.T developed further the rele- +vant theories and analyses of the project. B.E. designed and programmed the device and the +device’s calibration routine. D.C.T. and B.E. assembled, built, tested the components, and per- +formed simulations and experimental measurements. D.C.T and B.E. developed the numerical +examples. All the authors discussed the results. D.C.T wrote the first draft of the manuscript +and D.C.T, B.E. and N.E. discussed, developed, and edited the final version of the manuscript. +Data and materials availability: +All data needed to evaluate the conclusions in the paper are +present in the main text and the supplementary materials. +Supplementary Materials +Materials and Methods +Supplementary Text +Figs. S1 to S9 +References (1-15) +15 + +Figure 1: A reconfigurable wave-based analog computing metastructure: (A) A conceptual +representation that describes the main objective, i.e., a reconfigurable device that can provide +us with repeated matrix inversions of arbitrary matrices in order to achieve stationary and non- +stationary algorithms. The device can implement any given kernel K = I − A and give the +(I − K)−1 = A−1. (B) The central component of the design consists of a direct complex +matrix (DCM) (32) architecture of 5x5 elements. (C) The experimental realization of this design +for the 45 MHz operating frequency. (D) The essential element of the DCM is the multiplier +module, consisting of both a phase-shifting and an amplification part (which can also function +as attenuation part); controlled with an onboard microntroller unit. Finally, (E) and (F) depict +the performance of the DCM machine in the open-loop (matrix-vector multiplication) and the +closed-loop (matrix inversion) setups. In both we compare the experimentally obtained matrix +to one computed conventionally to see good agreement. +16 + +(A) +x = (I-K)"b +(C) +(E) +(out) +b (in) +K +=A +90 +K +n +0.25 +in +ino +w +180 +0 +- +K +n-2 +C +o exact +exp +270 +a, +(F) +(B) +a. +12 +couplers +2 +a. +ino +kernel +(D) +K=I-A +90 +41 +control +in +out +?2 +multiplier +phase shift +0.5 +52 +amp +in. +ino +180 +in. +in +kernel +exact +in +out +phase shift +amp +exp +K +multiplier mod +270Figure 2: Experimental verification of Newton’s root finding method with the proposed +metadevice: (A) The algorithmic steps implemented in Newton’s root finding method. A fixed +kernel is programmed into the DCM machine in each iteration. The measured results are used +for calculating the next steps of the algorithm. (B) A comparison between the experimental +results and a numerical implementation of the algorithm. The faint solid lines are cases where +additional stochastic noise has been added to the system. We observe that the experimental +trajectory is well contained within these simulated noisy paths. (C) A comparison between +the numerical (left column) and measured (right column) evaluation of the inverse Jacobian at +different iterations. +17 + +J-1(zn) +(A) +(C) +4×103 +Algorithm 1: Newton's Method +1: Initialize z, =[0,0,0,0,0] +numerical +experiment +2: for n = 1, ... , m do +3: +J(z), f(zn), α +> Jacobian, function, scaling +4: +K, = I - α,J(zh) +> kernel for DCM +n= 1 +5: +J-1(z.) = DCMnx(K.) +> matrix inverse with DCM +7: +d,= J-1(z.)f(zn) +8: +Zn+1 = Zn- αd. +9: end for +(B) +iterations +numerical +num + noise +?n=15 +n= 5 +experiment +0.2 +exact +@n=5 +imag +n= 1 +0 + n=15 +-0.2 +-0.2 +0 +0.2 +realFigure 3: A metastructure that designs a metastructure: Numerical and experimental +results: (A) Schematic of the numerical test case. A set of five two-dimensional (2D) scatterers +with unknown permittivities, to be determined via our analog metadevice, ε = [ε1, ..., ε5] subject +to a plane wave excitation. Pictured is the complex-valued scattered field. The scattered fields +at the observations points [o1, ..., o4] are used as the benchmark values of this problem in which +the objective fields are shown as the color in each torus. The algorithm tunes each permittivity +in order to match the scattered field (center of each torus) to the objective fields. (B) The relative +error E for the algorithm computed both numerically, and experimentally using the metadevice, +under various noise and filtering scenarios. The experimental device gives a minimum relative +error of 0.00172 at 87 iterations. (C) Objective permittivities (black rings) compared to those +computed numerically (blue cross) and experimentally via the metadevice (yellow rhombus) +using the described algorithm at iteration 87. (D) Evolution of scattered field vector up to +iteration 87 in comparison to objective fields for experiment and simulation under various noise +and filtering scenarios. +18 + +100 +(A) +(B) +experiment +num +num+filter +>X +2入 +num+noise+filter +W +min (87,0.0017) +scatterers 入w/25 +OP +10-1 +error +3 +102 +0.025 +025 +objectivepoints +103 +re +0 +20 +40 +60 +80 +100 +120 +iterations +(C) +(D) +X10-3 +numerical +04 +0 +experiment +exact +permittivity +3 +imag +-5 +O +2 +U +-10 +1 +sca:numerical +esca:experiment +-15 +2 +3 +4 +5 +numberof scaterer +-0.01 +-0.005 +0 +0.005 +0.01 +realSupporting Material +Dimitrios C. Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗ +1Department of Electrical and Systems Engineering, +School of Engineering and Applied Sciences, +University of Pennsylvania, Philadelphia, 19104, U.S.A. +†These authors contributed equally to this work. +‡Present address: Meta Materials Inc. (Europe), +Ap. Pavlou 10A, Marousi, 15123, Greece. +∗To whom correspondence should be addressed; e-mail: engheta@seas.upenn.edu +January 10, 2023 +1 +Details for the root finding algorithm +In the following the lowercase quantities are vectors, the capitalized ones are matrices while Greek +letters denote scalars - the subscripts follow a logical notation. The problem statement for the root +finding procedure is +f(z) = 0 +(1) +find z ∈ Cm×1 that satisfies the above equation (roots) of f ∈ Cm×1. The above problem is solved +using Newton’s method for finding the root of a vector polynomial function [1]. For example we have +f(z) = +� +� +� +� +� +� +f1(z1, z2, z3, z4, z5) +f2(z1, z2, z3, z4, z5) +f3(z1, z2, z3, z4, z5) +f4(z1, z2, z3, z4, z5) +f5(z1, z2, z3, z4, z5) +� +� +� +� +� +� +(2) +with z1·5 ∈ C or (equivalently) +f1(z) = (z1 − r1)(z2 − 4.2i)(z3 + 2)(z4 − 5i)(z5 − 3.5) +(3) +f2(z) = (z1 − 3.9)(z2 − r2)(z3 + 2.5i)(z4 − 3.2i)(z5 − 4.2) +(4) +f3(z) = (z1 + 5.2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.1) +(5) +f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i) +(6) +f5(z) = (z1 − 5.2i)(z2 − 4)(z3 + 4.75i)(z4 − 8)(z5 − r4) +(7) +where +r = 1 +4 (s1 + c1i, −s1 + c1i, −s2 − c2i, s2 − c2i, 1i)T +(8) +with c1 = cos(2π/5), c2 = cos(π/5), s1 = sin(2π/5), and s2 = sin(4π/5). The point corresponds to +the vertices of a regular pentagon. For the evaluation of Newton’s method we need to calculate the +Jacobian matrix, i.e., Jij = ∂fi +∂xj (here i and j are indexes) or +Jf(z) = +� +� +� +� +� +� +∂f1 +∂z1 +∂f1 +∂z2 +· · · +∂f1 +∂z5 +∂f2 +∂z1 +∂f2 +∂z2 +· · · +∂f2 +∂z5 +... +... +... +... +∂f5 +∂z1 +∂f5 +∂z2 +· · · +∂f5 +∂z5 +� +� +� +� +� +� +(9) +1 +arXiv:2301.02850v1 [physics.app-ph] 22 Dec 2022 + +therefore the root can be found as +zn+1 = zn − αJ−1 +f (zn)f(zn) +(10) +where α is a relaxation constant. Here we used α = 0.2. In terms of an algorithm, we have the +following routine +Algorithm 1 Root finding with Newton’s method +1: Initial guess for z1 +2: for n = 1, . . . , m do +3: +Jf(zn) +4: +αλ = +2 +λmin+λmax +▷ Scaling factor: λmim/max are the min/max eigenvalues of Jf(zn) +5: +Kn = I − αλJf(zn) +▷ Kernel that is fed to DCM machine +6: +dn = J−1 +f (zn)f(zn) +▷ Compute matrix inverse with the DCM machine +7: +zn+1 = zn − αdn +8: end for +2 +Details on the inverse design algorithm +In this section we present the details for the inverse design algorithm implemented in text. The al- +gorithm consist of a part of the DDA methodology for the quantification of the problem and an its +adaptation to a Lagrange formalism for solving the require inverse scattering problem, the determina- +tion of the permittivity of the scatterers. Note that both methods are arguably the simplest methods to +follow, since they offer an intuitive understanding on the formulated problem and the coorresponding +inverse-design (constraints optimization) problem. +2.1 +Notes on the DDA method +In this section we present a few details regarding the DDA method used in the main text. The details +can be found also in [2, 3, 4, 5]. A similar methodological approach was also used in [6]. +We start by assuming that each 2D scatterer (assuming a point in the x-y plane) acquires its +z-oriented dipole moment due to the local electric field, i.e., +p = αeloc +(11) +where the eloc is the vector of local z-polarized electric fields at the center of each point and α is the +polarizability that depends on the shape and the material composition of each 2D scatterer. The local +field is the sum of the incident field and the secondary fields generated from all the other dipoles such +that: +eloc = einc + Gp +(12) +where einc is the incident field vector, p is the induced polarization vector and G is the 2D Green’s +function. In our case we consider a two-dimensional (2D) problem with a transverse electric (TE) +excitation (the field is normal (z-direction) to the x-y plane). Therefore the corresponding Green’s +function reads +G = G(ri − rk) = −j k2 +0 +4πε0 +H(2) +0 (k0|ri − rk|) +(13) +where H(2) +0 (k0|ri − rk|) is the Hankel function of the second type (with the time harmonic convention +e+jωt) and 0-th order and k0 = ω0√µ0ε0 is the free-space wavenumber [7]. The G is a CN×N Toeplitz +matrix with zero diagonal entries since the |ri − rk| is treated as in (assuming a uniformly spaced +discrete grid) [2, 3, 4, 5]. +By combining Eqs. (11) and (12) we obtain the following expression, arranged using the matrix +formulation as follows +p = +� +A−1 − G +�−1 einc +(14) +2 + +where the lowercase quantities p = [p1, p2, ..., pN]T , A = diag(α), α = Acellε0[ε1 − 1, ε2 − 1, ..., εN − 1] +(Acell is the cross-sectional area of a cylinders) and einc = [einc +1 , einc +2 , ..., einc +N ]T are CN×1 vectors, diag(·) +is the diagonal matrix operator. +Finally, the scattered field observed at M specified discrete detection points (in general M ̸= N) +is given by: +esca = Gpr p = Gpr +� +diag(α−1) − G +�−1 einc +(15) +where Gpr ∈ CM×N is the “propagator" Green’s function matrix. This propagator function connects +the induced dipole polarization vectors of the scatterers with the desired detection (or objective) points. +The above matrix representation of the scattering problem allow us to have a clear inspection of the +unknown quantities. These quantities are the ones that will be formulated as a Lagrange multiplier +algorithm for the solution of the desired constrained optimization problem. We note here that, as seen +in Eq. (15), the forward scattering problem requires a matrix inversion to evaluate the polarization +density vectors induced in each scattering cell, as we have discussed in our previous work [6] in which +we utilized the same DDA approach for the evaluation Eq. (14) and the matrix-vector operation of +Eq. (15) for different excitation and for different scattering scenarios. +2.2 +Notes on the Lagrange multiplier algorithm +For this, we utilize the DDA algorithm (where we closely follow the contrast source inversion method![8]) +and the Lagrange multiplier method for applying the constraints and finding the optimal solution. +First, in terms of the defined problem, we have that the polarization is connected with the following +expressions +p = A(einc + Gp) +(16) +and +esca = Gprp +(17) +A typical constrained minimization problem (primal) can be written as [9, 1, 10] +min +x,y +y∈R +0≤y≤1 +f(x, y) +s.t. +g(x, y) ≤ 0 +(18) +where f(x, y) is the objective and g(x, y) are the constraints also subject to further requirements of +the problem such as y ∈ R and 0 ≥ y ≥ 1 For such problems the dual Lagrangian problem is expressed +as +max +λ +min +x,y +y∈R +0≤y≤1 +L(x, y, λ) = f(x, y) + λg(x, y) +(19) +which is essentially a dual unconstrained problem (since all the constraints are encapsulated to the λ +term). It is worth noting that the Lagrange multiplier should be positive real, λ ∈ R+. Finding an +approximate solution to the primal inverse scattering problem is therefore reduced to finding a solution +to the above dual problem. Notice that the Lagrange multiplier can be applied to either f(x, y) or +g(x, y) without affecting the outcome of the overall process. +The algorithm for solving the above dual problem is the following: +• Step 0: initial x0 and λ0 +• Step 1: minimize yn, i.e., via ∇yL(xn−1, yn, λn−1) = 0 +• Step 2: project yn into y ∈ R and 0 ≤ y ≤ 1 +• Step 3: minimize xn, i.e., ∇xL(x, yn, λn−1) = 0 +• Step 4: maximize λn, i.e., ∇λL(xn, yn, λ) = 0 +• Step 5: Repeat steps 1-4 until the error is minimized +3 + +For our particular example we have that x = p, y = A = diag(ε − 1), and f(p, A) = 1/2||(A−1 − +G)p − einc||2 and g(p) = 1/2||Gprp − eobj||2, and Lagrange function reads +L(p, A, λ) = ||(A−1 − G)p − Aeinc||2 + λ||Gpp − eobj||2 +(20) +The corresponding algorithmic steps are: +• Step 0: initial p0 and λ0 +• Step 1: minimize An via ∇AL(pn−1, A, λn−1) = 0 +– We have that ∇AL(pn−1, A, λn−1) = ∇f(p, A)∗||(A−1−G)p−einc|| (∗ is complex conjugate). +This expression lead to An = p/(Gp − einc). In practice this is a simple calculation since A +is a diagonal matrix, i.e., A = diag(ε − 1). +• Step 2: project An into An ∈ R and 0 ≤ A ≤ 4 (for the range ε ∈ [1, 5]) +– this is the point where essentially the required properties and bound of the permittivity can +be implemented. These bounds or constrains can be general +– the above projection is rather a simple projection that does not guarantee always the min- +imum within the projection domain. +A more accurate projection would be of the form +An = proj[An−1 − η∇A||(A−1 +n−1 − G)pn−1 − einc||2]. +• Step 3: minimize pn, i.e., ∇pL(p, An, λn−1) = 0 (DCM metadevice). +– pn = K−1 +n en +L +– Kn = (A−1 +n +− G)∗(A−1 +n +− G) + λn−1G∗ +prGpr +– eL +n = λn−1G∗ +preobj + (A−1 +n +− G)∗einc +– The matrix inversion pn is performed with our DCM metadevice +– Due to noise error a simple weighted average filtering is applied, i.e., pn = (1 − αF )pn−1 + +αF pn with αF = 0.25 +• Step 4: maximize λn, i.e., ∇λL(An, pn, λ) = 0 +– This maximization can be calculated by a simple gradient descent, i.e., λn = λn−1 + +η (∇λL(pn, An, λ) − δ) or λn = λn−1 + η +� +||Gppn − eobj||2 − δ +� +– Notice that this is an gradient ascent since we assume η > 0, therefore we maximize the +problem. +• Step 5: Repeat steps 1-4 until the error is minimized. In our case we used the following error +– ||esca − eobj||2/||eobj||2 +Note that the quantities η and δ are the step and minimal error quantities that are user determined. +The whole process stop either when λ reaches a plateau, or when the required error criterion is met. +The optimization goal was set as ||esca−eobj||2 +||eobj||2 +< δ, where esca = Gppm with pm = (A−1 +m − G)−1einc +being the final m-th evaluation of the iteration. +Notice that our approach has several similarities with the contrast source inversion method and +other similar inverse scattering methods [11, 8, 12, 13]. +Undoubtedly this approach is only one of the available methods for approximating the inverse design +problem. This is rather an attempt to showcase the ability of our device for performing inverse design +with desired objectives and constraints by exposing the crucial parts of the algorithm, such as the +matrix inversions. This part is usually implicit within commercially available FDTD or FEM software. +Hence here we developed our own methodology so we can have deeper inspection to quantities. As +a remark, the field of inverse design and inverse scattering is a very rich field with a plethora of +methodologies that try to address similar problems [14]. +4 + +Figure S1: Photograph of the experimental setup with the corresponding components. +3 +RF Design, PCB, Device Implementation +A photograph of the experimental setup is shown in Fig S1, where all parts are designated accordingly. +3.1 +Measurement +Measurements were performed using an ENA-5071C two port VNA. In order to avoid the saturation +of the amplifiers (multiplier module) the VNA power level was set to be −20dBm for the open loop +configuration and −10dBm for the closed loop configuration. The VNA was set to have an IF band- +width of 10 kHz. The single frequency measurements (1601 point) at 45MHz with averaging applied +after obtaining the measured signal from VNA. +3.2 +Multiplier +The schematic of the multiplier is depicted in Fig. S2. The multiplier was designed to perform multi- +plication on the incoming complex amplitude such that a new complex amplitude is rendered at the +output. In other words, the output is Vout = zVin. This involves changing both the amplitude and +phase of the incoming signal. Phase change was performed using a pair of serially connected Minicir- +cuit JSPHS-51+ Phase Shifters (PS). Each phase shifter provides slightly over 180 degrees of rotation. +The amplitude change was performed using the Analog Devices AD603ARZ Variable Gain Amplifier +(VGA). The Multiplier design contain the appropriate loads such that both the input and output of +the device externally appears as 50 Ohm. +Both of these devices are controlled using analog voltages with ranges of [−0.5V, +0.5V] and +[0V, 12V] for the VGA and PS, respectively. In order to create a common control mechanism, op- +amp level shifting circuits were used to put these on a common [0V, 5V] interface. The Multiplier +5 + +couplers +system in (trigger) +system out (read +DCMcontrol +kernel out +kernel.in +kernel(DCM +VNA port 2 +VNAport +1-6.switch +1-5 switch +DCMpower sourceFigure S2: Schematics for the multiplier: The PCB layout design (top figure), and the corresponding +subcircuit (pictures from Altium®). Bottom figure represent the AWR Microwave Office® schematic +with the realistic data +board has a connection that allows for a daughter board. The daughter board is supplied with 0V and ++5V and is responsible for returning two control voltages in the range of [0V, 5V]. +This simple interface allows for a number of possible control schematics. At its most simple scenario, +the control board can consist of a pair of potentiometers. However, we will present another control +board which utilizes a microcontroller to receive UART input and render the two analog voltages. +The VGA’s dynamic range could be shifted using an external resistor. This was set so that the +Multipliers’s range (including load elements, PS losses, etc) was [-30dB – +17dB]. The multiplier +effectively saturates if the input is greater than -10dBm. Therefore for all measurements the reference +input signal that was used was -30dBm for avoiding any saturation effects. +It should be stated that the VGA imparts a varying phase change and the PS pair imparts an +amplitude change. This will be addressed later. +3.3 +1-5 splitter (5-1 combiner) +The schematic of the 1-5 splitter is depicted in Fig. S3. An ideal passive n-way splitter is comprised of +a summation port and n feed ports. The scattering parameters are expected to be reciprocal such that +for the ith feed port |SSi|2 = |SiS|2 = 1/n and all other elements within the matrix are zero. Due to +losses, a real splitter will fall short of this precise definition. Our splitter was based on the Minicircuits +AD5PS-1+, which yielded good performance at 45MHz with approximately -7.2dB split ratio for all +outputs.Note that 1/5 ≈ −7.0dB. +6 + +: +15 +GND 15 +5 +C8 +GND +GND +JGA +RF +GND +NetU1_7 +NetU1_7 +GNDGND +R8 +2 : GND + 5 : GND +000 +R6 +00 + _2 : GND +1 : NetJ1_-1 +5 +R7 +1 : NetC2_2 +000 +000 +5 : GND +2 : GND +C2 ++15 +R1 +3 +R2 +GND +61300211121 +U2 +VGASub +VGASub SchDoo + RF_In RF_Out +> Cont_ In +CONMCX003.031 +CON ACX003.031 + PSInput +JSPHS-51+ +JSPHS-51+ +613b041142 +C6 +CNT O +10uF ++5 +ContProc +C7 +ContProcessor. SchDoc +10uF +J3 +[0.5] +> PS_In +PS_Out +[0.15] ++15 +4 +10uF +[0.5] + VGA_In VGA_Out +[-0.5, 0.51] +951103-8622-AR +P3 +61300411021 +CNT +RFIn +RFOut +100R1% +.7nF5% +>100R 1% +100R1% +LM6172IMX +1k 1% +[0.15] +PS Out +PS Im +[0,5] +VINP +VOUT7 ++5H +VPOS +O v[so's0-] +0.1uF 5% +GRES +DINOO +R7 +FDBK +4 +3.5k 1% +7.15k 1% +AD603ARZ +0.1uF 5% +VGAIn +[0.5] +1k 1% +Cont InSUBCKT +TLIN +ID=S2 +TLIN +SUBCKT +TLIN +PORT1 +ID=TL1 +NET='phase BLK" +ID=TL2 +ID=S1 +ID=TL3 +P=1 +Z0=50 Ohm +Vph1=V/ph +Z0=50 Ohm +NET="AD_603" +Z0=50 Ohm +Z=50 Ohm +EL=el Deg +Vph2=Vph +EL=el Deg +VG=Vg_test +EL=el Deg +Pwr=[-30] dBm +F0=45 MHz +F0=45 MHz +F0=45 MHz +PORT +P=2 +Z=50 OhmFigure S3: Schematic and layout for 1-5 splitter based on the Minicircuits AD5PS-1+ (pictures from +Altium®) +3.4 +Feedback coupler +Figure S4: Schematic and layout for the feedback coupler (pictures from Altium®)). +The schematic of the feedback coupler is depicted in Fig. S4 The Feedback coupler must perform +several tasks. +• Provide near unity feedback +• Introduce the input signal +• Sample the output signal +Therefore, the feedback coupler is a four-port device wherein the primary path has near unity trans- +mission such that the feedback is strong. +3.5 +Switches +In order to replicate having a 10 port VNA, we utilized two demo boards (EV1HMC253AQS24), which +acted as RF SP8T RF switches, i.e., an analog multiplexer. For one SP8T, we utilized five of these +ports for illuminating the bank of five couplers. The other SP8T was used to receive signals from the +couplers. The remaining three ports on each were used for system sanity checks. Note that the stock +high-pass 100pF capacitors on these boards were switched to 470pF for better transmission at 45MHz. +7 + +CONMCX003.031 +P1 +CONMCX003.031 +U1 +NelJi_ +5 : GND +P3 +PS +: Net/3 +GND +:NeJ4.1 +5 : GND +CONMCX003.031 +2 : GND +CONMCX003.031 +AD5PsJi+ +139-AD5PS-1 +CONMCX003.031 +2lst +CONMCX003.031J1 +J2 +OO +OO +1 : NetJ1_ +1 : NetJ1_1 +OGO +R3 +R1 +OGO +Nei1.1 +Nei.1 +Neia.1 +NeiJ4.1 +R4 +82 +J3 +J4 +2 +OGO +OGO +1 : NetJ3_1 +1 : NetJ4_1 +OO +OO +>R3 +>R1 +1k +1k +GND +GND +GND +R4 +R2 +GND +50R +50R +GNDWhile the off ports were nominally matched to 50 Ohm from DC-2.5GHz, there was significant +reflections. +Deeper inspection of the datasheet indicated that the "off" ports were only matched +above 500MHz. Measurements indicated that the off ports were approximately "open" at the design +frequency and therefore reflections from the off ports could be significantly reduced with parallel 50Ohm +terminations. However, this was not done as this it would have reduced power within the system on +the "on" port. Rather, we note that any polluting signal from these "open" off ports will have crossed +through the coupler twice. Due to the small coupling coefficient of the feedback coupler, these values +will have become very small. +The VNA was calibrated to the end of the switch ports. Measurements indicated that transmission +through each of the switch ports was similar enough as to not warrant individual calibrations on each. +Each of the switches was actuated by three digital inputs to address the 23 = 8 ports on each +switch. These digital signals were created by a micro-controller which was programmed to respond to +UART commands from an attached computer. Code is available at github.com/brianedw/RFMath/ +Arduino/mcu_control_V2/mcu_control_V2.ino. +3.6 +Micro Controller Unit (MCU) +The two analog input control voltages for each Multiplier was created by an MCU Control Board, +which attached directly to the Multiplier. The heart of this board is a Metro-Mini MCU. +Each control line was connected to both an 8-bit PWM DAC pin (labeled “fast”) and a 10-bit +PWM DAC pin (labeled “slow”). While both pins connected to the control line through a high-pass +filter, the fast DAC utilized a lower capacitance and resistance than that of the slow DAC. During a +set operation, both pins would drive to their appropriate values, during this time, the behavior of the +collective output would be dominated by the fast DAC and rapidly converge, but exhibit large ripples. +After 20ms, the fast PWM DAC pin would switch to a high-impedance state, leaving the voltage to +settle in the remaining difference utilizing the slow DAC alone. The high-pass filter was designed to +maintain accuracy of 10bit. Since the Metro-Mini is a 5V compliant device, the generated voltages +nicely matched to the expected inputs of the Multiplier. +Each MCU board had two 3-pin UART input connectors. These were shorted such that one could +be used to receive a command from "upstream" while the other would effectively passively repeat the +signal. Additionally, each MCU board had two 3-pin UART output connector which were similarly +shorted together, allowing it to transmit the same message to two devices. Each MCU was programmed +with a unique identification number. Upon receiving a UART command, it would either act on that +command or repeat the command on its output UART pins for downstream devices. This input/output +configuration created a lot of possibilities for control topologies. However, in practice we found that +we could use a single MCU board (no multiplier attached), as a bridge between the computer and the +array of MCU Boards and that this array could all be connected in parallel such that the output of +the bridge was effectively driving 25 inputs. Note that the required time complexity is of the order +of O(n2). +Possibly this complexity can be further reduced by implementing different connectivity +schemes than the simple serial one that we used. Code is available at github.com/brianedw/RFMath/ +Arduino/mcu_control_V2/mcu_control_V2.ino. +4 +Tuning/Calibration +As stated in 3.2, the VGA has a minor effect on the phase and the PS has a minor effect on the +amplitude. +In other words, the phase and amplitude responses are coupled. +Additionally, other +systematic errors are present such as nonidealities in the level shifting circuits due to resistor tolerances. +When connected in a network that includes RF jumper cables of varying length, there will also be phase +shifts that naturally arise. In short, the relationship between the control voltages and the response +of the Multiplier in situ, are repeatable, but difficult to predict without developing a more complex +model. +We found that an effective strategy to capture, model, and invert the relationship between control +voltages and system response goes as follows. +1. A collection of Multipliers are swept across their input values to map the relationship between +control voltage and complex multiplier response. +8 + +2. These responses were analyzed using Principle Component Analysis (PCA) [15]. +3. The multipliers were assembled into the open-loop configuration and the response of the entire +open-loop network was measured under many sets of input control voltages. +4. These results were compared to a theoretical model of the network wherein the weights of the +components could be adjusted until the theoretical results matched the experimental results. +5. With accurate PCA weights in hand, the Multipliers can be immediately adjusted to achieve a +desired multiplication factor by inverting the model to achieve any open-loop kernel. +6. Additional refinement can be obtained by changing the device configuration into the closed loop, +which now includes the feedback couplers. Again, we measure the response of the closed-loop +network under many input conditions. +7. We further refine the PCA weights of each multiplier to match this more demanding data set. +This becomes our final device model for both the open- and closed-loop configurations. +We will go into detail on each one of these items in the following sections. +4.1 +Multiplier PCA +A collection of 35 multipliers were each mapped using the MCU control boards, capable of 10-bit +resolution on both control voltages. The mapping occurred with a grid of values based on [0, 11, ..., +1012, 1023] on both controls. Ideally, the mapping of two Multipliers would yield identical responses. +However, for all the reasons stated above, they do not. All of the mappings were compared using a +complex domain PCA analysis. Typically, in PCA, one would examine deviations from the mean, but +here we take another approach. Rather, the collection of mappings were analyzed directly to yield a +set of 4 PCA components. The response of any individual Multiplier could then be found as the linear +superposition of these components given by: +m(dvga, dPS) = +3 +� +i=0 +wici(dvga, dPS) +The term c0(dvga, dPS) is effectively the “average” response scaled by a complex factor, while the next +several components represent likely deviations due to the systematic errors described above. Within a +PCA analysis the final PCA components (i.e. c34(dvga, dPS), not shown) should be nearly pure noise. +We found that only the first four terms were needed to effectively model any given Multiplier. +Given any randomly chosen multiplier, we can find the complex valued PCA weights wi through a +least-squares analysis. As opposed to the "deviation from the mean" approach, the above formulation +is particularly useful for RF engineering. While the Multipliers were measured directly at their input +and output ports and analyzed as such, the model can easily account for the addition of RF cables +which would provide attenuation and phase rotation. These will appear as a complex scaling of all +of the components weights and the Multiplier’s behavior (RF jumpers cables included) can still be +captured as the simple linear superposition of the PCA components. +In fact, any losses or phase +rotations along the Multipliers flow path can be incorporated into these weights. Therefore, we do +not characterize the individual multipliers, but delay this until the architecture is fully assembled, as +described in the next section. +Regardless, we will use least-squares to find the set of wi which characterizes the average multiplier +response. We call these the “base weights”. +4.2 +Open-Loop Device Fitting +The goal of the this section is to determine the PCA weights that characterize each Multiplier in +situ, so that the system errors can be captured and modeled. The open-loop DCM system was fully +assembled including jumper cables, splitters, and couplers. All multipliers within the array were set +to the same input value (dvga, dPS) pair. The transmission matrix of the system was then measured. +This was repeated for all possible combinations of 10 evenly spaced values in the range [0, 1023] to +9 + +Figure S5: +PCA Components and Average Multiplier Response. +The first four panels represent +c0(dvga, dPS), c1(dvga, dPS), c2(dvga, dPS), and c3(dvga, dPS) and have a maximum saturation of 0.05. +The final image shows the response of the “average” Multiplier with a maximum saturation at 7.5 +yield 100 measured transmission matrices, Tmeas. Note that not all of these 100 transmission matrices +represent "passive" operators. +The same system was modeled using Scikit-RF, wherein the following assumptions were made: +• The 5-1 splitters were ideal such that power was evenly split with no phase +• All jumpers were zero-length +• The coupler feedback path was ideal with no power removed +• The multipliers were all assumed to be “average” and the their responses were assumed to be +given by the “base weights”. +The system was simulated using SciKit-RF for each input pair (dvga, dPS) to yield 100 measured +transmission matrices Tsim(w), which are naturally a function of each Multipliers PCA weight. We +can then define an error error(w) = |Tsim(w) − Tmeas|2 and optimize w until that error is minimized. +It should be noted, that with only four PCA weights per Multiplier, in theory, only 4 transmission +matrices are required to fully define the system. Using 100 helps guarantee that normal measurement +noise does not unduly influence the fitting. Additionally, if a low error can be achieved across 100 +measurements using only 4 weights, then we can be confident that the model was sufficient to capture +the entire open loop system response, K. +4.3 +Setting the Open Loop System Response +Given a desired open-loop system response, K, we need to calculate the necessary multiplier values for +the DCM architecture, mi,j, gathered to form M. In this case, the simplicity of the DCM architecture +makes this trivial. If we assume an idealized passive five port splitters such that given an input of 1W +at the summation port, s, we will observe 1/5W on each branch port, i. Put in terms of Scattering +Parameters, Ss,i = 1/ +√ +5 and via reciprocity Si,s = 1/ +√ +5. Since we have such splitters at the input +and output of the Multiplier array, K = (1/ +√ +5)M(1/ +√ +5) and therefore M = 5K. Note that since we +fitted the PCA weights of the Multipliers under the assumption of ideal components, it is appropriate +to assume ideal components here. +With each of the desired mi,j in hand to achieve a given K, the next step is to determine the +required (dvga, dPS). This can be done using a number of function inversion schemes such a gradient +descent. In practice, this could be very fast as it is likely that in many applications, each new M will +be a small step from the previous M and therefore each multiplier will change only slightly. +4.4 +Closed-Loop Device Fitting +Due to the recursive nature of the closed-loop configuration (Matrix Inversion), the accuracy require- +ments are more stringent than for the open-loop configuration. Moreover, additional degrees of freedom +are introduced in the form of coupler coefficients. These can be considered part of w. In short, the +devices must be fitted again. +We will employ a similar strategy as was used in the Open-Loop Device Fitting. Using the open- +loop calibrated device models, a sequence of randomly generated passive transmission matrices, K, +10 + +co(dvga, dps) +Ci(dvga, dps) +C2(dvga, dps) +C3(dvga, dps) +mave(dvga, dps) +imag +imag +ima +imag +dps +real +real +real +real +realare shown to the system. Note, unlike the open-loop matrices, in order to guarantee convergence, +these matrices must be passive. We model the closed-loop system using Scikit-RF. Using the open- +loop weights as a starting point, we optimize the multiplier weights and coupling coefficients until the +simulated Tsim(w) matches the measured Tmeas. This represents a small, but necessary, refinement +from the open-loop device model and can be used for both open- and closed-loop applications. +4.5 +Setting the Closed Loop System Response +Setting the closed-loop system response, K is identical to setting the open-loop response. In both +cases, each desired mi,j is used to find the required (dvga, dPS) using a function inversion scheme. +5 +System Accuracy +We performed an open loop measurement on 100 complex-valued random matrices with (eigenvalues) +values within the unit circle. For these, we configured the open-loop with the target (or ideal kernel) +Ae and retrieved the measured results Am. We define as error the quantity +||Am − Ae||2 +||Ae||2 +100% +(21) +In Fig. S6, we can see the difference between the two matrices for 100 random cases. We observe +that all the results are within a 0.05 − 0.3% percent error. Similarly, we performed the same error +analysis for the same 100 random matrices, only this time on a closed-loop setup (matrix-inversion). +The results (Fig. S7) reveal that the error can climb up to 20%, but for most of the results, we get a +matrix inversion with less than 2% error. Finally, we assess the matrix inversion fidelity by evaluating +the trace of the A−1 +m Ae product. Ideally the trace of the product tr(A−1 +m Ae)/5 = 1. In Fig. S8, we +observe that this product spans between 0.5 − 1.5. However, for the particular examples we used in +the manuscript, this accuracy can be maintained at reasonably high levels once error-correcting and +filtering techniques are applied. Note that for the closed-loop case, the level of the measured voltage +is in the order of µV, very close to the noise floor of the VNA device we used. For the open loop +operation, the measured voltage was hundreds of mV. +0 +50 +100 +0 +0.1 +0.2 +0.3 +Figure S6: The error between the exact and the measured matrices, open loop configuration, for 100 +random complex matrices. +11 + +0 +50 +100 +0 +5 +10 +15 +20 +Figure S7: The error between the exact and the measured matrices, closed loop configuration (matrix +inversion), for 100 random complex matrices. +6 +System Transient Analysis +6.1 +Single Multiplier +In terms of the time response of the multiplier module the transient analysis reveal (Fig. S9) that +the module obtained the desired value approximately within 3-4 signal periods, i.e., T = 22.2ns. The +measurements were performed using the RIGOL DG4062 pulse generator (15 sinusoidal pulses at +45MHz), and the measured response extracted with the RIGOL DS1104 oscilloscope. +The open loop response is therefore assumed to be very close to the single multipliers response +since both splitters and connecting cables introduce a small phase shift to the signal. The closed loop +transient response is affected by both the multiplier timing and the condition number of the input +matrix (kernel) as shown in [16]. +12 + +0 +50 +100 +0 +0.5 +1 +1.5 +2 +Figure S8: The fidelity of the matrix inversion expressed in terms of the normalized trace of the A−1 +m Ae +product for 100 random complex matrices. +measurements +in +out +0 +50 +100 +150 +200 +ns +simulations +in +out +Figure S9: The transient response of a single multiplier module. +The blue curves correspond to +the input signal, while the red curves are the measured (top) and simulated (bottom) using AWR +Microwave Office® results. The agreement is excellent. It is evident that it takes approximately 3 to +4 signal periods for the multiplier to obtain the desired output signal. Here we assumed small signal +amplification (VGA voltage is +.05) and the phase shift voltage is 0V. +13 + +7 +De-embedding the solution +7.1 +Open Loop +Let us define the open-loop response as +Vout = KVin +Note that this includes not only the DCM architecture (multipliers, splitters, jumpers), but also the +through channel of the input/output coupler. In other words, the open-loop is defined using all of +the components of the closed-loop. However, the loop has been broken "open" just after the coupler +array and measured at this point. Since in the closed configuration, these measurement planes were +coincident, upon "closing" the loop, these measurement will then represent the complete response of +the loop. While a minor perturbation to the results, this definition assumes that the weakly coupled +additional ports on the coupler are properly terminated. +Let us further define response of only the DCM architecture as K′. When the system is in a closed +loop configuration, this relates the vector exiting the coupler array (V4) to the vector incident on the +coupler array (V2). +V2 = K′V4 +The coupler array introduces a small loss as the input is introduced and the output is measured. The +near unity transmission is named α1. It is clear then that K = α1K′. +7.2 +Closed Loop +The closed loop response is fully defined by the open-loop response and the definition of the scattering +parameters of the coupler. +V2 = K′V4 +(22) +V3 = α2V1 + βV2 +(23) +V4 = βV1 + α1V2 +(24) +Our goal is to solve the equations for V4, which represents the vectorial solution of the problem in +question. For the expected solution, this should be done such that the solution depends only on the +kernel K and the input vector (V1). For the measured solution, this should be only in terms of the +measured results (V3) and the known input (V1). +7.3 +Expected Solution +We begin by applying the definitions above +V4 = βV1 + α1V2 +V4 = βV1 + α1K′V4 +V4 = βV1 + KV4 +and then solve the final equation for the V4. +V4 = (I − K)−1βV1 +7.4 +Measured Solution +We begin with Eq 23: +V3 = α2V1 + βV2 +and then solve it for V2 +V2 = 1 +β V3 − α2 +β V1 +14 + +and then substitute the above into 24 +V4 = βV1 + α1( 1 +β V3 − α2 +β V1) +and then simplify +V4 = (β − α1α2 +β +)V1 + α1 +β V3 +Note that in many real world cases, the coupler will be defined such that we can assume α2 → 0 +V4 = βV1 + α1 +β V3 +References +[1] D. P. Bertsekas, Nonlinear programming (Athena Scientific„ Belmont, Mass. :, 1995). +[2] E. M. Purcell, C. R. Pennypacker, Astrophys. J. 186, 705 (1973). +[3] B. T. Draine, P. J. Flatau, J. Opt. Soc. Am. A 11, 1491 (1994). +[4] M. A. Yurkin, A. G. Hoekstra, J. Quant. Spectrosc. Radiat. Transf. 106, 558 (2007). +[5] S. P. Groth, A. G. Polimeridis, J. K. White, J. Quant. Spectrosc. Radiat. Transf. 240, 106689 +(2020). +[6] V. Nikkhah, D. C. Tzarouchis, A. Hoorfar, N. Engheta, ACS Photonics (2022). +[7] C. A. Balanis, Advanced engineering electromagnetics (John Wiley & Sons, 1999). +[8] P. M. van den Berg, R. E. Kleinman, Inverse Probl. 13, 1607 (1997). +[9] D. Bertsekas, Convex optimization theory, vol. 1 (Athena Scientific, 2009). +[10] S. Boyd, S. P. Boyd, L. Vandenberghe, Convex optimization (Cambridge University Press, 2004). +[11] R. E. Kleinman, P. den Berg, J. Comput. Appl. Math. 42, 17 (1992). +[12] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied +Mathematical Sciences (Springer New York, New York, NY, 2013). +[13] S. Boutami, S. Fan, Journal of the Optical Society of America B 36, 2378 (2019). +[14] Z. Li, R. Pestourie, Z. Lin, S. G. Johnson, F. Capasso, ACS Photonics 9, 2178 (2022). +[15] I. T. Jolliffe, Principal component analysis for special types of data (Springer, 2002). +[16] D. C. Tzarouchis, M. J. Mencagli, B. Edwards, N. Engheta, Light Sci. Appl. 11, 263 (2022). +15 + diff --git a/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/load_file.txt b/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08b15d6d4661182d9932a2ab3f5f77710b302a7d --- /dev/null +++ b/6dE1T4oBgHgl3EQfBQKx/content/tmp_files/load_file.txt @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf,len=1156 +page_content='Programmable wave-based analog computing machine: a metastructure that designs metastructures Dimitrios C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗ 1Department of Electrical and Systems Engineering, School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, 19104, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ‡Present address: Meta Materials Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (Europe), Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Pavlou 10A, Marousi, 15123, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' e-mail: engheta@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='edu January 10, 2023 Abstract: The ability to perform mathematical computations using metastruc- tures is an emergent paradigm that carries the potential of wave-based analog computing to the realm of near-speed-of-light, low-loss, compact devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We theoretically introduce and experimentally verify the concept of a reconfig- urable metastructure that performs analog complex mathematical computa- tions using electromagnetic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Reconfigurable, RF-based components en- dow our device with the ability to perform stationary and non-stationary iter- ative algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' After demonstrating matrix inversion (stationary problem), we use the machine to tackle two major non-stationary problems: root finding with Newton’s method and inverse design (constrained optimization) via the Lagrange multiplier method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The platform enables possible avenues for wave- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='02850v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='app-ph] 22 Dec 2022 based, analog computations for general linear algebraic problems and beyond in compact, ultrafast, and parallelized ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' One-Sentence Summary: A reconfigurable wave-based analog computing metastructure that can inverse-design a metastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Calculators of various kinds have emerged by forging numerical algorithms with corre- sponding technological platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While the algorithms describe the mathematical paths on how solutions to problems can be found, the platforms are responsible for the transliteration of this abstract path into measurable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The algorithms, the platforms, and their fusion define such systems’ features and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Following the ever-growing demand for ultrafast, compact, low/near-zero-power, and integrable cyber-physical devices for mathematical compu- tations, it is organic that significant research efforts focus on making these numerical systems as optimal and efficient as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This quest led to the exploration and development of unconventional analog computing sys- tems that exploit electromagnetic waves to deliver parallelized, ultrafast, compact, low-power computations (1–4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The two main categories in this domain involve systems that utilize free- space (scattering) elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' lenses (3)), and waveguides (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' photonic systems (5, 6) and phased arrays (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Sufficient free space propagation can act as dense matrix multiplica- tion (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Realized with traditional optics, this results in bulky devices (3,9), while metasurfaces can be more compact (10–12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, in both there can be major bottlenecks regarding photonic and electronic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Waveguiding systems offer more mature solutions for integrable and reconfigurable devices, at the expense of much larger footprints compared to their free-space counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In all cases, their main challenge is reconfigurability since its implementation requires some form of a-priori mathematical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For instance, meta- surfaces/complex media (13, 14) requires optimization, and photonic meshes require operator 2 decomposition (15–18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In terms of their mathematical abilities, the above examples demonstrate wave-based analog computing with functionalities such as integration/differentiation in space (19–22) and time (23), matrix-vector multiplication (24), emulating equations through physical phenomena (25, 26), or acting as platforms for neural network functionalities (3,6,27,28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The intersection with the metamaterial paradigm delivered a series of remarkable analog computing devices with matrix multiplication (4,19,29) and ultimately equation solving (matrix inversion) capabilities (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In most of the cases the matrix computations (especially the matrix inversion (30)) were performed through stationary algorithms (31), such as the Jacobi method, where the matrix (operator/kernel) does not change with the iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The fundamental and far-reaching question we address here is whether a wave-based analog metastructure can be reconfigurable simply and intuitively, without needing a-priori calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Most importantly, the resolution to this question endows one with the ability to implement stationary and non-stationary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We propose a device based on an RF waveguide archi- tecture with reconfigurable components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Regarding stationary problems, we use this device to perform matrix inversion of a statistically large number of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' As for non-stationary prob- lems, we demonstrate both root finding using Newton’s method and Inverse Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' All three examples are only possible due to the reconfigurability of the device and hint at the possibility of deeper explorations into the realm of advanced numerical algebra methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A conceptual representation of the main idea is pictorially summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The main property of our proof-of-concept system distinctively different from all previous metas- tructure approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (30)) is its reconfigurability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' the metastructure has the ability to rapidly take on different matrices (operators or kernels) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' To facilitate this, we employ a wave-based direct complex matrix (DCM) architecture, which offers an intuitive and simple implementation of any desired matrix (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Using waves instead of currents and voltages, it is analogous to the 3 crossbar architecture used in electronic analog computing systems (33–35), and it can be seen as a generalized phased array feed (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In this device, a collection of n × n tunable phase shifting and amplifying elements (which can also act as attenuating element) connect an input vector of n complex amplitudes on an array of transmission lines to a similar output vector through combiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This architecture can be seen schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (B) and its corresponding experimental implementation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The key component of the metadevice is the multiplier module (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (D)) so named be- cause given an input signal characterized by its complex amplitude at 45MHz, Vin, it will render a similar output Vout = zVin, where z is a complex multiplication factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This module consists of two basic components: (a) a voltage-controlled phase shifter with over 360 degrees of potential phase rotation, and (b) a voltage-controlled amplifier with ≈47dB of dynamic range (-30dB to +17dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Through the use of an embedded microcontroller unit (MCU) in each multiplier, each device can be controlled externally through a suitable communication network and a computer (see supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While the design frequency is 45MHz, the module could be im- plemented at RF (GHz) and photonic (THz) platforms platforms, following the same principle of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The experimental DCM implementation consists of 25 multipliers to yield 5 × 5 complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The ingress and egress stages (32) are implemented with five 1-to-5 power splitters (ingress stage) and five 5-to-1 signal combiners (egress stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The multipliers are clustered into five groups, one for each matrix row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1(B) we depict the planar schematic of the DCM suitable for photonic implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, for the RF implementation we stacked and routed the components vertically (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (C)), making the device compact for our particular wavelength and platform choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Different stacking or integrated circuitry approaches can potentially be used to further reduce its overall footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The metadevice can be operated in one of two configurations with dramatically different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' When the DCM is set in an open-loop configuration (Fig 1 (E) inset), it can be used 4 for rapidly calculating parallelized matrix-vector multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, a closed-loop con- figuration can be created by connecting the outputs and the inputs with a feedback loop using properly designed couplers (Fig 1 (F) inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' When the DCM is in a closed-loop configuration, the metadevice can rapidly calculate parallelized matrix inversion (equation solving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This is a unique feature of metadevices/metastructures (30,32) that incorporate feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' First we investigate the stationary analysis capabilities of our metadevice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the assess- ment of the open and closed loop operation, we performed a series of randomized trials, one instance of which is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (E) and (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For each trial, a random passive matrix A ∈ C5×5 was chosen and applied to the metadevice in both its configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each mea- surement was performed by exciting each input port in turn with all other inputs appropriately terminated and then observing the complex amplitudes on each of the output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the open-loop configuration, this corresponds to performing five matrix-vector multiplications, or as A · I where each column of I is progressively applied (one column at a time) as separate vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The closed-loop configuration was measured similarly, but in this case we are prob- ing the steady-state of the metadevice which corresponds to (I − K)−1 · I = A−1 · I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While this measurement technique fully characterizes both configurations, in practice dense complex vectors will be input and read to achieve parallelized results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The estimated relative error ||Aexact − Ameas||2/||Aexact||2 for both cases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1 (E) and (F)) revealed an error about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='005, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Despite the component imperfections, misalignments, measurement noise, and other stochastic errors, the measured results are in excellent agreement with the theoretical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The calibration procedure of the metadevice and the statistical analysis of the full trial set (100 values) are presented in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A single multiplier module has a rise time of approximately 80 ns to achieve its desired com- plex value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This value is approximately 4T assuming one-period duration of T = 1/45MHz ≈ 5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In the open loop configuration, the total response requires approximately 5T, including signal delays in connections and splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The duration of the closed-loop case is affected by the platform and the condition number of the inverted matrix (32), but in principle is in the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Possible photonic implementations may further reduce this time to the picosecond range (28) and below (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We now apply the implemented metadevice to two characteristic non-stationary problems that highlight its mathematical abilities: (i) root finding of a system of five equations with five unknowns using Newton’s iterative technique and (ii) implementing an inverse-design problem using the Lagrangian multiplier formalism for constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Both cases require that the kernel be reprogrammed in each iteration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that our approach is not restricted to these two problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' instead, we choose these to highlight the potential of the introduced metadevice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the first case we construct a simple nonlinear toy problem and we apply Newton’s algo- rithm (37) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2 (A)) for finding one possible root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The vector problem statement reads f(z) = [f1(z), f2(z), f3(z), f4(z), f5(z)]T = 0 (1) where f ∈ C5×1, z = [z1, z2, z3, z4, z5]T ∈ C5×1, and 0 is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We construct the vector function to have the following polynomial form f1(z) = (z1 − r1)(z2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z3 + 2)(z4 − 5i)(z5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5) (2) f2(z) = (z1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='9)(z2 − r2)(z3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5i)(z4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z5 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2) (3) f3(z) = (z1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1) (4) f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i) (5) f5(z) = (z1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z2 − 4)(z3 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='75i)(z4 − 8)(z5 − r5) (6) where r = [r1, r2, r3, r4, r5]T are the vertices of a regular pentagon with 1/4 radius (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2(B)) and the other factors represent additional extraneous roots far from the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 6 For the evaluation of Newton’s method we need to calculate the Jacobian matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', Jij = ∂fi ∂zj or Jf(z) = � � � � � ∂f1 ∂z1 ∂f1 ∂z2 · · ∂f1 ∂z5 ∂f2 ∂z1 ∂f2 ∂z2 · · ∂f2 ∂z5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ∂f5 ∂z1 ∂f5 ∂z2 · · ∂f5 ∂z5 � � � � � (7) therefore the root can be estimated by the following iterative process zn+1 = zn − αJ−1 f (zn)f(zn) (8) where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 is a relaxation constant (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2 (A), we can see the required algorithm steps that implement the iterative scheme described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that the Jacobian changes value in each iteration and it is required that its inverse is calculated anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This is traditionally a computationally expensive operation which is accelerated through the use of our metadevice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The results are then used to update the z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The method converges successfully after a few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A numerical version (using MATLAB) is compared with the experimental results illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2 (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We observe that for both MATLAB and the experiment, the estimation vector converges close to the exact roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Moreover, the estimated vector reaches a stationary point as the iteration count increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' After 15 iterations the relative error is ||z − r||2/||r||2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This is similar to the accuracy achieved for the stationary trials, thus representing the accuracy floor of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A similar picture is also visible by comparing three specific iterations, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2 (C), where a comparison of the full Jacobian is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The experimental results do not precisely follow the paths indicated by the numerical im- plementation realized using MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This can be explained by adding random noise to the Jacobian on each iteration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The added noise N ∈ C5×5 is a random complex matrix that fol- lows a normal distribution inside a disk with radius rN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='01λmax, where λmax is the maximum eigenvalue of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The noise creates many possible paths, all of which successfully 7 converge and we observe that our measured results comfortably lie within these families of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that some solution branches are more susceptible to this noise than others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' r1 (blue) and r3 (red) curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2(B)) and this is due to the details of the toy problem solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Generally, the numerical accuracy of the device has a threshold that depends on both the implementation and the measuring apparatus (vector network analyzer (VNA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' When higher precision computations are required, this device can be a part of a mixed-precision computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In these systems, part of the calculations are done in a fast, low-precision estimation stage and then fed and further refined at a higher precision stage, similar to the in-memory mixed-precision approaches in electronic platforms (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the second example, we chose the case of an inverse design problem (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3 (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We assume that our design consists of a collection of m = 5 two-dimensional (2D) scatterers with circular cross section at fixed known locations r = [r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', r5], each with an unknown bounded permittivity ε = [ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ε5] ∈ C5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The goal is to achieve a specific user-defined scattered field measured at a series of n = 4 detection (objective) points, o = [o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', o4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that in our case we assume a collection of cylindrical circular scatterers (2D) excited with a monochromatic incident field of λw wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The x-propagating incident field (kx) is a polarized in the z-direction (TE wave - Ez) with the ejωt convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The scatterers are coupled, making this a nonlinear problem modeled using the Lippmann– Schwinger (39) scheme, solved with a standard discrete dipole approximation (DDA) method- ology (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each scatterer will respond to the local (self-excluded) electric field, which consists of the known incident electric field, einc ∈ C5×1, and the scattered field from all other scatterers, esca ∈ C5×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The scatterers exhibits a complex polarization vector p = A(einc + esca) ∈ C5×1 where A is the normalized polarizabilitiy diagonal matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', A = diag(ε − εbackground).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The field interaction between the scatterers are expressed via the Greens matrix G ∈ C5×5 (hollow symmetric matrix) such that esca = Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We may express the polarization vector 8 p = A(einc + Gp), which indicates the mutual dependence of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Therefore, the polarization vector can be calculated as p = (A−1 − G)−1einc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, we use the four objective points o to measure the scattered field vector emeas = Gprp ∈ C4×1 where Gpr ∈ C4×5 is the propagator Greens function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The measured field is then compared to a (user-defined) objective eobj ∈ C4×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A typical constrained minimization problem (primal) can be written as (37) min x,y f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' g(x, y) ≤ 0 (9) where f(x, y) are the objectives and g(x, y) are the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For such problems the La- grangian (dual) problem is expressed as max λ min x,y L(x, y, λ) = f(x, y) + λg(x, y) (10) Note that x and y may be subject to further requirements such as domains and bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For our particular example we have that x = p, y = ε, and f(p, ε) = 1/2||Gprp − eobj||2 and g(p, ε) = 1/2||(A(ε)−1 − G)p − einc||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In our formulation, the objective is the scattered field at the observation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The constraints comprise the self-consistency of the polarization vector (physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Also, the permittivity vector is subject to specific bounds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ε ∈ R and ε ∈ [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that g(p, ε) is nonlinear with respect to p and ε and therefore requires a non-stationary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Following an initialization, our numerical evaluation of the above is implemented by a non- stationary algorithm that requires repeated application of the following three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' First, we minimize with respect to ε by examining ∇εL(p, ε, λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' At this step we project the resulting permittivity vector to the desired domain and bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Second, we minimize with respect to p by examining ∇pL(p, ε, λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' At this stage the required stationary matrix inversion is performed with our metadevice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally we maximize for λ by using ∇λL(p, ε, λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These steps are repeated until convergence is achieved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', E = ||eobj − emeas||2/||eobj||2 < δ (11) 9 (for more information see SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' As a numerical test case, the scatterers are assumed to be lossless with permittivity of ε = [ε1, ε2, ε3, ε4, ε5] = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The objective scattered field at the detection points o, as depicted in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3(A)), is eobj = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0086 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0078j, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0089 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0132j, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0066 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0120j, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0043 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0004j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The values were extracted from the DDA method and verified with a full-wave COMSOL simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that the Fig 3(A) depicts the complex (hue/saturation) of the electric scattered field (Ez), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' the difference between the total field and the incident excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Figure 3 (B) depicts a set of four cases for the same algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In the first case (black line), the idealized (noiseless, no filtering) computer evaluation of the algorithm is given - we observe that after only 20 iterations the error drops below 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The experimental results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3(B) as red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The measured results exhibit an optimal point (minimum error) after 87 iterations, with an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='00172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' As an analog device, there is an additional systematic/stochastic/experimental noise to the system which affects the fidelity of the matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We apply a simple averaging filtering scheme on the polarization estimation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', pnew = (1 − αF)p + αFpprevious, with αF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='25, as a way to partially mitigate this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The filter affects the convergence speed by increasing the iteration count but also significantly improves the accuracy/fidelity of the matrix inversion, hence the metadevice’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This feature is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3 (B), where the retrieved experimental results are compared to the idealized computer evaluation with the applied filter (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We also performed a series of 100 randomized cases of the idealized filtered computer evaluation with added noise to the estimated/measured polarization vector (faint blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The noise profile is similar to the one used in the first example (Newton’s method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The measured results are well contained within these error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that iteration count is not equivalent of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For a traditional computer evaluation, each iteration (with its required matrix-inversion) could 10 ultimately be slower than the convergence time of an optimized hardware implementation of the metadevice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Due to systematic/measurement noise, the error begins to grow after the experimental accu- racy floor is obtained - an indication that a termination criterion could be applied at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This result also agrees with the maximum accuracy we obtained in the previous non-stationary example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' More sophisticated error-correcting and filtering schemes can possibly push the accu- racy below this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For instance, αF could be adaptively tuned during the non-stationary evaluation to realize a mixed-precision computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' At the minimum error point (iteration 87), the extracted permittivity estimation is illustrated in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Notice that the values are very close to the numerical test case objectives and permittivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3 (D) illustrates the path of the scattering vector, emeas, for these 87 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Similar to the above example, the faint paths represent the added noise effects to the numerical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For both presented non-stationary examples, it is evident that our metadevice can act ei- ther as an ultrafast analog computing machine and mathematics calculator with waves, or in a broader sense as an electromagnetic emulator for inverse design (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It can be used for a plethora of realistic problems where the linear response of a system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' matrix-vector mul- tiplication) or the solution of a system of equations (stationary problems, matrix inversion) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Moreover, the intuitive reconfigurability of this metadevice also enables the per- formance of constrained optimization tasks, like the ones required in non-stationary problems such as inverse design, where the desired response of complex media requires intensive opti- mization (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In short, this metastructure can design metastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, an adaptation of the above proof-of-concept metadevice in RF-IC, photonic, or hybrid platforms can make it an excellent candidate for on-the-fly or computation-through-propagation ultrafast, parallelized calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 11 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' H.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 8, 1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Cordaro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 19, 8418 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Zheng, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Kravchenko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Valentine, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1, 246 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Pham, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Imaging 6, 727 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Yurkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Molesky, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Photonics 12, 659 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Horodynski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' K¨uhmayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Ferise, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Rotter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Davy, Nature 607, 281 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Acknowledgments The authors would like to thank Mario Junior Mencagli for useful discussions and preliminary experimental survey on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T acknowledges Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Chamon and Juan Cervi˜no for the useful inputs and discussions regarding the constrained optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Funding : This work is supported in part by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) grant numbers FA9550-17- 1-0002 and FA9550-21-1-0312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Competing Interests : N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' is a strategic scientific advisor/consultant to Meta Materials Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The authors have no competing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 14 Authors Contributions: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' conceived the idea for the reconfigurable device that solve equations, acquired the funds, and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T developed further the rele- vant theories and analyses of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' designed and programmed the device and the device’s calibration routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' assembled, built, tested the components, and per- formed simulations and experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' developed the numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' All the authors discussed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T wrote the first draft of the manuscript and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='T, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' discussed, developed, and edited the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the main text and the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Supplementary Materials Materials and Methods Supplementary Text Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S1 to S9 References (1-15) 15 Figure 1: A reconfigurable wave-based analog computing metastructure: (A) A conceptual representation that describes the main objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', a reconfigurable device that can provide us with repeated matrix inversions of arbitrary matrices in order to achieve stationary and non- stationary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The device can implement any given kernel K = I − A and give the (I − K)−1 = A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (B) The central component of the design consists of a direct complex matrix (DCM) (32) architecture of 5x5 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (C) The experimental realization of this design for the 45 MHz operating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (D) The essential element of the DCM is the multiplier module, consisting of both a phase-shifting and an amplification part (which can also function as attenuation part);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' controlled with an onboard microntroller unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, (E) and (F) depict the performance of the DCM machine in the open-loop (matrix-vector multiplication) and the closed-loop (matrix inversion) setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In both we compare the experimentally obtained matrix to one computed conventionally to see good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 16 (A) x = (I-K)"b (C) (E) (out) b (in) K =A 90 K n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='25 in ino w 180 0 K n-2 C o exact exp 270 a, (F) (B) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 12 couplers 2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ino kernel (D) K=I-A 90 41 control in out ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 multiplier phase shift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 52 amp in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ino 180 in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' in kernel exact in out phase shift amp exp K multiplier mod 270Figure 2: Experimental verification of Newton’s root finding method with the proposed metadevice: (A) The algorithmic steps implemented in Newton’s root finding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A fixed kernel is programmed into the DCM machine in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The measured results are used for calculating the next steps of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (B) A comparison between the experimental results and a numerical implementation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The faint solid lines are cases where additional stochastic noise has been added to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We observe that the experimental trajectory is well contained within these simulated noisy paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (C) A comparison between the numerical (left column) and measured (right column) evaluation of the inverse Jacobian at different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=" 17 J-1(zn) (A) (C) 4×103 Algorithm 1: Newton's Method 1: Initialize z, =[0,0,0,0,0] numerical experiment 2: for n = 1, ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' , m do 3: J(z), f(zn), α > Jacobian, function, scaling 4: K, = I - α,J(zh) > kernel for DCM n= 1 5: J-1(z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=') = DCMnx(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=') > matrix inverse with DCM 7: d,= J-1(z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' )f(zn) 8: Zn+1 = Zn- αd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 9: end for (B) iterations numerical num + noise ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='n=15 n= 5 experiment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 exact @n=5 imag n= 1 0 n=15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 realFigure 3: A metastructure that designs a metastructure: Numerical and experimental results: (A) Schematic of the numerical test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A set of five two-dimensional (2D) scatterers with unknown permittivities, to be determined via our analog metadevice, ε = [ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ε5] subject to a plane wave excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Pictured is the complex-valued scattered field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The scattered fields at the observations points [o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', o4] are used as the benchmark values of this problem in which the objective fields are shown as the color in each torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The algorithm tunes each permittivity in order to match the scattered field (center of each torus) to the objective fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (B) The relative error E for the algorithm computed both numerically, and experimentally using the metadevice, under various noise and filtering scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The experimental device gives a minimum relative error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='00172 at 87 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (C) Objective permittivities (black rings) compared to those computed numerically (blue cross) and experimentally via the metadevice (yellow rhombus) using the described algorithm at iteration 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (D) Evolution of scattered field vector up to iteration 87 in comparison to objective fields for experiment and simulation under various noise and filtering scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 18 100 (A) (B) experiment num num+filter >X 2入 num+noise+filter W min (87,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0017) scatterers 入w/25 OP 10-1 error 3 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='025 025 objectivepoints 103 re 0 20 40 60 80 100 120 iterations (C) (D) X10-3 numerical 04 0 experiment exact permittivity 3 imag 5 O 2 U 10 1 sca:numerical esca:experiment 15 2 3 4 5 numberof scaterer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='01 realSupporting Material Dimitrios C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗ 1Department of Electrical and Systems Engineering, School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, 19104, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ‡Present address: Meta Materials Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (Europe), Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Pavlou 10A, Marousi, 15123, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' e-mail: engheta@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='edu January 10, 2023 1 Details for the root finding algorithm In the following the lowercase quantities are vectors, the capitalized ones are matrices while Greek letters denote scalars - the subscripts follow a logical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The problem statement for the root finding procedure is f(z) = 0 (1) find z ∈ Cm×1 that satisfies the above equation (roots) of f ∈ Cm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The above problem is solved using Newton’s method for finding the root of a vector polynomial function [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For example we have f(z) = � � � � � � f1(z1, z2, z3, z4, z5) f2(z1, z2, z3, z4, z5) f3(z1, z2, z3, z4, z5) f4(z1, z2, z3, z4, z5) f5(z1, z2, z3, z4, z5) � � � � � � (2) with z1·5 ∈ C or (equivalently) f1(z) = (z1 − r1)(z2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z3 + 2)(z4 − 5i)(z5 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5) (3) f2(z) = (z1 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='9)(z2 − r2)(z3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5i)(z4 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z5 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2) (4) f3(z) = (z1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1) (5) f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i) (6) f5(z) = (z1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2i)(z2 − 4)(z3 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='75i)(z4 − 8)(z5 − r4) (7) where r = 1 4 (s1 + c1i, −s1 + c1i, −s2 − c2i, s2 − c2i, 1i)T (8) with c1 = cos(2π/5), c2 = cos(π/5), s1 = sin(2π/5), and s2 = sin(4π/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The point corresponds to the vertices of a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the evaluation of Newton’s method we need to calculate the Jacobian matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', Jij = ∂fi ∂xj (here i and j are indexes) or Jf(z) = � � � � � � ∂f1 ∂z1 ∂f1 ∂z2 · · ∂f1 ∂z5 ∂f2 ∂z1 ∂f2 ∂z2 · · ∂f2 ∂z5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ∂f5 ∂z1 ∂f5 ∂z2 · · ∂f5 ∂z5 � � � � � � (9) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='02850v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='app-ph] 22 Dec 2022 therefore the root can be found as zn+1 = zn − αJ−1 f (zn)f(zn) (10) where α is a relaxation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Here we used α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In terms of an algorithm, we have the following routine Algorithm 1 Root finding with Newton’s method 1: Initial guess for z1 2: for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' , m do 3: Jf(zn) 4: αλ = 2 λmin+λmax ▷ Scaling factor: λmim/max are the min/max eigenvalues of Jf(zn) 5: Kn = I − αλJf(zn) ▷ Kernel that is fed to DCM machine 6: dn = J−1 f (zn)f(zn) ▷ Compute matrix inverse with the DCM machine 7: zn+1 = zn − αdn 8: end for 2 Details on the inverse design algorithm In this section we present the details for the inverse design algorithm implemented in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The al- gorithm consist of a part of the DDA methodology for the quantification of the problem and an its adaptation to a Lagrange formalism for solving the require inverse scattering problem, the determina- tion of the permittivity of the scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that both methods are arguably the simplest methods to follow, since they offer an intuitive understanding on the formulated problem and the coorresponding inverse-design (constraints optimization) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Notes on the DDA method In this section we present a few details regarding the DDA method used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The details can be found also in [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A similar methodological approach was also used in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We start by assuming that each 2D scatterer (assuming a point in the x-y plane) acquires its z-oriented dipole moment due to the local electric field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', p = αeloc (11) where the eloc is the vector of local z-polarized electric fields at the center of each point and α is the polarizability that depends on the shape and the material composition of each 2D scatterer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The local field is the sum of the incident field and the secondary fields generated from all the other dipoles such that: eloc = einc + Gp (12) where einc is the incident field vector, p is the induced polarization vector and G is the 2D Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In our case we consider a two-dimensional (2D) problem with a transverse electric (TE) excitation (the field is normal (z-direction) to the x-y plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Therefore the corresponding Green’s function reads G = G(ri − rk) = −j k2 0 4πε0 H(2) 0 (k0|ri − rk|) (13) where H(2) 0 (k0|ri − rk|) is the Hankel function of the second type (with the time harmonic convention e+jωt) and 0-th order and k0 = ω0√µ0ε0 is the free-space wavenumber [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The G is a CN×N Toeplitz matrix with zero diagonal entries since the |ri − rk| is treated as in (assuming a uniformly spaced discrete grid) [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (11) and (12) we obtain the following expression, arranged using the matrix formulation as follows p = � A−1 − G �−1 einc (14) 2 where the lowercase quantities p = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', pN]T , A = diag(α), α = Acellε0[ε1 − 1, ε2 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', εN − 1] (Acell is the cross-sectional area of a cylinders) and einc = [einc 1 , einc 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', einc N ]T are CN×1 vectors, diag(·) is the diagonal matrix operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, the scattered field observed at M specified discrete detection points (in general M ̸= N) is given by: esca = Gpr p = Gpr � diag(α−1) − G �−1 einc (15) where Gpr ∈ CM×N is the “propagator" Green’s function matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This propagator function connects the induced dipole polarization vectors of the scatterers with the desired detection (or objective) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The above matrix representation of the scattering problem allow us to have a clear inspection of the unknown quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These quantities are the ones that will be formulated as a Lagrange multiplier algorithm for the solution of the desired constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We note here that, as seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (15), the forward scattering problem requires a matrix inversion to evaluate the polarization density vectors induced in each scattering cell, as we have discussed in our previous work [6] in which we utilized the same DDA approach for the evaluation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (14) and the matrix-vector operation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' (15) for different excitation and for different scattering scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 Notes on the Lagrange multiplier algorithm For this, we utilize the DDA algorithm (where we closely follow the contrast source inversion method!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' [8]) and the Lagrange multiplier method for applying the constraints and finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' First, in terms of the defined problem, we have that the polarization is connected with the following expressions p = A(einc + Gp) (16) and esca = Gprp (17) A typical constrained minimization problem (primal) can be written as [9, 1, 10] min x,y y∈R 0≤y≤1 f(x, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' g(x, y) ≤ 0 (18) where f(x, y) is the objective and g(x, y) are the constraints also subject to further requirements of the problem such as y ∈ R and 0 ≥ y ≥ 1 For such problems the dual Lagrangian problem is expressed as max λ min x,y y∈R 0≤y≤1 L(x, y, λ) = f(x, y) + λg(x, y) (19) which is essentially a dual unconstrained problem (since all the constraints are encapsulated to the λ term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It is worth noting that the Lagrange multiplier should be positive real, λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finding an approximate solution to the primal inverse scattering problem is therefore reduced to finding a solution to the above dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Notice that the Lagrange multiplier can be applied to either f(x, y) or g(x, y) without affecting the outcome of the overall process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The algorithm for solving the above dual problem is the following: Step 0: initial x0 and λ0 Step 1: minimize yn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', via ∇yL(xn−1, yn, λn−1) = 0 Step 2: project yn into y ∈ R and 0 ≤ y ≤ 1 Step 3: minimize xn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ∇xL(x, yn, λn−1) = 0 Step 4: maximize λn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' ∇λL(xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' λ) = 0 Step 5: Repeat steps 1-4 until the error is minimized 3 For our particular example we have that x = p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' y = A = diag(ε − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' and f(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A) = 1/2||(A−1 − G)p − einc||2 and g(p) = 1/2||Gprp − eobj||2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' and Lagrange function reads L(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' λ) = ||(A−1 − G)p − Aeinc||2 + λ||Gpp − eobj||2 (20) The corresponding algorithmic steps are: Step 0: initial p0 and λ0 Step 1: minimize An via ∇AL(pn−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' λn−1) = 0 – We have that ∇AL(pn−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' λn−1) = ∇f(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A)∗||(A−1−G)p−einc|| (∗ is complex conjugate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This expression lead to An = p/(Gp − einc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In practice this is a simple calculation since A is a diagonal matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', A = diag(ε − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Step 2: project An into An ∈ R and 0 ≤ A ≤ 4 (for the range ε ∈ [1, 5]) – this is the point where essentially the required properties and bound of the permittivity can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These bounds or constrains can be general – the above projection is rather a simple projection that does not guarantee always the min- imum within the projection domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A more accurate projection would be of the form An = proj[An−1 − η∇A||(A−1 n−1 − G)pn−1 − einc||2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Step 3: minimize pn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ∇pL(p, An, λn−1) = 0 (DCM metadevice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' – pn = K−1 n en L – Kn = (A−1 n − G)∗(A−1 n − G) + λn−1G∗ prGpr – eL n = λn−1G∗ preobj + (A−1 n − G)∗einc – The matrix inversion pn is performed with our DCM metadevice – Due to noise error a simple weighted average filtering is applied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', pn = (1 − αF )pn−1 + αF pn with αF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='25 Step 4: maximize λn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', ∇λL(An, pn, λ) = 0 – This maximization can be calculated by a simple gradient descent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', λn = λn−1 + η (∇λL(pn, An, λ) − δ) or λn = λn−1 + η � ||Gppn − eobj||2 − δ � – Notice that this is an gradient ascent since we assume η > 0, therefore we maximize the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Step 5: Repeat steps 1-4 until the error is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In our case we used the following error – ||esca − eobj||2/||eobj||2 Note that the quantities η and δ are the step and minimal error quantities that are user determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The whole process stop either when λ reaches a plateau, or when the required error criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The optimization goal was set as ||esca−eobj||2 ||eobj||2 < δ, where esca = Gppm with pm = (A−1 m − G)−1einc being the final m-th evaluation of the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Notice that our approach has several similarities with the contrast source inversion method and other similar inverse scattering methods [11, 8, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Undoubtedly this approach is only one of the available methods for approximating the inverse design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This is rather an attempt to showcase the ability of our device for performing inverse design with desired objectives and constraints by exposing the crucial parts of the algorithm, such as the matrix inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This part is usually implicit within commercially available FDTD or FEM software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Hence here we developed our own methodology so we can have deeper inspection to quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' As a remark, the field of inverse design and inverse scattering is a very rich field with a plethora of methodologies that try to address similar problems [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4 Figure S1: Photograph of the experimental setup with the corresponding components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3 RF Design, PCB, Device Implementation A photograph of the experimental setup is shown in Fig S1, where all parts are designated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Measurement Measurements were performed using an ENA-5071C two port VNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In order to avoid the saturation of the amplifiers (multiplier module) the VNA power level was set to be −20dBm for the open loop configuration and −10dBm for the closed loop configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The VNA was set to have an IF band- width of 10 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The single frequency measurements (1601 point) at 45MHz with averaging applied after obtaining the measured signal from VNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 Multiplier The schematic of the multiplier is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The multiplier was designed to perform multi- plication on the incoming complex amplitude such that a new complex amplitude is rendered at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In other words, the output is Vout = zVin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This involves changing both the amplitude and phase of the incoming signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Phase change was performed using a pair of serially connected Minicir- cuit JSPHS-51+ Phase Shifters (PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each phase shifter provides slightly over 180 degrees of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The amplitude change was performed using the Analog Devices AD603ARZ Variable Gain Amplifier (VGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The Multiplier design contain the appropriate loads such that both the input and output of the device externally appears as 50 Ohm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Both of these devices are controlled using analog voltages with ranges of [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5V, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5V] and [0V, 12V] for the VGA and PS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In order to create a common control mechanism, op- amp level shifting circuits were used to put these on a common [0V, 5V] interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The Multiplier 5 couplers system in (trigger) system out (read DCMcontrol kernel out kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='in kernel(DCM VNA port 2 VNAport 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='switch 1-5 switch DCMpower sourceFigure S2: Schematics for the multiplier: The PCB layout design (top figure), and the corresponding subcircuit (pictures from Altium®).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Bottom figure represent the AWR Microwave Office® schematic with the realistic data board has a connection that allows for a daughter board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The daughter board is supplied with 0V and +5V and is responsible for returning two control voltages in the range of [0V, 5V].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This simple interface allows for a number of possible control schematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' At its most simple scenario, the control board can consist of a pair of potentiometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, we will present another control board which utilizes a microcontroller to receive UART input and render the two analog voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The VGA’s dynamic range could be shifted using an external resistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This was set so that the Multipliers’s range (including load elements, PS losses, etc) was [-30dB – +17dB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The multiplier effectively saturates if the input is greater than -10dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Therefore for all measurements the reference input signal that was used was -30dBm for avoiding any saturation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It should be stated that the VGA imparts a varying phase change and the PS pair imparts an amplitude change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This will be addressed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='3 1-5 splitter (5-1 combiner) The schematic of the 1-5 splitter is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' An ideal passive n-way splitter is comprised of a summation port and n feed ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The scattering parameters are expected to be reciprocal such that for the ith feed port |SSi|2 = |SiS|2 = 1/n and all other elements within the matrix are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Due to losses, a real splitter will fall short of this precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Our splitter was based on the Minicircuits AD5PS-1+, which yielded good performance at 45MHz with approximately -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2dB split ratio for all outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='Note that 1/5 ≈ −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='0dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 6 : +15 GND 15 5 C8 GND GND JGA RF GND NetU1_7 NetU1_7 GNDGND R8 2 : GND 5 : GND 000 R6 00 _2 : GND 1 : NetJ1_-1 5 R7 1 : NetC2_2 000 000 5 : GND 2 : GND C2 +15 R1 3 R2 GND 61300211121 U2 VGASub VGASub SchDoo RF_In RF_Out > Cont_ In CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 CON ACX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 PSInput JSPHS-51+ JSPHS-51+ 613b041142 C6 CNT O 10uF +5 ContProc C7 ContProcessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' SchDoc 10uF J3 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5] > PS_In PS_Out [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='15] +15 4 10uF [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5] VGA_In VGA_Out [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='51] 951103-8622-AR P3 61300411021 CNT RFIn RFOut 100R1% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='7nF5% >100R 1% 100R1% LM6172IMX 1k 1% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content="15] PS Out PS Im [0,5] VINP VOUT7 +5H VPOS O v[so's0-] 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1uF 5% GRES DINOO R7 FDBK 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5k 1% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='15k 1% AD603ARZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1uF 5% VGAIn [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5] 1k 1% Cont InSUBCKT TLIN ID=S2 TLIN SUBCKT TLIN PORT1 ID=TL1 NET=\'phase BLK" ID=TL2 ID=S1 ID=TL3 P=1 Z0=50 Ohm Vph1=V/ph Z0=50 Ohm NET="AD_603" Z0=50 Ohm Z=50 Ohm EL=el Deg Vph2=Vph EL=el Deg VG=Vg_test EL=el Deg Pwr=[-30] dBm F0=45 MHz F0=45 MHz F0=45 MHz PORT P=2 Z=50 OhmFigure S3: Schematic and layout for 1-5 splitter based on the Minicircuits AD5PS-1+ (pictures from Altium®) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='4 Feedback coupler Figure S4: Schematic and layout for the feedback coupler (pictures from Altium®)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The schematic of the feedback coupler is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S4 The Feedback coupler must perform several tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Provide near unity feedback Introduce the input signal Sample the output signal Therefore, the feedback coupler is a four-port device wherein the primary path has near unity trans- mission such that the feedback is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 Switches In order to replicate having a 10 port VNA, we utilized two demo boards (EV1HMC253AQS24), which acted as RF SP8T RF switches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', an analog multiplexer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For one SP8T, we utilized five of these ports for illuminating the bank of five couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The other SP8T was used to receive signals from the couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The remaining three ports on each were used for system sanity checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that the stock high-pass 100pF capacitors on these boards were switched to 470pF for better transmission at 45MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 7 CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 P1 CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 U1 NelJi_ 5 : GND P3 PS : Net/3 GND :NeJ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 5 : GND CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 2 : GND CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 AD5PsJi+ 139-AD5PS-1 CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031 2lst CONMCX003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='031J1 J2 OO OO 1 : NetJ1_ 1 : NetJ1_1 OGO R3 R1 OGO Nei1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Nei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Neia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 NeiJ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 R4 82 J3 J4 2 OGO OGO 1 : NetJ3_1 1 : NetJ4_1 OO OO >R3 >R1 1k 1k GND GND GND R4 R2 GND 50R 50R GNDWhile the off ports were nominally matched to 50 Ohm from DC-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5GHz, there was significant reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Deeper inspection of the datasheet indicated that the "off" ports were only matched above 500MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Measurements indicated that the off ports were approximately "open" at the design frequency and therefore reflections from the off ports could be significantly reduced with parallel 50Ohm terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, this was not done as this it would have reduced power within the system on the "on" port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Rather, we note that any polluting signal from these "open" off ports will have crossed through the coupler twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Due to the small coupling coefficient of the feedback coupler, these values will have become very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The VNA was calibrated to the end of the switch ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Measurements indicated that transmission through each of the switch ports was similar enough as to not warrant individual calibrations on each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each of the switches was actuated by three digital inputs to address the 23 = 8 ports on each switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These digital signals were created by a micro-controller which was programmed to respond to UART commands from an attached computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Code is available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='com/brianedw/RFMath/ Arduino/mcu_control_V2/mcu_control_V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='ino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='6 Micro Controller Unit (MCU) The two analog input control voltages for each Multiplier was created by an MCU Control Board, which attached directly to the Multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The heart of this board is a Metro-Mini MCU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each control line was connected to both an 8-bit PWM DAC pin (labeled “fast”) and a 10-bit PWM DAC pin (labeled “slow”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While both pins connected to the control line through a high-pass filter, the fast DAC utilized a lower capacitance and resistance than that of the slow DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' During a set operation, both pins would drive to their appropriate values, during this time, the behavior of the collective output would be dominated by the fast DAC and rapidly converge, but exhibit large ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' After 20ms, the fast PWM DAC pin would switch to a high-impedance state, leaving the voltage to settle in the remaining difference utilizing the slow DAC alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The high-pass filter was designed to maintain accuracy of 10bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Since the Metro-Mini is a 5V compliant device, the generated voltages nicely matched to the expected inputs of the Multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each MCU board had two 3-pin UART input connectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These were shorted such that one could be used to receive a command from "upstream" while the other would effectively passively repeat the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Additionally, each MCU board had two 3-pin UART output connector which were similarly shorted together, allowing it to transmit the same message to two devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Each MCU was programmed with a unique identification number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Upon receiving a UART command, it would either act on that command or repeat the command on its output UART pins for downstream devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This input/output configuration created a lot of possibilities for control topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, in practice we found that we could use a single MCU board (no multiplier attached), as a bridge between the computer and the array of MCU Boards and that this array could all be connected in parallel such that the output of the bridge was effectively driving 25 inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that the required time complexity is of the order of O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Possibly this complexity can be further reduced by implementing different connectivity schemes than the simple serial one that we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Code is available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='com/brianedw/RFMath/ Arduino/mcu_control_V2/mcu_control_V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='ino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4 Tuning/Calibration As stated in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2, the VGA has a minor effect on the phase and the PS has a minor effect on the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In other words, the phase and amplitude responses are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Additionally, other systematic errors are present such as nonidealities in the level shifting circuits due to resistor tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' When connected in a network that includes RF jumper cables of varying length, there will also be phase shifts that naturally arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In short, the relationship between the control voltages and the response of the Multiplier in situ, are repeatable, but difficult to predict without developing a more complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We found that an effective strategy to capture, model, and invert the relationship between control voltages and system response goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' A collection of Multipliers are swept across their input values to map the relationship between control voltage and complex multiplier response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These responses were analyzed using Principle Component Analysis (PCA) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The multipliers were assembled into the open-loop configuration and the response of the entire open-loop network was measured under many sets of input control voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These results were compared to a theoretical model of the network wherein the weights of the components could be adjusted until the theoretical results matched the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' With accurate PCA weights in hand, the Multipliers can be immediately adjusted to achieve a desired multiplication factor by inverting the model to achieve any open-loop kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Additional refinement can be obtained by changing the device configuration into the closed loop, which now includes the feedback couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Again, we measure the response of the closed-loop network under many input conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We further refine the PCA weights of each multiplier to match this more demanding data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This becomes our final device model for both the open- and closed-loop configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We will go into detail on each one of these items in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Multiplier PCA A collection of 35 multipliers were each mapped using the MCU control boards, capable of 10-bit resolution on both control voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The mapping occurred with a grid of values based on [0, 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', 1012, 1023] on both controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Ideally, the mapping of two Multipliers would yield identical responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, for all the reasons stated above, they do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' All of the mappings were compared using a complex domain PCA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Typically, in PCA, one would examine deviations from the mean, but here we take another approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Rather, the collection of mappings were analyzed directly to yield a set of 4 PCA components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The response of any individual Multiplier could then be found as the linear superposition of these components given by: m(dvga, dPS) = 3 � i=0 wici(dvga, dPS) The term c0(dvga, dPS) is effectively the “average” response scaled by a complex factor, while the next several components represent likely deviations due to the systematic errors described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Within a PCA analysis the final PCA components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' c34(dvga, dPS), not shown) should be nearly pure noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We found that only the first four terms were needed to effectively model any given Multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Given any randomly chosen multiplier, we can find the complex valued PCA weights wi through a least-squares analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' As opposed to the "deviation from the mean" approach, the above formulation is particularly useful for RF engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While the Multipliers were measured directly at their input and output ports and analyzed as such, the model can easily account for the addition of RF cables which would provide attenuation and phase rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These will appear as a complex scaling of all of the components weights and the Multiplier’s behavior (RF jumpers cables included) can still be captured as the simple linear superposition of the PCA components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In fact, any losses or phase rotations along the Multipliers flow path can be incorporated into these weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Therefore, we do not characterize the individual multipliers, but delay this until the architecture is fully assembled, as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Regardless, we will use least-squares to find the set of wi which characterizes the average multiplier response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We call these the “base weights”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 Open-Loop Device Fitting The goal of the this section is to determine the PCA weights that characterize each Multiplier in situ, so that the system errors can be captured and modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The open-loop DCM system was fully assembled including jumper cables, splitters, and couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' All multipliers within the array were set to the same input value (dvga, dPS) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The transmission matrix of the system was then measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This was repeated for all possible combinations of 10 evenly spaced values in the range [0, 1023] to 9 Figure S5: PCA Components and Average Multiplier Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The first four panels represent c0(dvga, dPS), c1(dvga, dPS), c2(dvga, dPS), and c3(dvga, dPS) and have a maximum saturation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The final image shows the response of the “average” Multiplier with a maximum saturation at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 yield 100 measured transmission matrices, Tmeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that not all of these 100 transmission matrices represent "passive" operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The same system was modeled using Scikit-RF, wherein the following assumptions were made: The 5-1 splitters were ideal such that power was evenly split with no phase All jumpers were zero-length The coupler feedback path was ideal with no power removed The multipliers were all assumed to be “average” and the their responses were assumed to be given by the “base weights”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The system was simulated using SciKit-RF for each input pair (dvga, dPS) to yield 100 measured transmission matrices Tsim(w), which are naturally a function of each Multipliers PCA weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We can then define an error error(w) = |Tsim(w) − Tmeas|2 and optimize w until that error is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It should be noted, that with only four PCA weights per Multiplier, in theory, only 4 transmission matrices are required to fully define the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Using 100 helps guarantee that normal measurement noise does not unduly influence the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Additionally, if a low error can be achieved across 100 measurements using only 4 weights, then we can be confident that the model was sufficient to capture the entire open loop system response, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='3 Setting the Open Loop System Response Given a desired open-loop system response, K, we need to calculate the necessary multiplier values for the DCM architecture, mi,j, gathered to form M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In this case, the simplicity of the DCM architecture makes this trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' If we assume an idealized passive five port splitters such that given an input of 1W at the summation port, s, we will observe 1/5W on each branch port, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Put in terms of Scattering Parameters, Ss,i = 1/ √ 5 and via reciprocity Si,s = 1/ √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Since we have such splitters at the input and output of the Multiplier array, K = (1/ √ 5)M(1/ √ 5) and therefore M = 5K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that since we fitted the PCA weights of the Multipliers under the assumption of ideal components, it is appropriate to assume ideal components here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' With each of the desired mi,j in hand to achieve a given K, the next step is to determine the required (dvga, dPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This can be done using a number of function inversion schemes such a gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In practice, this could be very fast as it is likely that in many applications, each new M will be a small step from the previous M and therefore each multiplier will change only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='4 Closed-Loop Device Fitting Due to the recursive nature of the closed-loop configuration (Matrix Inversion), the accuracy require- ments are more stringent than for the open-loop configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Moreover, additional degrees of freedom are introduced in the form of coupler coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' These can be considered part of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In short, the devices must be fitted again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We will employ a similar strategy as was used in the Open-Loop Device Fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Using the open- loop calibrated device models, a sequence of randomly generated passive transmission matrices, K, 10 co(dvga, dps) Ci(dvga, dps) C2(dvga, dps) C3(dvga, dps) mave(dvga, dps) imag imag ima imag dps real real real real realare shown to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note, unlike the open-loop matrices, in order to guarantee convergence, these matrices must be passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We model the closed-loop system using Scikit-RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Using the open- loop weights as a starting point, we optimize the multiplier weights and coupling coefficients until the simulated Tsim(w) matches the measured Tmeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' This represents a small, but necessary, refinement from the open-loop device model and can be used for both open- and closed-loop applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 Setting the Closed Loop System Response Setting the closed-loop system response, K is identical to setting the open-loop response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In both cases, each desired mi,j is used to find the required (dvga, dPS) using a function inversion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 5 System Accuracy We performed an open loop measurement on 100 complex-valued random matrices with (eigenvalues) values within the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For these, we configured the open-loop with the target (or ideal kernel) Ae and retrieved the measured results Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We define as error the quantity ||Am − Ae||2 ||Ae||2 100% (21) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S6, we can see the difference between the two matrices for 100 random cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' We observe that all the results are within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='05 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='3% percent error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Similarly, we performed the same error analysis for the same 100 random matrices, only this time on a closed-loop setup (matrix-inversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S7) reveal that the error can climb up to 20%, but for most of the results, we get a matrix inversion with less than 2% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Finally, we assess the matrix inversion fidelity by evaluating the trace of the A−1 m Ae product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Ideally the trace of the product tr(A−1 m Ae)/5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S8, we observe that this product spans between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, for the particular examples we used in the manuscript, this accuracy can be maintained at reasonably high levels once error-correcting and filtering techniques are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Note that for the closed-loop case, the level of the measured voltage is in the order of µV, very close to the noise floor of the VNA device we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the open loop operation, the measured voltage was hundreds of mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 0 50 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='3 Figure S6: The error between the exact and the measured matrices, open loop configuration, for 100 random complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 11 0 50 100 0 5 10 15 20 Figure S7: The error between the exact and the measured matrices, closed loop configuration (matrix inversion), for 100 random complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 6 System Transient Analysis 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Single Multiplier In terms of the time response of the multiplier module the transient analysis reveal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' S9) that the module obtained the desired value approximately within 3-4 signal periods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=', T = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The measurements were performed using the RIGOL DG4062 pulse generator (15 sinusoidal pulses at 45MHz), and the measured response extracted with the RIGOL DS1104 oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The open loop response is therefore assumed to be very close to the single multipliers response since both splitters and connecting cables introduce a small phase shift to the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The closed loop transient response is affected by both the multiplier timing and the condition number of the input matrix (kernel) as shown in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 12 0 50 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='5 2 Figure S8: The fidelity of the matrix inversion expressed in terms of the normalized trace of the A−1 m Ae product for 100 random complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' measurements in out 0 50 100 150 200 ns simulations in out Figure S9: The transient response of a single multiplier module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The blue curves correspond to the input signal, while the red curves are the measured (top) and simulated (bottom) using AWR Microwave Office® results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The agreement is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It is evident that it takes approximately 3 to 4 signal periods for the multiplier to obtain the desired output signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Here we assumed small signal amplification (VGA voltage is +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='05) and the phase shift voltage is 0V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 13 7 De-embedding the solution 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='1 Open Loop Let us define the open-loop response as Vout = KVin Note that this includes not only the DCM architecture (multipliers, splitters, jumpers), but also the through channel of the input/output coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' In other words, the open-loop is defined using all of the components of the closed-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' However, the loop has been broken "open" just after the coupler array and measured at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Since in the closed configuration, these measurement planes were coincident, upon "closing" the loop, these measurement will then represent the complete response of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' While a minor perturbation to the results, this definition assumes that the weakly coupled additional ports on the coupler are properly terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Let us further define response of only the DCM architecture as K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' When the system is in a closed loop configuration, this relates the vector exiting the coupler array (V4) to the vector incident on the coupler array (V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' V2 = K′V4 The coupler array introduces a small loss as the input is introduced and the output is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' The near unity transmission is named α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' It is clear then that K = α1K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='2 Closed Loop The closed loop response is fully defined by the open-loop response and the definition of the scattering parameters of the coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' V2 = K′V4 (22) V3 = α2V1 + βV2 (23) V4 = βV1 + α1V2 (24) Our goal is to solve the equations for V4, which represents the vectorial solution of the problem in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the expected solution, this should be done such that the solution depends only on the kernel K and the input vector (V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' For the measured solution, this should be only in terms of the measured results (V3) and the known input (V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='3 Expected Solution We begin by applying the definitions above V4 = βV1 + α1V2 V4 = βV1 + α1K′V4 V4 = βV1 + KV4 and then solve the final equation for the V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' V4 = (I − K)−1βV1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content='4 Measured Solution We begin with Eq 23: V3 = α2V1 + βV2 and then solve it for V2 V2 = 1 β V3 − α2 β V1 14 and then substitute the above into 24 V4 = βV1 + α1( 1 β V3 − α2 β V1) and then simplify V4 = (β − α1α2 β )V1 + α1 β V3 Note that in many real world cases, the coupler will be defined such that we can assume α2 → 0 V4 = βV1 + α1 β V3 References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' Bertsekas, Nonlinear programming (Athena Scientific„ Belmont, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfBQKx/content/2301.02850v1.pdf'} +page_content=' :, 1995).' metadata={'source': 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Mariani +University of Milano-Bicocca +Milan, Italy +leonardo.mariani@unimib.it +Dejan Niˇckovi´c +Austrian Institute of Technology +Vienna, Austria +Dejan.Nickovic@ait.ac.at +Drishti Yadav +TU Wien +Vienna, Austria +drishti.yadav@tuwien.ac.at +©2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or +reuse of any copyrighted component of this work in other works. +Abstract—Mutation testing is an established software quality +assurance technique for the assessment of test suites. While it is +well-suited to estimate the general fault-revealing capability of a +test suite, it is not practical and informative when the software +under test must be validated against specific requirements. This +is often the case for embedded software, where the software is +typically validated against rigorously-specified safety properties. +In such a scenario (i) a mutant is relevant only if it can impact +the satisfaction of the tested properties, and (ii) a mutant is +meaningfully-killed with respect to a property only if it causes +the violation of that property. To address these limitations of +mutation testing, we introduce property-based mutation testing, a +method for assessing the capability of a test suite to exercise +the software with respect to a given property. We evaluate +our property-based mutation testing framework on Simulink +models of safety-critical Cyber-Physical Systems (CPS) from the +automotive and avionic domains and demonstrate how property- +based mutation testing is more informative than regular mutation +testing. These results open new perspectives in both mutation +testing and test case generation of CPS. +Index Terms—Cyber-Physical Systems, Mutation Testing, Sig- +nal Temporal Logic (STL), Simulink Models, Software Testing +I. INTRODUCTION +Software has a pivotal role in safety-critical applications, +from autonomous vehicles to medical devices. Inadequate soft- +ware quality assurance may result in potentially catastrophic +system failures. It is thus important to thoroughly test software, +checking that it does not violate its critical properties. +Mutation testing (MT) is a well-established technique to +measure the adequacy of a test suite w.r.t. a fault model [1]– +[4]: MT first injects some artificial defects in the software- +under-test, and then measures the thoroughness of the test suite +as the percentage of injected faults that the test suite can reveal. +The injection is performed through mutation operators that +modify the software according to well-defined patterns. The +resulting modified program is called a mutant. A test case kills +a mutant if its execution causes observable differences in the +behavior of the original and mutated programs. The ratio of +killed mutants w.r.t. the mutants that are not equivalent to the +original program is known as the mutation score. Ideally, a +test suite should reach a mutation score equal to one. +While MT is effective when the test suite has to be assessed +against a wide set of faults spread in the software, it loses its +effectiveness when the purpose of a test suite is to validate +the software against specific requirements. This is particularly +true in the embedded software domain, where software must +be often validated against rigorously-defined safety properties. +For example, the ATCS (Automatic Transmission Controller +System) we used in the experimental evaluation is annotated +with several safety properties expressed with Signal Temporal +Logic (STL) [5], and test cases are designed to validate the +software against these properties. +When applying mutation testing to assess the capability of a +test suite to thoroughly exercise a software w.r.t. a given prop- +erty, there are two challenges to take into consideration: the +relevance of the mutants and the relevance of the executions +that kill the mutants. +Relevance of the mutants w.r.t. a tested property. Not all +the mutants are relevant to assess the thoroughness of a test +suite against a property. In fact, only the mutants whose +effects propagate in a way that ultimately causes the property +violation are relevant. A mutant that does not impact a property +shall also not contribute to measuring the adequacy of a test +suite against that property. Regular MT does not distinguish +between these mutants, and hence does not consider the +difference between them when computing the mutation score. +Relevance of the execution that kills a mutant. Producing +different outputs for the original and the mutated programs is +insufficient to kill a mutant when a test suite is assessed against +a property. In fact, a test is thoroughly exercising the software +w.r.t. a property only if the difference in the two outputs +is severe and relevant enough to cause a violation of the +property under consideration. Otherwise, the test is generating +differences that are marginal w.r.t. the testing objective. For +instance, in our evaluation, we assessed the test cases for the +ATCS against the property that requires the engine speed and +the vehicle speed to remain below certain thresholds. Several +tests succeeded in exercising a mutant in the Transmission +component, causing differences in the outputs, but failed to +produce outputs that violate these properties, which is a clear +inadequacy of the test suite. This situation is visually illus- +trated in Fig. 1 (top), where the test is generating differences in +the engine and vehicle speeds without exceeding the threshold. +The mutant would be counted as killed according to regular +mutation testing, although the test does not make the software +to violate the property. In practice, if the fault would be present +in the original model, the test would not reveal it. This also +exemplifies how mutations could be easily killed according to +regular mutation testing in data-flow models, where most of +the components are activated in every computation and values +easily propagate through the blocks in the model. However, the +arXiv:2301.13615v1 [cs.SE] 31 Jan 2023 + +propagated values often result in minor and non-significant +output differences. Killing mutants while taking the tested +properties under consideration is a definitely harder challenge. +For instance, Fig. 1 (bottom) shows the case of a test that +reveal the mutant by violating the tested properties, obtained +in our experiments. +0 +10 +20 +30 +Time (seconds) +0 +20 +40 +60 +80 +100 +120 +Vehicle Speed (mph) +Output (Vehicle Speed) +0 +10 +20 +30 +Time (seconds) +0 +1000 +2000 +3000 +4000 +Engine Speed (RPM) +Output (Engine Speed) +Original model +Mutant +Threshold +0 +10 +20 +30 +Time (seconds) +0 +20 +40 +60 +80 +100 +120 +Vehicle Speed (mph) +Output (Vehicle Speed) +0 +10 +20 +30 +Time (seconds) +0 +1000 +2000 +3000 +4000 +Engine Speed (RPM) +Output (Engine Speed) +26 +28 +30 +116 +118 +120 +122 +Fig. 1. Output plots for the original and mutated models of ATCS: (top) for +a test case satisfying the property on the mutant, (bottom) for a test case +violating the property on the mutant. The portion of the output trace (vehicle +speed) responsible for property violation is highlighted. +In this paper, we address these challenges by defining the +notion of Property-Based Mutation Testing (PBMT) to assess +test suites against properties or specifications. To this end, +we revise the key notions of mutation testing to measure the +effectiveness of the test suites as their capability to exercise +the software against a property. We also define a search-based +test generation strategy for Simulink models to effectively and +automatically identify the relevant mutants that could be killed +with meaningful executions, from a set of injected mutants. We +provide empirical evidence that PBMT is more informative +than MT to assess the thoroughness of test suites, considering +two benchmarks in the domain of safety-critical CPS with +requirements expressed in STL formalism. +In summary, this paper makes the following contributions: +1) We introduce the novel notion of Property-Based Muta- +tion Testing for testing software against properties. +2) We define a search-based strategy to automatically iden- +tify the mutants that contribute to PBMT experiments. +3) We report empirical results for Simulink models, demon- +strating that PBMT is more informative than regular MT +when software is tested against properties. +4) We make tools and experimental data publicly available +for reproduction and to ease follow-up research1. +1https://gitlab.com/DrishtiYadav/mt +Paper Organization. Section II presents the overview of +regular mutation testing. Section III presents property-based +mutation testing, our proposed approach. Section IV describes +testing CPS Simulink models against STL specifications. Sec- +tion V presents our evaluation of two safety-critical industrial +benchmarks. Section VI discusses threats to validity. Sec- +tion VII describes the lessons learned. Section VIII presents +related work. Section IX concludes the paper. +II. MUTATION TESTING +In this section, we present the background and fundamental +concepts of regular mutation testing. +Mutation testing relies on two fundamental assumptions [1], +[2]: (1) the Competent Programmer Hypothesis that states +that programmers create programs that differ from the correct +one mostly by small syntactic errors, and (2) the Coupling +Effect that asserts that “complex faults are coupled to simple +faults in such a way that a test data set that detects all +simple faults in a program will detect a high percentage +of the complex faults” [6]. Several studies investigate these +hypotheses demonstrating that results obtained with mutation +testing can reliably predict the results obtained for the vast +majority of high-priority real bugs [7]–[10]. Although not +every bug couples with mutants, mutation testing can still be +considered a good tool to measure test suite quality. +We now introduce the key concepts of mutation testing. +Definition II.1 (Mutation operator). A mutation operator is +a source-code transformation that introduces a modification in +the program-under-test. More rigorously, given a program P, +a mutation operator op is a function that takes as inputs P and +a location k inside P and creates a syntactic alteration of P +at location k, if the location can be mutated with op. +Definition II.2 (Mutant). For a given program P and a set of +mutation operators O = {op1, op2, ..., opn}, a mutant p is the +result of the application of a mutation operator op ∈ O to P +at a specified location k. A mutant created by the application +of only one mutation operator to P is known as First Order +Mutant (FOM). The application of multiple mutation operators +to P results in a Higher Order Mutant (HOM) [11]. +Given a test suite T , and a test t ∈ T , we write t |= p when +the test passes on p and t ̸|= p when the test fails on p. We +denote with O(t, p) the output generated by p with t and with +T p +U the (universal) set of every possible valid test case for p. +Definition II.3 (Killed Mutant). A mutant p is said to be killed +by T if at least one test case t in T fails when exercising p, +i.e., ∃t ∈ T : t ̸|= p. +Definition II.4 (Live Mutant). Mutants that do not lead to the +failure of any test case t ∈ T are said to be live or survived. +Formally, p is said to be live if ∀t ∈ T , t |= p. +Definition II.5 (Equivalent Mutant). A mutant p is equivalent +to the original program P if they both generate the same +output for any possible input. Formally, p is equivalent to +P if ∀t ∈ T P +U , O(t, p) = O(t, P). In other words, no test + +case can distinguish an equivalent mutant from the original +program [12]. Note that the detection of equivalent mutants is +undecidable. +Definition II.6 (Invalid Mutant). A mutant p is considered +invalid if it cannot be compiled [13]. Such a mutant is not +included in the mutation coverage. +Definition II.7 (Mutation coverage). The adequacy of a +test suite T can be measured using the mutation coverage +(hereafter, mutation score MS): the ratio of mutants killed +w.r.t. the total number of non-equivalent and valid mutants: +Mutation coverage = +#killed mutants +#valid mutants − #equivalent mutants +T is said to achieve 100% mutation test adequacy if it kills all +non-equivalent valid mutants. Full mutation coverage ensures +that T is (i) robust against the modeled mutation types, and +(ii) sensitive to small changes in the program-under-test (P). +Definition II.8 (Redundant Mutant). Redundant mutants are +not beneficial as they consume resources without contributing +to the test process as they are killed whenever other mutants +are killed. This redundancy can be expressed by duplicate +and subsumed mutants [14]. Duplicate mutants are equivalent +with each other but not equivalent to the original program [3]. +Subsumed mutants are not equivalent with each other but are +killed by the same test cases. The subsumption relation is +defined as follows [15]: We say that pi subsumes pj, denoted +pi → pj, iff the following two properties hold: +1) ∃t ∈ T P +U : t ̸|= pi. In other words, there exists some test +case t s.t. pi and P yield different outputs on t, i.e., pi +is not equivalent to P. +2) ∀t ∈ T P +U , if t ̸|= pi, then t ̸|= pj. In other words, for +every possible test case t on P, if pi yields a different +output than P on t, then so does pj. +With Regular MT, for a test case t ∈ T to kill a mutant p, +the following three conditions must be satisfied [16], [17]: +1) Reachability: t must reach the mutated statement in p. +2) Necessity: t must infect the program state by causing +different program states for p and P. +3) Sufficiency: the incorrect program state must propagate to +the output of p and be checked by t, i.e., there is an +observable difference in the outputs of p and P for t. +The above three conditions are known as the RIP model. +The capability of a test case t ∈ T to kill a mutant p is +governed by the observability of the program state, leading to +following two common types of mutation testing: +1) Weak mutation testing: A mutant p is killed by a test suite +T if only the first two conditions of the RIP model are +satisfied. +2) Strong mutation testing: For a test case t ∈ T to kill a +mutant p, all three conditions of the RIP model must be +met. +Tests, in particular automated tests, usually include an +explicit comparison of the observed program behavior to the +expected behavior using an oracle. Thus, automated tests +usually examine specific portions of the output state. However, +the oracle will fail to identify the failure if it does not check the +specific part of the output state which contains the erroneous +value. Therefore, the oracle should also reveal the failure [18], +as proposed in the RIPR model. This paper further elaborates +this concept defining how mutation testing can be designed +to validate and measure the quality of a test suite w.r.t. a +requirement, in our case taking the form of a rigorously +defined STL property for a MathWorks® Simulink model. +III. PROPERTY-BASED MUTATION TESTING +In this section, we present PBMT, a mutation testing ap- +proach designed to validate test suites against programs and +properties. We assume that we have a program P expressed in +a language L as the software-under-test (SUT), a property φ +of the SUT, a test suite T and a set of mutation operators O. +PBMT measures how thoroughly the test suite T validates P +against the property φ, studying the capability of T to reveal +faults —of type defined in O—that may impact φ. +Definition III.1 (φ-killed mutant). A mutant p is said to be +φ-killed by a test suite T ⊂ T P +U iff ∃ a test case t ∈ T such +that the following conditions hold: +1) O(t, P) |= φ, i.e., t satisfies φ when executed on the +original program P, and +2) O(t, p) ̸|= φ, i.e., t violates φ when executed on the +mutant p. It follows that t exercises the mutation/fault +in p in such a way that its effect is propagated to the +output up to the violation of the property φ. +The above two conditions collectively guarantee that the +execution of p against t yields an output strong enough to +violate φ (i.e., O(t, p) ̸|= φ), while still passing in the original +program (i.e., O(t, P) |= φ). This implies that the test is +specifically good in exercising the software so that the fault, +if present, is propagated to the output, producing significant +behavioral differences up to the point of violating φ. +Similar to the concept of equivalent mutants in regular MT, +we introduce a refined version of equivalent mutants which +we call: φ-trivially different mutants. The intuition is that in +this context, a mutant is irrelevant not only if it is equivalent +(i.e., it shows no behavioral differences w.r.t. the original +program), but also if the introduced behavioral differences are +not relevant w.r.t. the property φ, that is, no test case t ∈ T P +U +can distinguish between p and P. +Definition III.2 (φ-trivially different mutant). A mutant p is +φ-trivially different from P iff ∄t ∈ T P +U +: O(t, P) |= φ ∧ +O(t, p) ̸|= φ. +The set of the φ-trivially different mutants include equiva- +lent mutants. The identification of φ-trivially different mutants +is undecidable. +Definition III.3 (φ-adequate test suite). A test suite T is φ- +adequate w.r.t. a set of mutation operators O if it kills all the +non φ-trivially different mutants that can be generated by O. + +Definition III.4 (Mutation score). If KDφ denotes the φ- +killed mutants and NTDφ denotes the non φ-trivially different +mutants, the mutation score assigned with a test suite T for a +program P and a set of mutation operators O is +MSφ = |KDφ| +|NTDφ| +(1) +The objective of implementing test suites that are adequate +according to PBMT results in the Mutant killing problem. That +is, given a program P, a mutant of P denoted by p and a +property φ, the mutant killing problem is the problem of +finding a test case t such that O(t, P) |= φ, and O(t, p) ̸|= φ. +PBMT is usually more challenging than regular MT since: +• Higher risks of introducing φ-trivially different mutants: +PBMT can potentially generate more irrelevant mutations +than mutation testing since, in addition to equivalent +mutants, there might be mutants that are not equivalent +but introduce irrelevant differences w.r.t. a property φ. +• Harder to kill mutants: The faults must be exercised in +such a way that it does not only propagate to the output +but also leads to the violation of φ. +IV. MUTATION TESTING OF SIMULINK CPS PROGRAMS +We instantiate PBMT in the context of safety-critical CPS +Simulink (data-flow) models where the system safety prop- +erties are expressed using STL. While extensive details of +Simulink models [19]–[21] and STL [5], [22], [23] are avail- +able elsewhere, we introduce below the key concepts to make +the paper self-contained. We conclude by presenting a novel +technique to automatically determine the mutants that could +be φ-killed by test suites. +A. Simulink models +The MathWorks® Simulink environment is widely used for +CPS model-based development [24], [25]. Simulink allows +non-software engineers to design complex systems, compile +them to low-level code, and simulate the designed models to +observe their behavior against some test inputs. In general, +a Simulink model is the block diagram representation of a +system using blocks and lines (aka connections) as in Fig. 2. +A block receives data via its input ports and performs a defined +operation on its input data depending on its functionality. After +processing the input data, a block transmits the output data +via its output ports, along (directed) lines. Each line in the +model can be uniquely identified using (1) the source block +and its associated output port, and (2) the target block and its +associated input port. The model receives its inputs from a set +of input blocks and emits the output through a set of output +blocks. Usually, a block can be either atomic (i.e., it does +not include any other block within it) or hierarchical (i.e., it +includes other blocks within it). +When creating a model, a tester can either use standard +blocks from built-in libraries or create new custom blocks +from scratch. After designing the model, a tester compiles and +simulates the model using a suitable solver and simulation +Fig. 2. A Simulink model with hierarchical blocks (b4, b8) and atomic blocks +(remaining), input ports (black nodes), output ports (white nodes), inputs (In1 +and In2) and outputs (Out1, Out2 and Out3). +mode. Simulink allows to execute the model using user- +specified sample times (either fixed-length or variable-length). +A Simulink model M when simulated against a test case t +yields the model simulation output as the set of traces of all +input-internal-output signals. We denote the model simulation +output with O(t, M). A Simulink model can have multiple +outputs (such as Fig. 2’s Out1, Out2 and Out3). +B. Signal Temporal Logic (STL) +In recent years, for the verification of safety-critical CPS, +researchers have used temporal logic formalisms to express +safety properties. Signal Temporal Logic (STL) [5] is a well- +known specification formalism used to express temporal prop- +erties of dense-time real-valued behaviors of hybrid (i.e., both +continuous and discrete dynamic) systems, including safety- +critical CPS. The syntax of STL is formally defined as follows: +Φ := f(x(j)) > 0 | ¬Φ | Φ1 ∧ Φ2 | □IΦ | ♦IΦ | Φ1UIΦ2 +Here, the formula of the form f(x(j)) > 0 represents a signal +predicate, where x(j) is the value of a signal x at time instant +j, and f is a function from signal domain D to R. I ⊆ R≥0 is +an arbitrary time-interval. The propositional logic operators ¬ +and ∧ follow the obvious logical semantics, i.e., ¬ indicates +logical negation and ∧ indicates logical conjunction. Other +temporal operators are as follows: +• □IΦ (always operator) indicates that Φ must be true for +all samples in I. +• ♦IΦ (eventually operator) indicates that Φ must be true +at least once for samples in I. +• Φ1UIΦ2 means that Φ1 must be true in I until Φ2 +becomes true. UI refers to as until operator. +The Boolean satisfaction semantics aka qualitative seman- +tics of STL offers a boolean witness of the property Φ. The +Boolean satisfaction of the signal predicate is simply ⊤ if it +is satisfied; otherwise ⊥. We use the operators U, ♦, and □ +to denote UI, ♦I, and □I with I = [0, ∞). +Besides the qualitative semantics, STL also offers quanti- +tative semantics [23] that allows to compute the degree of +satisfaction of Φ by the traces generated by a system after +executing it against a test input. The degree of satisfaction of + +b1 +In1 +I1nO +Dut2 +In2 +Out3Φ for a trace q is measured using a robust satisfaction function +ρ(q, Φ) that computes a real value that indicates the distance +of the trace q from satisfying (|=s) the property Φ. Formally, +ρ(q, Φ) > 0 ⇒ q |=s Φ, and ρ(q, Φ) < 0 ⇒ q ̸|=s Φ. +C. Mutations in Simulink +From a conceptual perspective, mutations are simply mod- +ifications to the behavior of the Simulink model. Usually, +alterations can be made in a Simulink model in two ways: +1) Line mutations: changing the behavior of the signals that +propagate through lines from one block to another block +(see ‘Fault in line’ in Fig. 3), or +2) Block mutations: changing the behavior of a block (see +‘Fault in block’ in Fig. 3), for instance, by making changes +in its functionality. +Fig. 3. Mutations in a SUT (the seeded fault blocks F are highlighted in red). +A, B and C are blocks of original SUT. Internal signals s and s′ provide +knowledge of the fault location. +D. Robustness Measure +The notion of robustness function ρ becomes useful when +we need to search for a test t that passes the execution of the +model M w.r.t. an STL requirement φ. We use the following +notations [23]: +1) ρ(O(t, M), φ) < ϵ ⇒ O(t, M) ̸|= φ, i.e., t fails on M +with respect to the specification φ +2) ρ(O(t, M), φ) > ϵ ⇒ O(t, M) |= φ, i.e., t passes on M +with respect to the specification φ. +Here, the parameter ϵ represents the degree of violation of +the property as assessed by the robustness function ρ. The +standard choice is ϵ = 0 which implies that the identification +of passing or failing test case i.e., satisfaction or violation is +based on even a small (non-zero) deviation in the observed +behavior of M from the expected behavior w.r.t. φ. +E. Search-based generation of mutation adequate test cases +A key challenge in mutation testing, including PBMT, is +accurately computing the mutation score, due to the undecid- +able problem of identifying the equivalent mutants. In PBMT, +this problem is even harder due to the need of identifying +the φ-trivially different mutants, which include but are not +limited to the equivalent mutants. To address this challenge, +we defined a search-based test generation strategy that exploits +the knowledge of the mutants and their locations to generate +targeted executions that demonstrate if a mutant can be φ- +killed. Although nothing could be said about the mutants not +killed according to this procedure, the experimental results +show that assuming this procedure can identify every φ- +killable mutant may give an accurate approximation of the +mutation score. +Note that the proposed test strategy cannot be used to gen- +erate tests in a real situation, since it exploits the knowledge of +the fault location that is normally unknown when a software is +tested. However, the proposed test generation strategy is useful +in the context of PBMT to collect accurate empirical data. +In particular, we formulate the ‘Property-based test search +problem’, an optimization problem of finding a φ-adequate +test case as: +Property-based test search problem +INPUT: a Simulink model M, a first-order mutant M′ +(with signal s changed into signal s′ or a block b with +output s changed into a block b′ with output s′), and +a property φ. +PROBLEM: +Find +t +s.t. +ρ(O(t, M), φ) +> +0, +ρ(O(t, M′), φ) < 0 and D(s, s′) is maximum. +The proposed ‘Property-based test search problem’ com- +bines three key features, two deriving from the definition of +φ-killed mutant and one guiding the search toward the mutant, +and toward producing an execution that exploits the mutant to +significantly alter the state of the system: +• ρ(O(t, M), φ) > 0 requires finding a test that passes on +the original program, +• ρ(O(t, M′), φ) < 0 requires finding a test that violates +φ in the modified program, and +• D(s, s′) is maximum requires the mutation to impact on +the internal signal as much as possible. +We choose the Euclidean distance (aka L2 norm) as the +metric to compute the distance between s and s′. Since CPS +models involve continuous real-valued variables, Euclidean +distance, a prominent metric for real vector spaces, is a +good candidate for computing the distance. More rigorously, +given two finite-length signals s = (s1, · · · , sk) and s′ = +(s′ +1, · · · , s′ +k), each with k samples, the Euclidean distance +between s and s′ is mathematically expressed as: +D(s, s′) = ||s − s′||2 = +� +� +� +� +k +� +i=1 +(si − s′ +i)2 +The optimization task is to maximize D(s, s′) subject to +the constraints ρ(O(t, M), φ) > 0 and ρ(O(t, M′), φ) < 0. +To solve the formulated test search problem, we exploit +BCA [26], a recently developed global optimizer as outlined +in Algorithm 1. We chose BCA over other available optimizers +on account of its superior convergence and speed. While being +a global search with BCA in essence, Algorithm 1 introduces +two differences w.r.t. standard BCA: (1) The initial population +(Line 2) is a set of test cases randomly generated in their +valid numerical input domain. (2) Fitness (Line 3) corresponds +to the value of the test objective function for the given +population of test cases. The test objective function is obtained + +Original SUT +Mutated +(Faulty) +SUT +B +Fault in line +B +Faults in block +(block mutation)by converting the constrained optimization problem into an +unconstrained problem using the scalar penalty constraint +handling method [27]. The algorithm updates the test cases +(Line 6-8) and finds the best solution for the new population +depending on their fitness values (Lines 9-10). The candidate +fittest amongst all others in the population is accepted as the +new global best solution (Lines 11-14). The algorithm returns +the best solution if all the constraints are satisfied. Algorithm 1 +terminates (loop at Line 5) if either a test case satisfying the +optimization constraints is found, or the budget is exhausted +(time budget or the maximum number of iterations). +Algorithm 1: Search-based test generation. +Input : M : A Simulink model. +M′ : A mutant of M. +φ : An STL specification. +Output: tbest : A test case that φ-kills M′. +1 Initialize optimizer parameters +2 IP ← GENERATEINITIALPOPULATION() +3 FP ← Fitness(IP, M, M′, φ) +4 tbest, Fbest ← BestFound(FP) +5 while TimeOut() do +6 +for each candidate k ∈ IP do +7 +knew ← Update(k) +8 +end for +9 +FP ← Fitness(IP, M, M′, φ) +10 +tnew, F ← BestFound(FP) +11 +if F > Fbest then +12 +Fbest ← F ; +// update best fitness +13 +tbest ← tnew ; +// update best test +14 +end if +15 end while +16 return tbest +For each mutant, we solve the formulated ‘Property-based +test search problem’ to find a test case that φ-kills it. The +resulting test suite is a fault-directed test suite that is likely to +reveal all the non φ-trivially different mutants. +F. Test suite reduction +To maintain a small and practical fault-directed test suite, +we reduce its size automatically. We consider a test case tr +φ-redundant w.r.t. a fault-directed test suite T if the set of +φ-killed mutants by T remains unchanged after the inclusion +of tr in T , i.e., |KDφ|T = |KDφ|T ∪ tr. +A φ-non-redundant test suite does not contain φ-redundant +test cases. Usually, a test suite can contain redundant test cases +while retaining the same testing power in the sense that they +are capable of killing the same mutants w.r.t. φ. In other words, +a single test case can cover more than one mutation. +In our experiments, we use the greedy algorithm similar +to the one proposed in [28] for test suite reduction. In the +worst-case scenario, p test cases are required to cover all p +non φ-trivially different mutations. In practice, fewer tests are +usually necessary. +V. EVALUATION +Our evaluation aims to study Property-Based Mutation +Testing (PBMT) for testing CPS Simulink models against STL +properties, also w.r.t. regular Mutation Testing (MT). +A. Research Questions +Our experiments address the following research questions: +RQ1. Does PBMT measure the adequacy of a test suite +better than MT when a safety property is targeted? To answer +this research question, we assess the adequacy of multiple +test suites using both PBMT and MT, and discuss how the +resulting scores reflect the intrinsic capability of the test cases +to exercise the software based on the target property. +RQ2. Are mutation operators equally contributing in +PBMT? To answer this research question, we study the impact +of different mutation operators on the mutation score, aiming +at discovering operators that tend to generate mutants that are +either trivial or particularly hard to detect. +B. Experimental Setup +We performed our experiments on a MacBook Pro with +Apple M1 chip, 16 GB RAM, macOS Monterey with MAT- +LAB™ R2018b. For our evaluation, we developed a prototype +implementation of both PBMT and MT with CPS Simulink +models in MATLAB. We used the RTAMT library [29] for +offline evaluation of STL properties. +We limit the scope of the evaluation to FOMs. Moreover, we +use a fixed-length sampling when running Simulink models +with faults active from the beginning to the end of the +simulation. In the following, we describe our experimental +subjects, mutants and test suites. +1) Experimental subjects: We evaluate PBMT on Simulink +models of two industrial benchmarks across the safety-critical +domain, each one publicly available in the Simulink/Stateflow +online documentation of MathWorks® [30], [31]: ATCS, an +Automatic Transmission Controller System, and AECS, an +Aircraft Elevator Control System. +ATCS is a typical automotive drivetrain with the two inputs +throttle and brake governing the vehicle speed v (mph) and +the engine speed ω (RPM). Both user inputs are in the range +[0, 100] for all time instants. As one of the safety properties, +ATCS requires that v and ω must always remain below their +thresholds ¯v and ¯ω, respectively. This is represented in STL +in Table I where ¯v = 120 mph and ¯ω = 4500 RPM. +AECS from the avionics-aerospace domain controls the +positions of the left and right elevators of an aircraft using +the pilot command. In general, the elevator position should +maintain a constant value if the aircraft is flying at the desired +level. Among the safety requirements, the AECS requires that +whenever the Pilot Command cmd goes beyond a threshold +m, the measured elevator position pos must stabilize (should +not exceed cmd by more than n units) within T + a time +units. This is formally expressed with the STL specification +in Table I where m = 0.09, T = 2, a = 1 and n = 0.02. + +TABLE I +DETAILS OF SIMULINK MODELS OF OUR CASE STUDIES. +Model +Ref. +#Blocks +#Lines +φ (STL specification) +qT +Sample time +#Samples +ATCS +[32] +65 +92 +□((v ≤ ¯v) ∧ (ω ≤ ¯ω)) +30 +0.04 +751 +AECS +[33] +825 +577 +□(↑ (cmd ≥ m) → ♦[0,T ]□[0,a](|cmd − pos| ≤ n)) +10 +0.01 +1001 +2) Fault seeding and mutant generation: For each experi- +mental subject, we generated mutants using the FIM prototype +tool [34] that supports the following mutation operators for +Simulink models: Negate, Stuck-at, Absolute, Noise, +Bias/Offset, Time Delay, Package Drop, ROR (Re- +lational Operator Replacement), LOR (Logical Operator Re- +placement), S2P (Sum to Product mutation), P2S (Product to +Sum mutation) and ASR (Arithmetic Sign Replacement). The +detailed description of these operators can be found in [34]. +Since FIM does not support the injection of faults in look- +up tables (LUTs), we extended the tool implementing two +additional operators: (1) Stuck-at 0 fault in any one entry, and +(2) swapped entries (from two randomly chosen neighbors). +Table II reports, for each subject, the number of mutants +generated for the specific mutation operator. Table III indicates +the total number of mutants generated for every subject and +their generation time. Mutant generation is fast: On an average +(across ATCS and AECS), the generation of a mutant takes +1.74 seconds. +TABLE II +NUMBER OF MUTANTS OF OUR EXPERIMENTAL SUBJECTS. +Type +# Mutants +ATCS +AECS +Noise +13 +17 +Bias/Offset +13 +17 +Negate +13 +17 +Absolute +13 +17 +ROR +0 +10 +S2P +1 +3 +P2S +2 +6 +ASR +3 +8 +LUT +2 +5 +TABLE III +INFORMATION OF GENERATED MUTANTS. +Subject +Mutants generated +Mutant generation time (seconds) +ATCS +60 +68.76 +AECS +100 +261.64 +3) Test Suite: To compare PBMT to MT, we assess test +suites generated according to two different strategies: Adap- +tive Random Testing (ART) [35] and Falsification Testing +(FT) [36], [37]. ART is a baseline strategy that generates +evenly distributed test cases (within valid input ranges), +thereby ensuring adequate diversity in the test inputs. On the +other hand, FT generates counterexamples i.e., test cases that +violate a property for a given model [38], [39]. Note that ART +and FT work in radically complementary ways. ART quickly +generates many test inputs, considering diversity, but ignoring +the property under test. On the contrary, FT specifically targets +the generation of a test that violates the property under test. +In particular, for each mutant M′, FT attempts to generate a +test case t such that O(t, M′) ̸|= φ. The hypothesis is that +ART could obtain higher MS, but smaller MSφ since the +generated tests do not depend on φ. On the contrary, FT should +kill fewer mutants in general, but more mutants relevant to φ, +and thus obtain higher MSφ. +In our evaluation, we generated 30 and 50 test cases +with ART for ATCS and AECS, respectively. FT generates +a property-violating test per mutant, if successful. +For collecting data to address our research questions, we +have executed all the test cases in the test suite for every +subject and every generated mutant. To perform our exper- +iments, we executed multiple simulations in parallel using +the Parallel Computing Toolbox™ in the MATLAB/Simulink® +environment. Table IV provides, for each subject, the total +number of test cases executed (including both test suites) and +the total execution time. +TABLE IV +SCALE OF EXPERIMENTS. +Subject +Total test cases executed +Total execution time (seconds) +ATCS +90 +2,490 +AECS +150 +25,912 +C. Results +RQ1 studies the extent to which PBMT-based testing can +better capture the thoroughness of a test suite w.r.t. a safety +property that the software-under-test must fulfil. To this end, +we apply both MT and PBMT to our experimental subjects +and compute the mutation scores MS and MSφ. Note that we +use exactly the same mutants to compute both scores. Table V +reports the results. +We report the results for regular mutation testing (MT) +and Property-Based Mutation Testing (PBMT) in two different +rows, while columns ATCS and AECS correspond to the two +subject systems. For each subject system, we indicate the +scores achieved by the test suites generated with Adaptive +Random Testing (TART ) and Falsification Testing (TF T ). In +details, we report the number of mutants that have been +generated, the number of killable and φ-killable mutants, the +number of mutants killed by each test suite according to MT +and PBMT, and finally the mutation scores MS and MSφ. + +TABLE V +RESULTS OF MUTATION TESTING. +Approach +ATCS +AECS +TART +TF T +TART +TF T +MT +# Mutants +60 +60 +100 +100 +# Killable mutants +47 +47 +83 +83 +# Killed mutants +47 +46 +74 +70 +Mutation Score MS (in %) +100% +97.87% +89.15% +84.33% +PBMT +# Mutants +60 +60 +100 +100 +# φ-killable mutants +47 +47 +83 +83 +# φ-killed mutants +25 +27 +39 +35 +MSφ (in %) +53.19% +57.44% +46.98% +42.16% +To identify the killable mutants, we had to identify the +equivalent ones. To this end, we inspected the non-killed +mutants to determine if a mutation generated a variant that +cannot be distinguished from the original program. We could +identify every equivalent mutant with high-confidence. In fact, +the 13 equivalent mutants in the ATCS model all belong +to the Absolute fault type injected in the ‘Transmission’ +component and all try to change into positive values some +signals that could not be negative. The exact same situation +happened for the 17 equivalent mutants found in the AECS +model. To determine the φ-killable mutants, we used the +Search-based test generation (SBTG) technique presented in +Algorithm 1. Note that the SBTG strategy is more computa- +tionally expensive than ART and FT due to the optimization +constraints. Our procedure automatically identified every φ- +killable mutant with thirty independent runs of our search +algorithm and a maximum number of iterations (set to 1000) +as the stopping criterion. The remaining φ-trivially different +mutants are all equivalent mutants that cannot be killed. This +result provides confidence on the capability of our approach +to support fully automated experiments with Simulink models +by assuming that the mutants not killed with our strategy are +φ-trivially different mutants that do not need to be killed, and +thus can be excluded from the computation of MSφ. +By comparing the results obtained for MT to the results +obtained with PBMT, we can notice the mutation score ob- +tained with MT is significantly higher than the mutation score +obtained with PBMT. In fact, the value of MS ranges between +84.33% and 100% for the four test suites and the two subject +systems. On the other hand, the value of MSφ ranges between +42.16% and 57.44%. This is also due to the intrinsic nature of +both Simulink models and data-flow computations, where it is +generally easy to activate every component (i.e., to generate a +sequence of inputs that exercise every element in a program), +but it is definitely harder to activate these components while +guaranteeing they contribute to the computation propagating +the fault to the output, finally causing observable issues. That +is, it is relatively easy to reach faults, but it is still hard +to meaningfully propagate and detect faults. This result is +confirmed across the test suites generated with two alternative +strategies. +These results demonstrate that MT may mislead testers +when there are important properties to be validated. For in- +stance, referring to Fig. 1 (top), the test case can kill the mutant +but cannot φ-kill it. In fact, the test suites generated with ART +and FT achieve high mutation score (MS), possibly inducing +testers to believe the test suites are thoroughly exercising +software. On the contrary, it turns out that the test cases are not +good enough to guarantee that even the simple faults (e.g., like +the ones we injected) that may affect the property are actually +detected. +It is also interesting that FT, which targets the falsification of +the property, in comparison to ART, which addresses diversity +neglecting the existence of the property, does not kill more +mutants. Combined with the evidence that almost half of the +killable mutants have not been φ-killed, this suggests that more +research is needed to exercise software thoroughly w.r.t. a +target property, at least for Simulink programs. +We finally checked for the capability of the generated +tests to kill and φ-kill mutants. Interestingly, there is often +high redundancy across tests, that is, each test can kill many +mutants. For instance, all the mutants that have been killed +with ART could be killed by a single test. This reinforces the +idea that there are some surface faults that are easy to reveal, +but at the same time there are other faults that, even if simple +in structure, require more sophisticated tests to be revealed. +On the other hand, we found that four test cases, derived +with our SBTG technique are needed to reveal all 47 φ-killable +mutations of ATCS. Likewise, all 83 φ-killable mutations of +AECS could be revealed with 12 test cases. This suggests +that compact but effective test suites could be designed to +reveal faults according to PBMT. Yet, PBMT requires a higher +number of tests than regular MT to φ-kill and kill mutants, +respectively. +RQ2 assesses the contribution of individual mutation oper- +ators in PBMT. The goal is to identify the operators that tend +to generate easy-to-kill mutants (simple mutants), which do +not contribute much to measuring the adequacy of a test suite, +and the operators that tend to generate hard-to-kill mutants +(stubborn mutants), which can contribute more in measuring +the thoroughness of a test suite. +Table VI reports the following results for each mutation +operator: (1) the number of mutants generated, (2) the number +(and percentage) of φ-trivially different mutants, (3) the num- + +ber (and percentage) of NTDφ (i.e., non φ-trivially different +mutants), (4) mutation score achieved by ART, (5) mutation +score achieved by FT, and (6) number (and percentage) of +NTDφ mutants not killed by any test generation technique +(neither ART nor FT). Note that Table VI reports the combined +results for our two experimental subjects (ATCS and AECS). +At least half of the mutations generated by the Negate, +ROR, S2P and ASR operators have been killed neither by +ART nor by FT. This may suggest that these operators might +be more useful than others for PBMT because they tend to +generate faults that are not easy to propagate to the output. +For instance, all the mutants of AECS with the Negate +operator were generated by alterations in the Right Outer +Hydraulic Actuator component. The available test cases can +easily infect the execution (e.g., they change the output of the +‘Line resistance’ block), but fail to propagate the infection due +to the presence of an intermediate signal (e.g., ‘Piston Force’) +that masks changes if differences are not large enough. +None of the mutants generated by ROR has been detected +by TART and TF T . In particular, we observe that for all +available test cases t ∈ TART ∪ TF T , with the execution of +the ROR mutations, the robustness value evaluated for the STL +property for every mutant is the same as that obtained for the +original model. However, there exist test cases that produce +visible differences in the outputs and φ-kill the mutants as +demonstrated by the tests obtained with our SBTG technique. +Mutations generated by S2P have been also hard to φ- +kill. Besides, some mutations with ASR operator could not be +detected by test cases in TART and TF T . Though these mu- +tants alter the internal signal, the data-flow computations and +propagation of signals do not affect the property. For instance, +the ASR mutation in the ‘Hydraulic Actuator’ component of +Right Inner Hydraulic Actuator unit of AECS (−+ replaced +by +−) creates significant variations in the local signal but is +not strong enough to φ-kill the mutant. +On the other hand, two operators have not been particularly +useful. The Absolute operator only generated equivalent +mutants. This suggests that this operator must be used care- +fully, only with systems known to process negative values, and +possibly controlling the locations where the fault is injected. +This case is quite infrequent in CPS. In fact, we have not +observed any useful mutation in our two subjects. All the +mutations generated by LUT were easy to φ-kill, with only +one exception, which generates values hard to propagate to the +output (but still feasible to propagate as demonstrated by the +test suites generated with our SBTG approach). Although this +operator is the only one targeting look-up tables, testers might +consider skipping it when there are strong time constraints on +the testing process. +VI. THREATS TO VALIDITY +We now discuss the threats to validity centered around the +following perspectives of validity and threats: +External validity. The main threat to external validity +concerns with the generalization of our results. Indeed, the +reported evidence may not generalize to every software +system. In fact, we experimented in the domain of data- +flow oriented computations (i.e., Simulink models), and our +observations may not hold in other contexts (e.g., object- +oriented programs). However, results are already quite clear +and explainable in the domain of safety-critical CPS Simulink +programs, where testing software against safety properties is +particularly relevant. Moreover, the size of the experiment +made affordable the manual analysis of mutations to identify +equivalent mutants. +Another threat to validity is the representativeness of the +injected faults. The results reported in this study are based on +typical mutation operators for Simulink models. In particular, +we used the FIM tool [34] and its mutation operators, extended +with additional mutation operator to address lookup tables. +Internal +validity. In our experiments, we considered +only FOMs, i.e., faulty Simulink models with only one +fault/mutation. Models can have multiple faults/mutations that +may influence each other. Hence, the results might differ when +tested with multi-fault Simulink models. Nevertheless, since +most of the existing research on mutation testing focuses +on FOMs of software artifacts [40], [41], we assessed our +technique with single-fault models, leaving the study of HOMs +for future work. +Conclusion validity. Random variations is the main threat +to conclusion validity. We mitigate this threat by making thirty +independent runs of the test generation algorithms. +VII. LESSONS LEARNED +We now discuss the lessons learned from our experiments. +Lesson 1 - It is challenging to generate PBMT-adequate +test suites. Our study shows how none of the two state-of- +the-art test generation strategies for Simulink programs we +experimented with achieved high mutation score with PBMT. +Indeed, PBMT is more laborious than regular MT: a test +case that can kill a mutant might not φ-kill the same mutant. +The embedded software industry heavily relies on properties +for verification and validation activities, and it is important +to design testing tools that thoroughly exercise the software. +The definition of PBMT is a relevant advance to the state-of- +the-practice that may influence and guide the design of more +sophisticated and effective test generation strategies. +Lesson 2 - MT does not capture well the thoroughness +of a test suite. MT can still be applied to Simulink programs. +However, test generation techniques could easily kill mutants +as long as properties are not considered. This reveals that it is +important to not only design executions that cover mutants, but +that also propagate the errors produced by mutants, amplifying +its visibility on the outputs. These characteristics of a test are +not well assessed with MT. +Lesson 3 - PBMT-driven test case generation can result +in effective test cases. We defined a SBTG technique to find +test cases that demonstrate that mutants could be φ-killed. +Such a strategy has been highly effective in φ-killing mutants +and could be the basis for the design of a mutation-based test +case generation strategy. + +TABLE VI +SUMMARY OF RESULTS OF PBMT FOR INDIVIDUAL OPERATORS. +Noise +Negate +Bias +Absolute +ROR +S2P +P2S +ASR +LUT +# Mutants generated +30 +30 +30 +30 +10 +4 +8 +11 +7 +# (%) of φ-trivially different mutants +0 (0%) +0 (0%) +0 (0%) +30 (100%) +0 (0%) +0 (0%) +0 (0%) +0 (0%) +0 (0%) +# (%) NTDφ +30 (100%) +30 (100%) +30 (100%) +0 (0%) +10 (100%) +4 (100%) +8 (100%) +11 (100%) +7 (100%) +MSφART (in %) +66.67% +43.33% +46.66% +0% +0% +25% +62.5% +45.45% +85.71% +MSφF T (in %) +70% +43.33% +50% +0% +0% +25% +62.5% +45.45% +28.57% +# (%) NTDφ not killed by ART+FT +9 (30%) +17 (56.66%) +15 (50%) +0 (0%) +10 (100%) +3 (75%) +3 (37.5%) +6 (54.54%) +1 (14.28%) +Lesson 4 - Not all mutations are equally useful to +test CPS Simulink models. Based on our results, we might +deduce that some operators are more likely to generate φ- +trivially different mutants. For instance, the Absolute oper- +ator always generated equivalent mutants. On the other hand, +some operators (e.g., Negate, ROR, and ASR) generated +mutants that were hard to φ-kill, calling for test case generation +techniques that exercise the software in non-trivial ways. +VIII. RELATED WORK +Mutation Testing. From the software engineering perspec- +tive, mutation analysis is one of the powerful software testing +techniques that can evaluate the test suite quality [1], [2]. The +mutation testing and analysis literature includes a large number +of theoretical studies and empirical investigations of various +kinds of software artifacts [42], [43]. +The work in [44] combines symbolic execution, concolic +execution, and evolutionary testing to automate the test gener- +ation for weak mutation testing of programs. Along a similar +line of research, the work in [45] proposes a path selection +strategy to pick up test cases capable of killing the mutants. +Related research on test suite minimization include techniques +based on Integer Linear Programming (ILP) [46], Greedy +algorithms [28], [47], formal concept analysis [48], etc. +The most prominent works concerning the applicability of +mutation testing to safety-critical industrial systems include +the empirical investigations reported in [3], [49]–[51]. Al- +though the work in [3] proposes a well-defined mutation +analysis pipeline for test suite quality assessment of embedded +software, it misses to address the importance of properties +associated with the software and the ways to handle them +during mutation testing. Contrary to the existing research on +regular MT, we use properties—which allow us to express +software requirements and specifications—to formalize the +notion of killing the mutants. +Mutations with Simulink models. Mutation mainly relies +on alterations in the Simulink model by seeding defects using +mutation operators [52]. Researchers have proposed several +tools for creating mutants: SIMULTATE [53], MODIFI [54], +ErrorSim [55], FIBlock [56], and FIM [34]. We also mention +SLforge [19], a tool for automatically generating random valid +Simulink models for differential testing. In our experiments, +we used FIM since it provides a higher degree of automation +compared to the other tools. +Mutation-based test case generation. With regular MT, +the mutation-based test case generation approaches exploit the +mutants to generate test cases that can pick up the errors and +discover the mutants. Some approaches considered generating +tests that can reveal mutations introduced in the specification +(e.g., in UML models) [57]–[62]. PBMT is different in many +ways: it does not target mutations in the specification and it +introduces a novel notion of mutation testing. +The approaches designed to address Simulink models focus +on targeted test-data generation either using search-based test- +ing [63], [64] or behavioral analysis approaches (for instance, +bounded reachability) [65], [66]. In essence, the main objective +of these techniques is to generate a mutation-adequate test +suite that achieves full mutation coverage based on the RIP +model. Inspired by these techniques, we designed our search +strategy to automatically φ-kill mutants. Further, PBMT intro- +duces a novel instance of mutation testing that assesses the +mutation adequacy of test suites w.r.t. properties, which has +not been considered in mutation-based testing so far. +IX. CONCLUSION +We presented Property-based Mutation Testing (PBMT), +a novel approach to mutation testing that promises efficient +evaluation of test suites concerning software properties. Our +formalization of mutant killability concerns with the satisfac- +tion (and violation) of a property for the original program (and +its mutated version). We provide rigorous semantics for PBMT +and its associated mutant killing problem, enabling search- +based generation of test cases using a global optimizer. We +used different test generation strategies for creating test suites +and observed their impact on mutant killability. +We studied PBMT on two Simulink models across the +safety-critical CPS domain, providing evidence that testing +software against properties is more challenging and relevant +than opting for regular MT, in which mutants can be easily +killed. 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Kroening, “Test-case generation for embedded +simulink via formal concept analysis,” in Proceedings of the 48th Design +Automation Conference, 2011, pp. 224–229. + diff --git a/89FRT4oBgHgl3EQfqDcj/content/tmp_files/load_file.txt b/89FRT4oBgHgl3EQfqDcj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd2c596fea50959536e768f99cfc7d1f6d01603c --- /dev/null +++ b/89FRT4oBgHgl3EQfqDcj/content/tmp_files/load_file.txt @@ -0,0 +1,1109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf,len=1108 +page_content='Property-Based Mutation Testing Ezio Bartocci TU Wien Vienna, Austria ezio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='bartocci@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='at Leonardo Mariani University of Milano-Bicocca Milan, Italy leonardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='mariani@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='it Dejan Niˇckovi´c Austrian Institute of Technology Vienna, Austria Dejan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='Nickovic@ait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='at Drishti Yadav TU Wien Vienna, Austria drishti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='yadav@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='at ©2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Abstract—Mutation testing is an established software quality assurance technique for the assessment of test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' While it is well-suited to estimate the general fault-revealing capability of a test suite, it is not practical and informative when the software under test must be validated against specific requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This is often the case for embedded software, where the software is typically validated against rigorously-specified safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In such a scenario (i) a mutant is relevant only if it can impact the satisfaction of the tested properties, and (ii) a mutant is meaningfully-killed with respect to a property only if it causes the violation of that property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To address these limitations of mutation testing, we introduce property-based mutation testing, a method for assessing the capability of a test suite to exercise the software with respect to a given property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We evaluate our property-based mutation testing framework on Simulink models of safety-critical Cyber-Physical Systems (CPS) from the automotive and avionic domains and demonstrate how property- based mutation testing is more informative than regular mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' These results open new perspectives in both mutation testing and test case generation of CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Index Terms—Cyber-Physical Systems, Mutation Testing, Sig- nal Temporal Logic (STL), Simulink Models, Software Testing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' INTRODUCTION Software has a pivotal role in safety-critical applications, from autonomous vehicles to medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Inadequate soft- ware quality assurance may result in potentially catastrophic system failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' It is thus important to thoroughly test software, checking that it does not violate its critical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutation testing (MT) is a well-established technique to measure the adequacy of a test suite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a fault model [1]– [4]: MT first injects some artificial defects in the software- under-test, and then measures the thoroughness of the test suite as the percentage of injected faults that the test suite can reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The injection is performed through mutation operators that modify the software according to well-defined patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The resulting modified program is called a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A test case kills a mutant if its execution causes observable differences in the behavior of the original and mutated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The ratio of killed mutants w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' the mutants that are not equivalent to the original program is known as the mutation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Ideally, a test suite should reach a mutation score equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' While MT is effective when the test suite has to be assessed against a wide set of faults spread in the software, it loses its effectiveness when the purpose of a test suite is to validate the software against specific requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This is particularly true in the embedded software domain, where software must be often validated against rigorously-defined safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For example, the ATCS (Automatic Transmission Controller System) we used in the experimental evaluation is annotated with several safety properties expressed with Signal Temporal Logic (STL) [5], and test cases are designed to validate the software against these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' When applying mutation testing to assess the capability of a test suite to thoroughly exercise a software w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a given prop- erty, there are two challenges to take into consideration: the relevance of the mutants and the relevance of the executions that kill the mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Relevance of the mutants w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a tested property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Not all the mutants are relevant to assess the thoroughness of a test suite against a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, only the mutants whose effects propagate in a way that ultimately causes the property violation are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant that does not impact a property shall also not contribute to measuring the adequacy of a test suite against that property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Regular MT does not distinguish between these mutants, and hence does not consider the difference between them when computing the mutation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Relevance of the execution that kills a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Producing different outputs for the original and the mutated programs is insufficient to kill a mutant when a test suite is assessed against a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, a test is thoroughly exercising the software w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a property only if the difference in the two outputs is severe and relevant enough to cause a violation of the property under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Otherwise, the test is generating differences that are marginal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' the testing objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, in our evaluation, we assessed the test cases for the ATCS against the property that requires the engine speed and the vehicle speed to remain below certain thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Several tests succeeded in exercising a mutant in the Transmission component, causing differences in the outputs, but failed to produce outputs that violate these properties, which is a clear inadequacy of the test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This situation is visually illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1 (top), where the test is generating differences in the engine and vehicle speeds without exceeding the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The mutant would be counted as killed according to regular mutation testing, although the test does not make the software to violate the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In practice, if the fault would be present in the original model, the test would not reveal it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This also exemplifies how mutations could be easily killed according to regular mutation testing in data-flow models, where most of the components are activated in every computation and values easily propagate through the blocks in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='13615v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='SE] 31 Jan 2023 propagated values often result in minor and non-significant output differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Killing mutants while taking the tested properties under consideration is a definitely harder challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1 (bottom) shows the case of a test that reveal the mutant by violating the tested properties, obtained in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 0 10 20 30 Time (seconds) 0 20 40 60 80 100 120 Vehicle Speed (mph) Output (Vehicle Speed) 0 10 20 30 Time (seconds) 0 1000 2000 3000 4000 Engine Speed (RPM) Output (Engine Speed) Original model Mutant Threshold 0 10 20 30 Time (seconds) 0 20 40 60 80 100 120 Vehicle Speed (mph) Output (Vehicle Speed) 0 10 20 30 Time (seconds) 0 1000 2000 3000 4000 Engine Speed (RPM) Output (Engine Speed) 26 28 30 116 118 120 122 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Output plots for the original and mutated models of ATCS: (top) for a test case satisfying the property on the mutant, (bottom) for a test case violating the property on the mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The portion of the output trace (vehicle speed) responsible for property violation is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In this paper, we address these challenges by defining the notion of Property-Based Mutation Testing (PBMT) to assess test suites against properties or specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To this end, we revise the key notions of mutation testing to measure the effectiveness of the test suites as their capability to exercise the software against a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We also define a search-based test generation strategy for Simulink models to effectively and automatically identify the relevant mutants that could be killed with meaningful executions, from a set of injected mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We provide empirical evidence that PBMT is more informative than MT to assess the thoroughness of test suites, considering two benchmarks in the domain of safety-critical CPS with requirements expressed in STL formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In summary, this paper makes the following contributions: 1) We introduce the novel notion of Property-Based Muta- tion Testing for testing software against properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2) We define a search-based strategy to automatically iden- tify the mutants that contribute to PBMT experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 3) We report empirical results for Simulink models, demon- strating that PBMT is more informative than regular MT when software is tested against properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 4) We make tools and experimental data publicly available for reproduction and to ease follow-up research1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='com/DrishtiYadav/mt Paper Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section II presents the overview of regular mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section III presents property-based mutation testing, our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section IV describes testing CPS Simulink models against STL specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Sec- tion V presents our evaluation of two safety-critical industrial benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section VI discusses threats to validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Sec- tion VII describes the lessons learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section VIII presents related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Section IX concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' MUTATION TESTING In this section, we present the background and fundamental concepts of regular mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutation testing relies on two fundamental assumptions [1], [2]: (1) the Competent Programmer Hypothesis that states that programmers create programs that differ from the correct one mostly by small syntactic errors, and (2) the Coupling Effect that asserts that “complex faults are coupled to simple faults in such a way that a test data set that detects all simple faults in a program will detect a high percentage of the complex faults” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Several studies investigate these hypotheses demonstrating that results obtained with mutation testing can reliably predict the results obtained for the vast majority of high-priority real bugs [7]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Although not every bug couples with mutants, mutation testing can still be considered a good tool to measure test suite quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We now introduce the key concepts of mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='1 (Mutation operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutation operator is a source-code transformation that introduces a modification in the program-under-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' More rigorously, given a program P, a mutation operator op is a function that takes as inputs P and a location k inside P and creates a syntactic alteration of P at location k, if the location can be mutated with op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='2 (Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For a given program P and a set of mutation operators O = {op1, op2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', opn}, a mutant p is the result of the application of a mutation operator op ∈ O to P at a specified location k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant created by the application of only one mutation operator to P is known as First Order Mutant (FOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The application of multiple mutation operators to P results in a Higher Order Mutant (HOM) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Given a test suite T , and a test t ∈ T , we write t |= p when the test passes on p and t ̸|= p when the test fails on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We denote with O(t, p) the output generated by p with t and with T p U the (universal) set of every possible valid test case for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='3 (Killed Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant p is said to be killed by T if at least one test case t in T fails when exercising p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', ∃t ∈ T : t ̸|= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='4 (Live Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutants that do not lead to the failure of any test case t ∈ T are said to be live or survived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Formally, p is said to be live if ∀t ∈ T , t |= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='5 (Equivalent Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant p is equivalent to the original program P if they both generate the same output for any possible input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Formally, p is equivalent to P if ∀t ∈ T P U , O(t, p) = O(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In other words, no test case can distinguish an equivalent mutant from the original program [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that the detection of equivalent mutants is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='6 (Invalid Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant p is considered invalid if it cannot be compiled [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Such a mutant is not included in the mutation coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='7 (Mutation coverage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The adequacy of a test suite T can be measured using the mutation coverage (hereafter, mutation score MS): the ratio of mutants killed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' the total number of non-equivalent and valid mutants: Mutation coverage = #killed mutants #valid mutants − #equivalent mutants T is said to achieve 100% mutation test adequacy if it kills all non-equivalent valid mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Full mutation coverage ensures that T is (i) robust against the modeled mutation types, and (ii) sensitive to small changes in the program-under-test (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='8 (Redundant Mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Redundant mutants are not beneficial as they consume resources without contributing to the test process as they are killed whenever other mutants are killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This redundancy can be expressed by duplicate and subsumed mutants [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Duplicate mutants are equivalent with each other but not equivalent to the original program [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Subsumed mutants are not equivalent with each other but are killed by the same test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The subsumption relation is defined as follows [15]: We say that pi subsumes pj, denoted pi → pj, iff the following two properties hold: 1) ∃t ∈ T P U : t ̸|= pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In other words, there exists some test case t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' pi and P yield different outputs on t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', pi is not equivalent to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2) ∀t ∈ T P U , if t ̸|= pi, then t ̸|= pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In other words, for every possible test case t on P, if pi yields a different output than P on t, then so does pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' With Regular MT, for a test case t ∈ T to kill a mutant p, the following three conditions must be satisfied [16], [17]: 1) Reachability: t must reach the mutated statement in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2) Necessity: t must infect the program state by causing different program states for p and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 3) Sufficiency: the incorrect program state must propagate to the output of p and be checked by t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', there is an observable difference in the outputs of p and P for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The above three conditions are known as the RIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The capability of a test case t ∈ T to kill a mutant p is governed by the observability of the program state, leading to following two common types of mutation testing: 1) Weak mutation testing: A mutant p is killed by a test suite T if only the first two conditions of the RIP model are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2) Strong mutation testing: For a test case t ∈ T to kill a mutant p, all three conditions of the RIP model must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Tests, in particular automated tests, usually include an explicit comparison of the observed program behavior to the expected behavior using an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Thus, automated tests usually examine specific portions of the output state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, the oracle will fail to identify the failure if it does not check the specific part of the output state which contains the erroneous value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Therefore, the oracle should also reveal the failure [18], as proposed in the RIPR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This paper further elaborates this concept defining how mutation testing can be designed to validate and measure the quality of a test suite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a requirement, in our case taking the form of a rigorously defined STL property for a MathWorks® Simulink model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' PROPERTY-BASED MUTATION TESTING In this section, we present PBMT, a mutation testing ap- proach designed to validate test suites against programs and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We assume that we have a program P expressed in a language L as the software-under-test (SUT), a property φ of the SUT, a test suite T and a set of mutation operators O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' PBMT measures how thoroughly the test suite T validates P against the property φ, studying the capability of T to reveal faults —of type defined in O—that may impact φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='1 (φ-killed mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant p is said to be φ-killed by a test suite T ⊂ T P U iff ∃ a test case t ∈ T such that the following conditions hold: 1) O(t, P) |= φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', t satisfies φ when executed on the original program P, and 2) O(t, p) ̸|= φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', t violates φ when executed on the mutant p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' It follows that t exercises the mutation/fault in p in such a way that its effect is propagated to the output up to the violation of the property φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The above two conditions collectively guarantee that the execution of p against t yields an output strong enough to violate φ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', O(t, p) ̸|= φ), while still passing in the original program (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', O(t, P) |= φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This implies that the test is specifically good in exercising the software so that the fault, if present, is propagated to the output, producing significant behavioral differences up to the point of violating φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Similar to the concept of equivalent mutants in regular MT, we introduce a refined version of equivalent mutants which we call: φ-trivially different mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The intuition is that in this context, a mutant is irrelevant not only if it is equivalent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', it shows no behavioral differences w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' the original program), but also if the introduced behavioral differences are not relevant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' the property φ, that is, no test case t ∈ T P U can distinguish between p and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='2 (φ-trivially different mutant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A mutant p is φ-trivially different from P iff ∄t ∈ T P U : O(t, P) |= φ ∧ O(t, p) ̸|= φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The set of the φ-trivially different mutants include equiva- lent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The identification of φ-trivially different mutants is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='3 (φ-adequate test suite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A test suite T is φ- adequate w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a set of mutation operators O if it kills all the non φ-trivially different mutants that can be generated by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='4 (Mutation score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' If KDφ denotes the φ- killed mutants and NTDφ denotes the non φ-trivially different mutants, the mutation score assigned with a test suite T for a program P and a set of mutation operators O is MSφ = |KDφ| |NTDφ| (1) The objective of implementing test suites that are adequate according to PBMT results in the Mutant killing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' That is, given a program P, a mutant of P denoted by p and a property φ, the mutant killing problem is the problem of finding a test case t such that O(t, P) |= φ, and O(t, p) ̸|= φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' PBMT is usually more challenging than regular MT since: Higher risks of introducing φ-trivially different mutants: PBMT can potentially generate more irrelevant mutations than mutation testing since, in addition to equivalent mutants, there might be mutants that are not equivalent but introduce irrelevant differences w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a property φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Harder to kill mutants: The faults must be exercised in such a way that it does not only propagate to the output but also leads to the violation of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' MUTATION TESTING OF SIMULINK CPS PROGRAMS We instantiate PBMT in the context of safety-critical CPS Simulink (data-flow) models where the system safety prop- erties are expressed using STL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' While extensive details of Simulink models [19]–[21] and STL [5], [22], [23] are avail- able elsewhere, we introduce below the key concepts to make the paper self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We conclude by presenting a novel technique to automatically determine the mutants that could be φ-killed by test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Simulink models The MathWorks® Simulink environment is widely used for CPS model-based development [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Simulink allows non-software engineers to design complex systems, compile them to low-level code, and simulate the designed models to observe their behavior against some test inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In general, a Simulink model is the block diagram representation of a system using blocks and lines (aka connections) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A block receives data via its input ports and performs a defined operation on its input data depending on its functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' After processing the input data, a block transmits the output data via its output ports, along (directed) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Each line in the model can be uniquely identified using (1) the source block and its associated output port, and (2) the target block and its associated input port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The model receives its inputs from a set of input blocks and emits the output through a set of output blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Usually, a block can be either atomic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', it does not include any other block within it) or hierarchical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', it includes other blocks within it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' When creating a model, a tester can either use standard blocks from built-in libraries or create new custom blocks from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' After designing the model, a tester compiles and simulates the model using a suitable solver and simulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A Simulink model with hierarchical blocks (b4, b8) and atomic blocks (remaining), input ports (black nodes), output ports (white nodes), inputs (In1 and In2) and outputs (Out1, Out2 and Out3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Simulink allows to execute the model using user- specified sample times (either fixed-length or variable-length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A Simulink model M when simulated against a test case t yields the model simulation output as the set of traces of all input-internal-output signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We denote the model simulation output with O(t, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A Simulink model can have multiple outputs (such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 2’s Out1, Out2 and Out3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Signal Temporal Logic (STL) In recent years, for the verification of safety-critical CPS, researchers have used temporal logic formalisms to express safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Signal Temporal Logic (STL) [5] is a well- known specification formalism used to express temporal prop- erties of dense-time real-valued behaviors of hybrid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', both continuous and discrete dynamic) systems, including safety- critical CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The syntax of STL is formally defined as follows: Φ := f(x(j)) > 0 | ¬Φ | Φ1 ∧ Φ2 | □IΦ | ♦IΦ | Φ1UIΦ2 Here, the formula of the form f(x(j)) > 0 represents a signal predicate, where x(j) is the value of a signal x at time instant j, and f is a function from signal domain D to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' I ⊆ R≥0 is an arbitrary time-interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The propositional logic operators ¬ and ∧ follow the obvious logical semantics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', ¬ indicates logical negation and ∧ indicates logical conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Other temporal operators are as follows: □IΦ (always operator) indicates that Φ must be true for all samples in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ♦IΦ (eventually operator) indicates that Φ must be true at least once for samples in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Φ1UIΦ2 means that Φ1 must be true in I until Φ2 becomes true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' UI refers to as until operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The Boolean satisfaction semantics aka qualitative seman- tics of STL offers a boolean witness of the property Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The Boolean satisfaction of the signal predicate is simply ⊤ if it is satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' otherwise ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We use the operators U, ♦, and □ to denote UI, ♦I, and □I with I = [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Besides the qualitative semantics, STL also offers quanti- tative semantics [23] that allows to compute the degree of satisfaction of Φ by the traces generated by a system after executing it against a test input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The degree of satisfaction of b1 In1 I1nO Dut2 In2 Out3Φ for a trace q is measured using a robust satisfaction function ρ(q, Φ) that computes a real value that indicates the distance of the trace q from satisfying (|=s) the property Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Formally, ρ(q, Φ) > 0 ⇒ q |=s Φ, and ρ(q, Φ) < 0 ⇒ q ̸|=s Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutations in Simulink From a conceptual perspective, mutations are simply mod- ifications to the behavior of the Simulink model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Usually, alterations can be made in a Simulink model in two ways: 1) Line mutations: changing the behavior of the signals that propagate through lines from one block to another block (see ‘Fault in line’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 3), or 2) Block mutations: changing the behavior of a block (see ‘Fault in block’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 3), for instance, by making changes in its functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutations in a SUT (the seeded fault blocks F are highlighted in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A, B and C are blocks of original SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Internal signals s and s′ provide knowledge of the fault location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Robustness Measure The notion of robustness function ρ becomes useful when we need to search for a test t that passes the execution of the model M w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' an STL requirement φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We use the following notations [23]: 1) ρ(O(t, M), φ) < ϵ ⇒ O(t, M) ̸|= φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', t fails on M with respect to the specification φ 2) ρ(O(t, M), φ) > ϵ ⇒ O(t, M) |= φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', t passes on M with respect to the specification φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Here, the parameter ϵ represents the degree of violation of the property as assessed by the robustness function ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The standard choice is ϵ = 0 which implies that the identification of passing or failing test case i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', satisfaction or violation is based on even a small (non-zero) deviation in the observed behavior of M from the expected behavior w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Search-based generation of mutation adequate test cases A key challenge in mutation testing, including PBMT, is accurately computing the mutation score, due to the undecid- able problem of identifying the equivalent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In PBMT, this problem is even harder due to the need of identifying the φ-trivially different mutants, which include but are not limited to the equivalent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To address this challenge, we defined a search-based test generation strategy that exploits the knowledge of the mutants and their locations to generate targeted executions that demonstrate if a mutant can be φ- killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Although nothing could be said about the mutants not killed according to this procedure, the experimental results show that assuming this procedure can identify every φ- killable mutant may give an accurate approximation of the mutation score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that the proposed test strategy cannot be used to gen- erate tests in a real situation, since it exploits the knowledge of the fault location that is normally unknown when a software is tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, the proposed test generation strategy is useful in the context of PBMT to collect accurate empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In particular, we formulate the ‘Property-based test search problem’, an optimization problem of finding a φ-adequate test case as: Property-based test search problem INPUT: a Simulink model M, a first-order mutant M′ (with signal s changed into signal s′ or a block b with output s changed into a block b′ with output s′), and a property φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' PROBLEM: Find t s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ρ(O(t, M), φ) > 0, ρ(O(t, M′), φ) < 0 and D(s, s′) is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The proposed ‘Property-based test search problem’ com- bines three key features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' two deriving from the definition of φ-killed mutant and one guiding the search toward the mutant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' and toward producing an execution that exploits the mutant to significantly alter the state of the system: ρ(O(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' φ) > 0 requires finding a test that passes on the original program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ρ(O(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' M′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' φ) < 0 requires finding a test that violates φ in the modified program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' and D(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' s′) is maximum requires the mutation to impact on the internal signal as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We choose the Euclidean distance (aka L2 norm) as the metric to compute the distance between s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Since CPS models involve continuous real-valued variables, Euclidean distance, a prominent metric for real vector spaces, is a good candidate for computing the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' More rigorously, given two finite-length signals s = (s1, · · · , sk) and s′ = (s′ 1, · · · , s′ k), each with k samples, the Euclidean distance between s and s′ is mathematically expressed as: D(s, s′) = ||s − s′||2 = � � � � k � i=1 (si − s′ i)2 The optimization task is to maximize D(s, s′) subject to the constraints ρ(O(t, M), φ) > 0 and ρ(O(t, M′), φ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To solve the formulated test search problem, we exploit BCA [26], a recently developed global optimizer as outlined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We chose BCA over other available optimizers on account of its superior convergence and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' While being a global search with BCA in essence, Algorithm 1 introduces two differences w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' standard BCA: (1) The initial population (Line 2) is a set of test cases randomly generated in their valid numerical input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' (2) Fitness (Line 3) corresponds to the value of the test objective function for the given population of test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The test objective function is obtained Original SUT Mutated (Faulty) SUT B Fault in line B Faults in block (block mutation)by converting the constrained optimization problem into an unconstrained problem using the scalar penalty constraint handling method [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The algorithm updates the test cases (Line 6-8) and finds the best solution for the new population depending on their fitness values (Lines 9-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The candidate fittest amongst all others in the population is accepted as the new global best solution (Lines 11-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The algorithm returns the best solution if all the constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Algorithm 1 terminates (loop at Line 5) if either a test case satisfying the optimization constraints is found, or the budget is exhausted (time budget or the maximum number of iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Algorithm 1: Search-based test generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Input : M : A Simulink model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' M′ : A mutant of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' φ : An STL specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Output: tbest : A test case that φ-kills M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1 Initialize optimizer parameters 2 IP ← GENERATEINITIALPOPULATION() 3 FP ← Fitness(IP, M, M′, φ) 4 tbest, Fbest ← BestFound(FP) 5 while TimeOut() do 6 for each candidate k ∈ IP do 7 knew ← Update(k) 8 end for 9 FP ← Fitness(IP, M, M′, φ) 10 tnew, F ← BestFound(FP) 11 if F > Fbest then 12 Fbest ← F ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' // update best fitness 13 tbest ← tnew ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' // update best test 14 end if 15 end while 16 return tbest For each mutant, we solve the formulated ‘Property-based test search problem’ to find a test case that φ-kills it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The resulting test suite is a fault-directed test suite that is likely to reveal all the non φ-trivially different mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Test suite reduction To maintain a small and practical fault-directed test suite, we reduce its size automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We consider a test case tr φ-redundant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a fault-directed test suite T if the set of φ-killed mutants by T remains unchanged after the inclusion of tr in T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', |KDφ|T = |KDφ|T ∪ tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A φ-non-redundant test suite does not contain φ-redundant test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Usually, a test suite can contain redundant test cases while retaining the same testing power in the sense that they are capable of killing the same mutants w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In other words, a single test case can cover more than one mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In our experiments, we use the greedy algorithm similar to the one proposed in [28] for test suite reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In the worst-case scenario, p test cases are required to cover all p non φ-trivially different mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In practice, fewer tests are usually necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' EVALUATION Our evaluation aims to study Property-Based Mutation Testing (PBMT) for testing CPS Simulink models against STL properties, also w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' regular Mutation Testing (MT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Research Questions Our experiments address the following research questions: RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Does PBMT measure the adequacy of a test suite better than MT when a safety property is targeted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To answer this research question, we assess the adequacy of multiple test suites using both PBMT and MT, and discuss how the resulting scores reflect the intrinsic capability of the test cases to exercise the software based on the target property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Are mutation operators equally contributing in PBMT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To answer this research question, we study the impact of different mutation operators on the mutation score, aiming at discovering operators that tend to generate mutants that are either trivial or particularly hard to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Experimental Setup We performed our experiments on a MacBook Pro with Apple M1 chip, 16 GB RAM, macOS Monterey with MAT- LAB™ R2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For our evaluation, we developed a prototype implementation of both PBMT and MT with CPS Simulink models in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We used the RTAMT library [29] for offline evaluation of STL properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We limit the scope of the evaluation to FOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Moreover, we use a fixed-length sampling when running Simulink models with faults active from the beginning to the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In the following, we describe our experimental subjects, mutants and test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1) Experimental subjects: We evaluate PBMT on Simulink models of two industrial benchmarks across the safety-critical domain, each one publicly available in the Simulink/Stateflow online documentation of MathWorks® [30], [31]: ATCS, an Automatic Transmission Controller System, and AECS, an Aircraft Elevator Control System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ATCS is a typical automotive drivetrain with the two inputs throttle and brake governing the vehicle speed v (mph) and the engine speed ω (RPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Both user inputs are in the range [0, 100] for all time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' As one of the safety properties, ATCS requires that v and ω must always remain below their thresholds ¯v and ¯ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This is represented in STL in Table I where ¯v = 120 mph and ¯ω = 4500 RPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' AECS from the avionics-aerospace domain controls the positions of the left and right elevators of an aircraft using the pilot command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In general, the elevator position should maintain a constant value if the aircraft is flying at the desired level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Among the safety requirements, the AECS requires that whenever the Pilot Command cmd goes beyond a threshold m, the measured elevator position pos must stabilize (should not exceed cmd by more than n units) within T + a time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This is formally expressed with the STL specification in Table I where m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='09, T = 2, a = 1 and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' TABLE I DETAILS OF SIMULINK MODELS OF OUR CASE STUDIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Model Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' #Blocks #Lines φ (STL specification) qT Sample time #Samples ATCS [32] 65 92 □((v ≤ ¯v) ∧ (ω ≤ ¯ω)) 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='04 751 AECS [33] 825 577 □(↑ (cmd ≥ m) → ♦[0,T ]□[0,a](|cmd − pos| ≤ n)) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='01 1001 2) Fault seeding and mutant generation: For each experi- mental subject, we generated mutants using the FIM prototype tool [34] that supports the following mutation operators for Simulink models: Negate, Stuck-at, Absolute, Noise, Bias/Offset, Time Delay, Package Drop, ROR (Re- lational Operator Replacement), LOR (Logical Operator Re- placement), S2P (Sum to Product mutation), P2S (Product to Sum mutation) and ASR (Arithmetic Sign Replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The detailed description of these operators can be found in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Since FIM does not support the injection of faults in look- up tables (LUTs), we extended the tool implementing two additional operators: (1) Stuck-at 0 fault in any one entry, and (2) swapped entries (from two randomly chosen neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Table II reports, for each subject, the number of mutants generated for the specific mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Table III indicates the total number of mutants generated for every subject and their generation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutant generation is fast: On an average (across ATCS and AECS), the generation of a mutant takes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='74 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' TABLE II NUMBER OF MUTANTS OF OUR EXPERIMENTAL SUBJECTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Type # Mutants ATCS AECS Noise 13 17 Bias/Offset 13 17 Negate 13 17 Absolute 13 17 ROR 0 10 S2P 1 3 P2S 2 6 ASR 3 8 LUT 2 5 TABLE III INFORMATION OF GENERATED MUTANTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Subject Mutants generated Mutant generation time (seconds) ATCS 60 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='76 AECS 100 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='64 3) Test Suite: To compare PBMT to MT, we assess test suites generated according to two different strategies: Adap- tive Random Testing (ART) [35] and Falsification Testing (FT) [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ART is a baseline strategy that generates evenly distributed test cases (within valid input ranges), thereby ensuring adequate diversity in the test inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the other hand, FT generates counterexamples i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', test cases that violate a property for a given model [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that ART and FT work in radically complementary ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ART quickly generates many test inputs, considering diversity, but ignoring the property under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the contrary, FT specifically targets the generation of a test that violates the property under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In particular, for each mutant M′, FT attempts to generate a test case t such that O(t, M′) ̸|= φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The hypothesis is that ART could obtain higher MS, but smaller MSφ since the generated tests do not depend on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the contrary, FT should kill fewer mutants in general, but more mutants relevant to φ, and thus obtain higher MSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In our evaluation, we generated 30 and 50 test cases with ART for ATCS and AECS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' FT generates a property-violating test per mutant, if successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For collecting data to address our research questions, we have executed all the test cases in the test suite for every subject and every generated mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To perform our exper- iments, we executed multiple simulations in parallel using the Parallel Computing Toolbox™ in the MATLAB/Simulink® environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Table IV provides, for each subject, the total number of test cases executed (including both test suites) and the total execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' TABLE IV SCALE OF EXPERIMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Subject Total test cases executed Total execution time (seconds) ATCS 90 2,490 AECS 150 25,912 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Results RQ1 studies the extent to which PBMT-based testing can better capture the thoroughness of a test suite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a safety property that the software-under-test must fulfil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To this end, we apply both MT and PBMT to our experimental subjects and compute the mutation scores MS and MSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that we use exactly the same mutants to compute both scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Table V reports the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We report the results for regular mutation testing (MT) and Property-Based Mutation Testing (PBMT) in two different rows, while columns ATCS and AECS correspond to the two subject systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For each subject system, we indicate the scores achieved by the test suites generated with Adaptive Random Testing (TART ) and Falsification Testing (TF T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In details, we report the number of mutants that have been generated, the number of killable and φ-killable mutants, the number of mutants killed by each test suite according to MT and PBMT, and finally the mutation scores MS and MSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' TABLE V RESULTS OF MUTATION TESTING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Approach ATCS AECS TART TF T TART TF T MT # Mutants 60 60 100 100 # Killable mutants 47 47 83 83 # Killed mutants 47 46 74 70 Mutation Score MS (in %) 100% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='87% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='15% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='33% PBMT # Mutants 60 60 100 100 # φ-killable mutants 47 47 83 83 # φ-killed mutants 25 27 39 35 MSφ (in %) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='19% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='44% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='98% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='16% To identify the killable mutants, we had to identify the equivalent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To this end, we inspected the non-killed mutants to determine if a mutation generated a variant that cannot be distinguished from the original program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We could identify every equivalent mutant with high-confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, the 13 equivalent mutants in the ATCS model all belong to the Absolute fault type injected in the ‘Transmission’ component and all try to change into positive values some signals that could not be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The exact same situation happened for the 17 equivalent mutants found in the AECS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' To determine the φ-killable mutants, we used the Search-based test generation (SBTG) technique presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that the SBTG strategy is more computa- tionally expensive than ART and FT due to the optimization constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Our procedure automatically identified every φ- killable mutant with thirty independent runs of our search algorithm and a maximum number of iterations (set to 1000) as the stopping criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The remaining φ-trivially different mutants are all equivalent mutants that cannot be killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This result provides confidence on the capability of our approach to support fully automated experiments with Simulink models by assuming that the mutants not killed with our strategy are φ-trivially different mutants that do not need to be killed, and thus can be excluded from the computation of MSφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' By comparing the results obtained for MT to the results obtained with PBMT, we can notice the mutation score ob- tained with MT is significantly higher than the mutation score obtained with PBMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, the value of MS ranges between 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='33% and 100% for the four test suites and the two subject systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the other hand, the value of MSφ ranges between 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='16% and 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This is also due to the intrinsic nature of both Simulink models and data-flow computations, where it is generally easy to activate every component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', to generate a sequence of inputs that exercise every element in a program), but it is definitely harder to activate these components while guaranteeing they contribute to the computation propagating the fault to the output, finally causing observable issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' That is, it is relatively easy to reach faults, but it is still hard to meaningfully propagate and detect faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This result is confirmed across the test suites generated with two alternative strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' These results demonstrate that MT may mislead testers when there are important properties to be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For in- stance, referring to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' 1 (top), the test case can kill the mutant but cannot φ-kill it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, the test suites generated with ART and FT achieve high mutation score (MS), possibly inducing testers to believe the test suites are thoroughly exercising software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the contrary, it turns out that the test cases are not good enough to guarantee that even the simple faults (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', like the ones we injected) that may affect the property are actually detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' It is also interesting that FT, which targets the falsification of the property, in comparison to ART, which addresses diversity neglecting the existence of the property, does not kill more mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Combined with the evidence that almost half of the killable mutants have not been φ-killed, this suggests that more research is needed to exercise software thoroughly w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' a target property, at least for Simulink programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We finally checked for the capability of the generated tests to kill and φ-kill mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Interestingly, there is often high redundancy across tests, that is, each test can kill many mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, all the mutants that have been killed with ART could be killed by a single test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This reinforces the idea that there are some surface faults that are easy to reveal, but at the same time there are other faults that, even if simple in structure, require more sophisticated tests to be revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the other hand, we found that four test cases, derived with our SBTG technique are needed to reveal all 47 φ-killable mutations of ATCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Likewise, all 83 φ-killable mutations of AECS could be revealed with 12 test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This suggests that compact but effective test suites could be designed to reveal faults according to PBMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Yet, PBMT requires a higher number of tests than regular MT to φ-kill and kill mutants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' RQ2 assesses the contribution of individual mutation oper- ators in PBMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The goal is to identify the operators that tend to generate easy-to-kill mutants (simple mutants), which do not contribute much to measuring the adequacy of a test suite, and the operators that tend to generate hard-to-kill mutants (stubborn mutants), which can contribute more in measuring the thoroughness of a test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Table VI reports the following results for each mutation operator: (1) the number of mutants generated, (2) the number (and percentage) of φ-trivially different mutants, (3) the num- ber (and percentage) of NTDφ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', non φ-trivially different mutants), (4) mutation score achieved by ART, (5) mutation score achieved by FT, and (6) number (and percentage) of NTDφ mutants not killed by any test generation technique (neither ART nor FT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Note that Table VI reports the combined results for our two experimental subjects (ATCS and AECS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' At least half of the mutations generated by the Negate, ROR, S2P and ASR operators have been killed neither by ART nor by FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This may suggest that these operators might be more useful than others for PBMT because they tend to generate faults that are not easy to propagate to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, all the mutants of AECS with the Negate operator were generated by alterations in the Right Outer Hydraulic Actuator component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The available test cases can easily infect the execution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', they change the output of the ‘Line resistance’ block), but fail to propagate the infection due to the presence of an intermediate signal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', ‘Piston Force’) that masks changes if differences are not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' None of the mutants generated by ROR has been detected by TART and TF T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In particular, we observe that for all available test cases t ∈ TART ∪ TF T , with the execution of the ROR mutations, the robustness value evaluated for the STL property for every mutant is the same as that obtained for the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, there exist test cases that produce visible differences in the outputs and φ-kill the mutants as demonstrated by the tests obtained with our SBTG technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutations generated by S2P have been also hard to φ- kill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Besides, some mutations with ASR operator could not be detected by test cases in TART and TF T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Though these mu- tants alter the internal signal, the data-flow computations and propagation of signals do not affect the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, the ASR mutation in the ‘Hydraulic Actuator’ component of Right Inner Hydraulic Actuator unit of AECS (−+ replaced by +−) creates significant variations in the local signal but is not strong enough to φ-kill the mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the other hand, two operators have not been particularly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The Absolute operator only generated equivalent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This suggests that this operator must be used care- fully, only with systems known to process negative values, and possibly controlling the locations where the fault is injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This case is quite infrequent in CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, we have not observed any useful mutation in our two subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' All the mutations generated by LUT were easy to φ-kill, with only one exception, which generates values hard to propagate to the output (but still feasible to propagate as demonstrated by the test suites generated with our SBTG approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Although this operator is the only one targeting look-up tables, testers might consider skipping it when there are strong time constraints on the testing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' THREATS TO VALIDITY We now discuss the threats to validity centered around the following perspectives of validity and threats: External validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The main threat to external validity concerns with the generalization of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Indeed, the reported evidence may not generalize to every software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In fact, we experimented in the domain of data- flow oriented computations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', Simulink models), and our observations may not hold in other contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', object- oriented programs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, results are already quite clear and explainable in the domain of safety-critical CPS Simulink programs, where testing software against safety properties is particularly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Moreover, the size of the experiment made affordable the manual analysis of mutations to identify equivalent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Another threat to validity is the representativeness of the injected faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The results reported in this study are based on typical mutation operators for Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In particular, we used the FIM tool [34] and its mutation operators, extended with additional mutation operator to address lookup tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Internal validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In our experiments, we considered only FOMs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', faulty Simulink models with only one fault/mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Models can have multiple faults/mutations that may influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Hence, the results might differ when tested with multi-fault Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Nevertheless, since most of the existing research on mutation testing focuses on FOMs of software artifacts [40], [41], we assessed our technique with single-fault models, leaving the study of HOMs for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Conclusion validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Random variations is the main threat to conclusion validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We mitigate this threat by making thirty independent runs of the test generation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' LESSONS LEARNED We now discuss the lessons learned from our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Lesson 1 - It is challenging to generate PBMT-adequate test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Our study shows how none of the two state-of- the-art test generation strategies for Simulink programs we experimented with achieved high mutation score with PBMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Indeed, PBMT is more laborious than regular MT: a test case that can kill a mutant might not φ-kill the same mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The embedded software industry heavily relies on properties for verification and validation activities, and it is important to design testing tools that thoroughly exercise the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The definition of PBMT is a relevant advance to the state-of- the-practice that may influence and guide the design of more sophisticated and effective test generation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Lesson 2 - MT does not capture well the thoroughness of a test suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' MT can still be applied to Simulink programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' However, test generation techniques could easily kill mutants as long as properties are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' This reveals that it is important to not only design executions that cover mutants, but that also propagate the errors produced by mutants, amplifying its visibility on the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' These characteristics of a test are not well assessed with MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Lesson 3 - PBMT-driven test case generation can result in effective test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We defined a SBTG technique to find test cases that demonstrate that mutants could be φ-killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Such a strategy has been highly effective in φ-killing mutants and could be the basis for the design of a mutation-based test case generation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' TABLE VI SUMMARY OF RESULTS OF PBMT FOR INDIVIDUAL OPERATORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Noise Negate Bias Absolute ROR S2P P2S ASR LUT # Mutants generated 30 30 30 30 10 4 8 11 7 # (%) of φ-trivially different mutants 0 (0%) 0 (0%) 0 (0%) 30 (100%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%) # (%) NTDφ 30 (100%) 30 (100%) 30 (100%) 0 (0%) 10 (100%) 4 (100%) 8 (100%) 11 (100%) 7 (100%) MSφART (in %) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='67% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='33% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='66% 0% 0% 25% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='5% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='45% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='71% MSφF T (in %) 70% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='33% 50% 0% 0% 25% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='5% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='45% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='57% # (%) NTDφ not killed by ART+FT 9 (30%) 17 (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='66%) 15 (50%) 0 (0%) 10 (100%) 3 (75%) 3 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='5%) 6 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='54%) 1 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='28%) Lesson 4 - Not all mutations are equally useful to test CPS Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Based on our results, we might deduce that some operators are more likely to generate φ- trivially different mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' For instance, the Absolute oper- ator always generated equivalent mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' On the other hand, some operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', Negate, ROR, and ASR) generated mutants that were hard to φ-kill, calling for test case generation techniques that exercise the software in non-trivial ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' RELATED WORK Mutation Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' From the software engineering perspec- tive, mutation analysis is one of the powerful software testing techniques that can evaluate the test suite quality [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The mutation testing and analysis literature includes a large number of theoretical studies and empirical investigations of various kinds of software artifacts [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The work in [44] combines symbolic execution, concolic execution, and evolutionary testing to automate the test gener- ation for weak mutation testing of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Along a similar line of research, the work in [45] proposes a path selection strategy to pick up test cases capable of killing the mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Related research on test suite minimization include techniques based on Integer Linear Programming (ILP) [46], Greedy algorithms [28], [47], formal concept analysis [48], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The most prominent works concerning the applicability of mutation testing to safety-critical industrial systems include the empirical investigations reported in [3], [49]–[51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Al- though the work in [3] proposes a well-defined mutation analysis pipeline for test suite quality assessment of embedded software, it misses to address the importance of properties associated with the software and the ways to handle them during mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Contrary to the existing research on regular MT, we use properties—which allow us to express software requirements and specifications—to formalize the notion of killing the mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutations with Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutation mainly relies on alterations in the Simulink model by seeding defects using mutation operators [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Researchers have proposed several tools for creating mutants: SIMULTATE [53], MODIFI [54], ErrorSim [55], FIBlock [56], and FIM [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We also mention SLforge [19], a tool for automatically generating random valid Simulink models for differential testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In our experiments, we used FIM since it provides a higher degree of automation compared to the other tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Mutation-based test case generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' With regular MT, the mutation-based test case generation approaches exploit the mutants to generate test cases that can pick up the errors and discover the mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Some approaches considered generating tests that can reveal mutations introduced in the specification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=', in UML models) [57]–[62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' PBMT is different in many ways: it does not target mutations in the specification and it introduces a novel notion of mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' The approaches designed to address Simulink models focus on targeted test-data generation either using search-based test- ing [63], [64] or behavioral analysis approaches (for instance, bounded reachability) [65], [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' In essence, the main objective of these techniques is to generate a mutation-adequate test suite that achieves full mutation coverage based on the RIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Inspired by these techniques, we designed our search strategy to automatically φ-kill mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Further, PBMT intro- duces a novel instance of mutation testing that assesses the mutation adequacy of test suites w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' properties, which has not been considered in mutation-based testing so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' CONCLUSION We presented Property-based Mutation Testing (PBMT), a novel approach to mutation testing that promises efficient evaluation of test suites concerning software properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Our formalization of mutant killability concerns with the satisfac- tion (and violation) of a property for the original program (and its mutated version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We provide rigorous semantics for PBMT and its associated mutant killing problem, enabling search- based generation of test cases using a global optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We used different test generation strategies for creating test suites and observed their impact on mutant killability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We studied PBMT on two Simulink models across the safety-critical CPS domain, providing evidence that testing software against properties is more challenging and relevant than opting for regular MT, in which mutants can be easily killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Finally, our evaluation shows that state-of-the-art Adap- tive Random Testing and Falsification Testing techniques are still weak in terms of their capability of generating test suites that can effectively kill mutants when tested against properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' Future work concerns adapting PBMT to closely related CPS modeling languages, including Simulink models inte- grated with Stateflow Charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' We further plan to conduct additional investigations with HOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' ACKNOWLEDGMENT This work has been supported by the Doctoral College Resilient Embedded Systems, which is run jointly by the TU Wien’s Faculty of Informatics and the UAS Technikum Wien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FRT4oBgHgl3EQfqDcj/content/2301.13615v1.pdf'} diff --git a/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/2301.01009v1.pdf.txt b/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/2301.01009v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f38540ed337e1e4765c3ac2cef0b1efaadde19e3 --- /dev/null +++ b/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/2301.01009v1.pdf.txt @@ -0,0 +1,790 @@ +Reduced Reference Quality Assessment for Point +Cloud Compression +Yipeng Liu +Cooperative MediaNet Innovation Center +Shanghai Jiao Tong University +Shanghai, China +liuyipeng@sjtu.edu.cn +Qi Yang +Cooperative MediaNet Innovation Center +Shanghai Jiao Tong University +Shanghai, China +yang littleqi@sjtu.edu.cn +Yiling Xu +Cooperative MediaNet Innovation Center +Shanghai Jiao Tong University +Shanghai, China +yl.xu@sjtu.edu.cn +Abstract—In this paper, we propose a reduced reference (RR) +point cloud quality assessment (PCQA) model named R-PCQA +to quantify the distortions introduced by the lossy compression. +Specifically, we use the attribute and geometry quantization +steps of different compression methods (i.e., V-PCC, G-PCC and +AVS) to infer the point cloud quality, assuming that the point +clouds have no other distortions before compression. First, we +analyze the compression distortion of point clouds under separate +attribute compression and geometry compression to avoid their +mutual masking, for which we consider 5 point clouds as +references to generate a compression dataset (PCCQA) containing +independent attribute compression and geometry compression +samples. Then, we develop the proposed R-PCQA via fitting the +relationship between the quantization steps and the perceptual +quality. We evaluate the performance of R-PCQA on both the +established dataset and another independent dataset. The results +demonstrate that the proposed R-PCQA can exhibit reliable +performance and high generalization ability. +I. +INTRODUCTION +Recently, point cloud has emerged as a promising represen- +tation format in prevalent 3D applications (e.g., autonomous +driving [1] and augmented reality [2]), for which the point +cloud compression (PCC) is of great interest for providing +efficient service in practices. Currently, the Moving Picture +Experts Group (MPEG) has applied the separable measure- +ments of geometry and attribute distortion in the course of +lossy PCC. For the geometry distortion, MPEG proposes to use +the point-to-point (p2point) [3], or point-to-plane (p2plane) [4] +to quantify the spatial perturbation; while for the attribute +part, the PSNRyuv is proposed to measure the differences +between corresponding color channels. Besides these metrics +which have already been applied in MPEG PCC standardiza- +tion, some other metrics which consider more human visual +characteristics and present better performance on public PCQA +databases are also developed, such as [5]–[15]. However, they +are full reference (FR) metrics which require both the reference +and distorted samples and have high computational complexity +for real-time quality prediction. +In many practical cases, e.g., transmission, the timely +feedback is expected, and only the compressed samples and +the meta data are available, in which the reduced reference +(RR) PCQA metrics are indispensable. Only a few researches +explore the RR methods for PCQA. [16] uses the statistical in- +formation (e.g., the luminance histogram) as the substitute for +the complete samples, but still requires the backend processing. +[17] applies the quantization parameters in V-PCC to estimate +the quality of compressed samples and guide rate control, +but other prevalent compression strategies (e.g., G-PCC) are +ignored. Therefore, in this paper, we propose a general RR +PCQA model for compression distortions named R-PCQA +which only takes the attribute and geometry quantization steps +of compression schemes (including V-PCC [18], G-PCC [19] +and AVS [20], V-PCC and G-PCC are provided by MPEG +while AVS is recommended by China Audio-Video Coding +Standard) as variables, since the quality of compressed point +clouds is highly related to the compression parameters. To fully +study the relationship between the compression parameters +and perceptual quality, we first establish a complete subjective +database for PCC, named PCC quality assessment (PCCQA) +database. +The reason why we establish PCCQA while many PCQA +datasets [7], [21]–[24] have been proposed is that current +databases only consider the superimposed compression distor- +tion, i.e. lossy-geometry (G)-lossy-attribute (A) compression, +which is recommended in the Common Test Conditions (CTC) +[25]–[27]. The separate compression strategies, i.e. lossless- +G-lossy-A and lossy-G-lossless-A compression, which are +not included in the CTC are often ignored. Considering the +mutual masking between geometry and attribute distortions +[28], they are useful for exploring the relationship between the +perceptual quality and compression parameters. In PCCQA, +the reference point clouds are compressed by V-PCC, G-PCC +and AVS under lossless-G-lossy-A condition, lossy-G-lossless- +A condition, and lossy-G-lossy-A condition respectively. +To model the relationship between the perceptual quality +and the compression parameters, we first convert all the +compression parameters to the quantization steps. Then, we +model the relationship between the perceptual quality and the +attribute/geometry quantization steps respectively via using the +least square fitting. Finally, the proposed R-PCQA combines +the attribute compression model and geometry compression +model to predict the final quality scores. +The rest of this paper is organized as follows: section II +introduces the established PCCQA dataset; section III presents +the proposed R-PCQA; section IV illustrates the experiment +results; the conclusion is summarized in section V. +978-1-6654-7592-1/22/$31.00 © 2022 IEEE +arXiv:2301.01009v1 [eess.IV] 3 Jan 2023 + +II. +PCCQA DATABASE +To better explore the relationship between the perceptual +quality and the attribute/geometry compression parameters, +we first establish a database called PCCQA under several +compression conditions. +Five reference point clouds are selected from MPEG and +AVS point cloud datasets. These reference point clouds are +ensured to have no holes and other distortions under 1080P +presentation with size-2 primitives. The reference point clouds +are then distorted with 3 compression algorithms, i.e. V- +PCC [18], G-PCC [19] and AVS [20]. Each compression is +conducted under 3 conditions, i.e. lossless-G-lossy-A, lossy-G- +lossless-A, and lossy-G-lossy-A. The compression parameters +are shown in Table I. In total, 225 compressed point clouds +are generated. +TABLE I: Compression parameters used for distorted point +cloud generation. +Conditions +Parameters +GPCC lossy-G-lossless-A +VPCC lossy-G-lossless-A +AVS lossy-G-lossless-A +(positionQuantizationScale) +0.75 0.5 0.25 0.125 0.0625 +(geomQP) +22 32 37 42 51 +(geom quant step) +2 4 8 12 16 +GPCC lossless-G-lossy-A +VPCC lossless-G-lossy-A +AVS lossless-G-lossy-A +(qp) +35 39 43 47 51 +(textureQP) +32 37 42 47 51 +(attr quant param) +24 32 40 44 48 +GPCC lossy-G-lossy-A +VPCC lossy-G-lossy-A +AVS lossy-G-lossy-A +(positionQuantizationScale, qp) +0.75,35 0.5,39 0.25,43 0.125,47 0.0625,51 +(geomQP, textureQP) +24,32 28,37 32,42 36,47 40,51 +(geom quant step, attr quant param) +2,24 4,32 8,40 12,44 16,48 +To annotate the compressed point clouds, a subjective +experiment is organized to collect the Mean Opinion Scores +(MOS). We adopt the double stimulus method since it can +obtain more stable results for minor impairments. The ex- +periment process and environment setting strictly follow the +ITU-R Recommendation BT. 500 [29]. Such a method for +collecting subjective MOS is also adopted in other researches, +such as [23], [30]–[32]. +III. +PROPOSED QUALITY ASSESSMENT MODEL R-PCQA +A. Unifying the Compression Parameters +The compression parameters with different meanings are +used in V-PCC, G-PCC and AVS, but the used compression +parameters can all be converted to the quantization steps. +Thus, to better explore the relationship between the perceptual +quality and the quantization, we first convert these compres- +sion parameters to quantization steps, denoted as Qs, before +proposing the R-PCQA. +In V-PCC, the parameters textureQP and geomQP, de- +noted as QP, are used to control the attribute compression +and geometry compression respectively, which apply +Qs = round(2 +QP −4 +6 +), +(1) +where round(·) means converting a number to the nearest +integer. +In attribute compression of G-PCC, the compression pa- +rameter qp has the same meaning as QP in V-PCC, following +(a) +(b) +Fig. 1: Variation of Qs as function of MOS for different +samples. (a) under V-PCC lossless-G-lossy-A condition; (b) +under V-PCC lossy-G-lossless-A condition. +the same conversion formula in Eq. (1). The parameter posi- +tionQuantizationScale, denoted as S, is used to control the +geometry quantization, which can be converted to Qs by +Qs = 1 +S . +(2) +In AVS, the parameter attr quant param, denoted as +QPa, is used to control the attribute quantization, which can +use the following formulation to convert it to Qs +Qs = 2 +QPa +8 . +(3) +For the geometry compression of AVS, the parameter +geom quant step shares the same meaning with the quan- +tization step Qs. +B. Overall Quality Model +We use the average MOS in PCCQA to fit the mathematical +model for quality prediction. The relationships between MOS +and Qs under V-PCC lossless-G-lossy-A condition and V-PCC +lossy-G-lossless-A condition are shown in Fig. 1. We can see +that under the same compression condition, different samples +share the fitting model with basically the same shape but +are added to different additive factors. Therefore, we assume +MOS and Qs satisfy the following relationship under a certain +compression condition: +MOS = F(Qs) + c(pc), +(4) + +5 +4.5 +4 +3.5 +SO +3 +2.5 +2 +→ Sample 1 Sample 2 ^ Sample 3 +1.5 +Sample 4 X Sample 5 +1 +50 +100 +150 +200 +250 +05 +4.5 +4 +3.5 +OS +3 +2.5 +2 +1.5 +← Sample 1 + Sample 2 ^ Sample 3 +Sample 4 × Sample 5 +1 +50 +100 +150 +200 +250 +0 +Oswhere F denotes the fitting function which is related to the +quantization step Qs. c denotes the additive factor which is +related to the intrinsic characteristics of the point cloud pc. +On the whole, different samples share the same relationship +model under a certain compression condition, but they are +added to an additive sample factor. To deal with the addi- +tive sample factor, we use Qs and average MOS which is +denoted as MOS to build up the relationship model for each +compression condition: +MOSf = MOS = F(Qs) + c(pc), +(5) +where MOSf is the final predicted quality score and c denotes +the average value of additive factors. +C. Modeling the Attribute Compression +The relationships between MOS and Qs are illustrated in +Fig. 2. For the attribute compression of all V-PCC, G-PCC and +AVS compression algorithms, the relationship between MOS +and Qs follows the linear model, i.e., +MOSa = c1,a ∗ Qsa + c2,a, +(6) +where Qsa denotes the quantization steps for attribute com- +pression. c1,a and c2,a are the model parameters, whose fitting +values are shown in Table II. +TABLE II: Fitting parameters in the attribute compression +model. +V-PCC +G-PCC +AVS +c1,a +-0.0089 +-0.01 +-0.0519 +c2,a +4.4862 +5.3515 +5.1337 +D. Modeling the Geometry Compression +For geometry compression of V-PCC, the relationship +between MOS and Qs follows the natural logarithm function, +i.e., +MOSg,V −P CC = c1,g ∗ lnQsg + c2,g, +(7) +where Qsg denotes the quantization steps for geometry com- +pression. c1,g and c2,g denote the model parameters. +For geometry compression of G-PCC and AVS compres- +sion algorithms, the relationship between MOS and Qs fol- +lows the linear model, i.e., +MOSg,G−P CC,AV S = c1,g ∗ Qsg + c2,g. +(8) +The fitted parameters in the geometry compression models +are shown in Table III: +TABLE III: Fitting parameters in the geometry compression +model. +V-PCC +G-PCC +AVS +c1,g +-0.559 +-0.2381 +-0.273 +c2,g +5.4165 +5.3818 +5.5034 +E. Combining the Attribute Model and Geometry Model +The point clouds are often compressed in both attribute +and geometry, and the attribute degradation and geometry +degradation are superimposed on the point clouds at the same +time. As explored in Section IV-B, the linear combination +of the attribute model and geometry model can accurately +estimate the quality. We take the weighted summation of +MOSa and MOSg to predict the final quality scores. +For V-PCC, the established model is +MOSf = p1,a ∗ Qsa + p1,g ∗ lnQsg + P. +(9) +For G-PCC and AVS, the established model is +MOSf = p1,a ∗ Qsa + p1,g ∗ Qsg + P, +(10) +where MOSf is the predicted quality scores, Qsa is the +quantization steps for attribute compression, and Qsg is the +quantization steps for geometry compression. p1,a = 1 +2 ∗ c1,a, +p1,g = 1 +2 ∗ c1,g, and P = 1 +2 ∗ (c2,a + c2,g) to cast the predicted +quality scores under the same range of subjective scores. +F. Analysis +Some findings can be made in the experiment: i) Eq. 6 +and Eq. 7 demonstrate that for the V-PCC distortion, the +geometry distortion is more annoying compared with the +attribute distortion, but the human eyes are more sensitive to +the quantization change in the attribute compression; ii) for +the geometry compression, the fitting curves of V-PCC and G- +PCC are different, which derives from that the quantization of +V-PCC is conducted on the projection while the quantization +of G-PCC and AVS is conducted on octree; iii) for the attribute +compression, all the three compression algorithms follow the +linear model, since their quantization is all conducted on RGB, +resulting in the similar perceptual pattern. +A potential concern is whether the obtained relation func- +tion is generic for different datasets. As discussed in Section +III-B, the difference of reference samples will only affect +the additive factors, as P in Eq. 9 and Eq. 10 which is a +predefined constant. Thus, the obtained relation function can +still accurately predict the quality rank of samples in other +datasets, which is demonstrated by the cross-dataset evaluation +in Section IV-C. +IV. +EXPERIMENTS +In this section, we evaluate the performance of the pro- +posed R-PCQA on the established PCCQA and WPC [24] +dataset. Specifically, we use PCCQA to fit the model parame- +ters and evaluate the fitting errors. Then, we evaluate on WPC +dataset as cross check to verify the performance of R-PCQA +and its generalization ability. +A. Error Analysis +The proposed PCCQA consists of three parts, part 1: +lossless-G-lossy-A, part 2: lossy-G-lossless-A and part 3: +lossy-G-lossy-A. The proposed R-PCQA is fitted on the +lossless-G-lossy-A and lossy-G-lossless-A parts, and we use +the remaining lossy-G-lossy-A part to evaluate the perfor- +mance. Especially, we note the former two parts as the training + +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 2: Variation of Qs as function of average MOS for each compression condition. The top row is under lossless-G-lossy-A, +and the bottom row is under lossy-G-lossless-A. (a) (d) is for V-PCC, (b) (e) is for G-PCC, and (c) (f) is for AVS. +set and the latter part as the testing set. The mean, standard +deviation and 95% quantile of fitting errors MOS − MOSf +on the testing set are shown in Table IV. The correlation +performance on the testing set is shown in Table V. +TABLE IV: Mean, standard deviation and 95% quantile of the +fitting errors on the testing set. +Mean +Standard deviation +95% quantile +V-PCC +0.0378 +0.5885 +0.7561 +G-PCC +-0.5794 +0.5113 +0.0969 +AVS +-0.1356 +0.2845 +0.2531 +We can see from Table IV and Table V that the proposed +model can not only fit the dataset accurately, but also conforms +to the characteristics of human visual system. +B. Combination Analysis +The correlation performance of four combination schemes +of the attribute model and geometry model on the testing set +is shown in Table V. +TABLE V: Correlation performance of four combination +schemes on the testing set. +PLCC +SROCC +RMSE +PLCC +SROCC +RMSE +Linear Combination +Multiplicative Combination +V-PCC +0.8360 +0.8554 +0.5070 +0.8360 +0.8554 +0.5070 +G-PCC +0.9854 +0.9582 +0.2098 +0.9853 +0.9582 +0.2100 +AVS +0.9917 +0.9854 +0.1650 +0.9913 +0.9854 +0.1691 +GA Combination +AG Combination +V-PCC +0.8351 +0.8554 +0.5082 +0.8356 +0.8554 +0.5075 +G-PCC +0.9444 +0.9582 +0.4046 +0.9767 +0.9582 +0.2644 +AVS +0.9881 +0.9854 +0.1978 +0.9862 +0.9854 +0.2127 +We can see from Table V that: i) the linear combination +is determined due to its slightly better performance and sim- +pler calculation for two relationship model mixing; ii) the +combination schemes hardly affect the performance, which +indicates that the obtained relationship models for attribute +and geometry are independent. Due to the removal of mutual +masking, it is not necessary to consider the interaction of +attribute and geometry components in the mixing. +C. Cross-dataset Evaluation +After the model is established on the proposed dataset, we +evaluate its generalization performance on another independent +dataset, the V-PCC part of WPC [24], which contains 400 +distorted samples derived from 16 reference point clouds with +25 different quantization parameters. The results are shown in +Table VI. +TABLE VI: Cross-dataset performance on WPC dataset. +PLCC +SROCC +PLCC +SROCC +M-p2po (FR) [3] +0.61 +0.58 +H-PSNRyuv (FR) [33] +0.29 +0.23 +M-p2pl (FR) [34] +0.63 +0.59 +PCQM (FR) [12] +0.74 +0.75 +H-p2po (FR) [3] +0.51 +0.46 +GraphSIM (FR) [13] +0.74 +0.75 +H-p2pl (FR) [34] +0.55 +0.48 +MPED (FR) [35] +0.60 +0.59 +PSNRyuv (FR) [33] +0.46 +0.47 +PCM RR (RR) [16] +0.42 +0.38 +R-PCQA (RR) +0.88 +0.88 +We can see from Table VI that: i) for the compression +distortions, the proposed R-PCQA exhibits the state-of-the- +art performance which only needs the assistance of two +compression parameters, even compared with the existing FR +metrics; ii) the model parameters derived from PCCQA still +exhibit robust performance on another independent dataset, +which demonstrates the generalization ability of the proposed +RR metric R-PCQA; iii) the massive increase in points after +reconstruction may interfere with the measurement of point- +wise FR metrics, resulting in the poor performance of some +FR metrics. +V. +CONCLUSION +In this paper, we analyze the compression distortions of +point clouds under separate attribute compression and geom- +etry compression to avoid their mutual masking. Then by +fitting the relationship between the quantization steps and the + +5 +4.5 +SO +4 +3.5 +3 +2.5 +2 +1.5 +1 +0 +100 +200 +300 +s5 +4.5 +SO) +4 +3.5 +3 +2.5 +2 +1.5 +1 +0 +100 +200 +300 +Os5 +4.5 +SOI +4 +3.5 +Average +3 +2.5 +2 +1.5 +1 +0 +20 +40 +60 +80 +Qs5 +4.5 +so +4 +7 +3.5 +3 +2.5 +2 +1.5 +1 +0 +100 +200 +300 +Os5 +4.5 +SOI +4 +M +3.5 +Average +3 +2.5 +2 +1.5 +1 +0 +5 +10 +15 +20 +Qs5 +4.5 +SOI +4 +3.5 +Average +3 +2.5 +2 +1.5 +1 +0 +5 +10 +15 +20 +Qsperceptual quality, we propose a RR PCQA model, called R- +PCQA, for evaluating V-PCC, G-PCC and AVS distortions. +The experiment results have demonstrated that the proposed +R-PCQA exhibits reliable and robust performance. +VI. +ACKNOWLEDGEMENT +This paper is supported in part by National Key R&D +Program of China (2018YFE0206700), National Natural Sci- +ence Foundation of China (61971282, U20A20185). The cor- +responding author is Yiling Xu (e-mail: yl.xu@sjtu.edu.cn). +REFERENCES +[1] +Y. Li, L. Ma, Z. Zhong, F. Liu, M. A. 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Chen et al., “MPED: Quantifying point cloud +distortion based on multiscale potential energy discrepancy,” arXiv +preprint arXiv:2103.02850, 2021. + diff --git a/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/load_file.txt b/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73c0e21dfc677863c6d395865770ab5b484c8353 --- /dev/null +++ b/9tAzT4oBgHgl3EQfFPrO/content/tmp_files/load_file.txt @@ -0,0 +1,481 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf,len=480 +page_content='Reduced Reference Quality Assessment for Point Cloud Compression Yipeng Liu Cooperative MediaNet Innovation Center Shanghai Jiao Tong University Shanghai, China liuyipeng@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='cn Qi Yang Cooperative MediaNet Innovation Center Shanghai Jiao Tong University Shanghai, China yang littleqi@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='cn Yiling Xu Cooperative MediaNet Innovation Center Shanghai Jiao Tong University Shanghai, China yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='xu@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='cn Abstract—In this paper, we propose a reduced reference (RR) point cloud quality assessment (PCQA) model named R-PCQA to quantify the distortions introduced by the lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Specifically, we use the attribute and geometry quantization steps of different compression methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', V-PCC, G-PCC and AVS) to infer the point cloud quality, assuming that the point clouds have no other distortions before compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' First, we analyze the compression distortion of point clouds under separate attribute compression and geometry compression to avoid their mutual masking, for which we consider 5 point clouds as references to generate a compression dataset (PCCQA) containing independent attribute compression and geometry compression samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Then, we develop the proposed R-PCQA via fitting the relationship between the quantization steps and the perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' We evaluate the performance of R-PCQA on both the established dataset and another independent dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The results demonstrate that the proposed R-PCQA can exhibit reliable performance and high generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' INTRODUCTION Recently, point cloud has emerged as a promising represen- tation format in prevalent 3D applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', autonomous driving [1] and augmented reality [2]), for which the point cloud compression (PCC) is of great interest for providing efficient service in practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Currently, the Moving Picture Experts Group (MPEG) has applied the separable measure- ments of geometry and attribute distortion in the course of lossy PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' For the geometry distortion, MPEG proposes to use the point-to-point (p2point) [3], or point-to-plane (p2plane) [4] to quantify the spatial perturbation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' while for the attribute part, the PSNRyuv is proposed to measure the differences between corresponding color channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Besides these metrics which have already been applied in MPEG PCC standardiza- tion, some other metrics which consider more human visual characteristics and present better performance on public PCQA databases are also developed, such as [5]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' However, they are full reference (FR) metrics which require both the reference and distorted samples and have high computational complexity for real-time quality prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' In many practical cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', transmission, the timely feedback is expected, and only the compressed samples and the meta data are available, in which the reduced reference (RR) PCQA metrics are indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Only a few researches explore the RR methods for PCQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' [16] uses the statistical in- formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', the luminance histogram) as the substitute for the complete samples, but still requires the backend processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' [17] applies the quantization parameters in V-PCC to estimate the quality of compressed samples and guide rate control, but other prevalent compression strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', G-PCC) are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Therefore, in this paper, we propose a general RR PCQA model for compression distortions named R-PCQA which only takes the attribute and geometry quantization steps of compression schemes (including V-PCC [18], G-PCC [19] and AVS [20], V-PCC and G-PCC are provided by MPEG while AVS is recommended by China Audio-Video Coding Standard) as variables, since the quality of compressed point clouds is highly related to the compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' To fully study the relationship between the compression parameters and perceptual quality, we first establish a complete subjective database for PCC, named PCC quality assessment (PCCQA) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The reason why we establish PCCQA while many PCQA datasets [7], [21]–[24] have been proposed is that current databases only consider the superimposed compression distor- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' lossy-geometry (G)-lossy-attribute (A) compression, which is recommended in the Common Test Conditions (CTC) [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The separate compression strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' lossless- G-lossy-A and lossy-G-lossless-A compression, which are not included in the CTC are often ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Considering the mutual masking between geometry and attribute distortions [28], they are useful for exploring the relationship between the perceptual quality and compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' In PCCQA, the reference point clouds are compressed by V-PCC, G-PCC and AVS under lossless-G-lossy-A condition, lossy-G-lossless- A condition, and lossy-G-lossy-A condition respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' To model the relationship between the perceptual quality and the compression parameters, we first convert all the compression parameters to the quantization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Then, we model the relationship between the perceptual quality and the attribute/geometry quantization steps respectively via using the least square fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Finally, the proposed R-PCQA combines the attribute compression model and geometry compression model to predict the final quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The rest of this paper is organized as follows: section II introduces the established PCCQA dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' section III presents the proposed R-PCQA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' section IV illustrates the experiment results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' the conclusion is summarized in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 978-1-6654-7592-1/22/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='00 © 2022 IEEE arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='01009v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='IV] 3 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' PCCQA DATABASE To better explore the relationship between the perceptual quality and the attribute/geometry compression parameters, we first establish a database called PCCQA under several compression conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Five reference point clouds are selected from MPEG and AVS point cloud datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' These reference point clouds are ensured to have no holes and other distortions under 1080P presentation with size-2 primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The reference point clouds are then distorted with 3 compression algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' V- PCC [18], G-PCC [19] and AVS [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Each compression is conducted under 3 conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' lossless-G-lossy-A, lossy-G- lossless-A, and lossy-G-lossy-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The compression parameters are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' In total, 225 compressed point clouds are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' TABLE I: Compression parameters used for distorted point cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Conditions Parameters GPCC lossy-G-lossless-A VPCC lossy-G-lossless-A AVS lossy-G-lossless-A (positionQuantizationScale) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0625 (geomQP) 22 32 37 42 51 (geom quant step) 2 4 8 12 16 GPCC lossless-G-lossy-A VPCC lossless-G-lossy-A AVS lossless-G-lossy-A (qp) 35 39 43 47 51 (textureQP) 32 37 42 47 51 (attr quant param) 24 32 40 44 48 GPCC lossy-G-lossy-A VPCC lossy-G-lossy-A AVS lossy-G-lossy-A (positionQuantizationScale, qp) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='75,35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5,39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='25,43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='125,47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0625,51 (geomQP, textureQP) 24,32 28,37 32,42 36,47 40,51 (geom quant step, attr quant param) 2,24 4,32 8,40 12,44 16,48 To annotate the compressed point clouds, a subjective experiment is organized to collect the Mean Opinion Scores (MOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' We adopt the double stimulus method since it can obtain more stable results for minor impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The ex- periment process and environment setting strictly follow the ITU-R Recommendation BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 500 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Such a method for collecting subjective MOS is also adopted in other researches, such as [23], [30]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' PROPOSED QUALITY ASSESSMENT MODEL R-PCQA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Unifying the Compression Parameters The compression parameters with different meanings are used in V-PCC, G-PCC and AVS, but the used compression parameters can all be converted to the quantization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Thus, to better explore the relationship between the perceptual quality and the quantization, we first convert these compres- sion parameters to quantization steps, denoted as Qs, before proposing the R-PCQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' In V-PCC, the parameters textureQP and geomQP, de- noted as QP, are used to control the attribute compression and geometry compression respectively, which apply Qs = round(2 QP −4 6 ), (1) where round(·) means converting a number to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' In attribute compression of G-PCC, the compression pa- rameter qp has the same meaning as QP in V-PCC, following (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 1: Variation of Qs as function of MOS for different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (a) under V-PCC lossless-G-lossy-A condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (b) under V-PCC lossy-G-lossless-A condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' the same conversion formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The parameter posi- tionQuantizationScale, denoted as S, is used to control the geometry quantization, which can be converted to Qs by Qs = 1 S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (2) In AVS, the parameter attr quant param, denoted as QPa, is used to control the attribute quantization, which can use the following formulation to convert it to Qs Qs = 2 QPa 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (3) For the geometry compression of AVS, the parameter geom quant step shares the same meaning with the quan- tization step Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Overall Quality Model We use the average MOS in PCCQA to fit the mathematical model for quality prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The relationships between MOS and Qs under V-PCC lossless-G-lossy-A condition and V-PCC lossy-G-lossless-A condition are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' We can see that under the same compression condition, different samples share the fitting model with basically the same shape but are added to different additive factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Therefore, we assume MOS and Qs satisfy the following relationship under a certain compression condition: MOS = F(Qs) + c(pc), (4) 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SO 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 → Sample 1 Sample 2 ^ Sample 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 Sample 4 X Sample 5 1 50 100 150 200 250 05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 OS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 ← Sample 1 Sample 2 ^ Sample 3 Sample 4 × Sample 5 1 50 100 150 200 250 0 Oswhere F denotes the fitting function which is related to the quantization step Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' c denotes the additive factor which is related to the intrinsic characteristics of the point cloud pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' On the whole, different samples share the same relationship model under a certain compression condition, but they are added to an additive sample factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' To deal with the addi- tive sample factor, we use Qs and average MOS which is denoted as MOS to build up the relationship model for each compression condition: MOSf = MOS = F(Qs) + c(pc), (5) where MOSf is the final predicted quality score and c denotes the average value of additive factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Modeling the Attribute Compression The relationships between MOS and Qs are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' For the attribute compression of all V-PCC, G-PCC and AVS compression algorithms, the relationship between MOS and Qs follows the linear model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', MOSa = c1,a ∗ Qsa + c2,a, (6) where Qsa denotes the quantization steps for attribute com- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' c1,a and c2,a are the model parameters, whose fitting values are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' TABLE II: Fitting parameters in the attribute compression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' V-PCC G-PCC AVS c1,a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0519 c2,a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='4862 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='3515 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='1337 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Modeling the Geometry Compression For geometry compression of V-PCC, the relationship between MOS and Qs follows the natural logarithm function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', MOSg,V −P CC = c1,g ∗ lnQsg + c2,g, (7) where Qsg denotes the quantization steps for geometry com- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' c1,g and c2,g denote the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' For geometry compression of G-PCC and AVS compres- sion algorithms, the relationship between MOS and Qs fol- lows the linear model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=', MOSg,G−P CC,AV S = c1,g ∗ Qsg + c2,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (8) The fitted parameters in the geometry compression models are shown in Table III: TABLE III: Fitting parameters in the geometry compression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' V-PCC G-PCC AVS c1,g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='273 c2,g 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='4165 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='3818 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5034 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Combining the Attribute Model and Geometry Model The point clouds are often compressed in both attribute and geometry, and the attribute degradation and geometry degradation are superimposed on the point clouds at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' As explored in Section IV-B, the linear combination of the attribute model and geometry model can accurately estimate the quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' We take the weighted summation of MOSa and MOSg to predict the final quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' For V-PCC, the established model is MOSf = p1,a ∗ Qsa + p1,g ∗ lnQsg + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (9) For G-PCC and AVS, the established model is MOSf = p1,a ∗ Qsa + p1,g ∗ Qsg + P, (10) where MOSf is the predicted quality scores, Qsa is the quantization steps for attribute compression, and Qsg is the quantization steps for geometry compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' p1,a = 1 2 ∗ c1,a, p1,g = 1 2 ∗ c1,g, and P = 1 2 ∗ (c2,a + c2,g) to cast the predicted quality scores under the same range of subjective scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Analysis Some findings can be made in the experiment: i) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 7 demonstrate that for the V-PCC distortion, the geometry distortion is more annoying compared with the attribute distortion, but the human eyes are more sensitive to the quantization change in the attribute compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' ii) for the geometry compression, the fitting curves of V-PCC and G- PCC are different, which derives from that the quantization of V-PCC is conducted on the projection while the quantization of G-PCC and AVS is conducted on octree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' iii) for the attribute compression, all the three compression algorithms follow the linear model, since their quantization is all conducted on RGB, resulting in the similar perceptual pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' A potential concern is whether the obtained relation func- tion is generic for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' As discussed in Section III-B, the difference of reference samples will only affect the additive factors, as P in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 9 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 10 which is a predefined constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Thus, the obtained relation function can still accurately predict the quality rank of samples in other datasets, which is demonstrated by the cross-dataset evaluation in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' EXPERIMENTS In this section, we evaluate the performance of the pro- posed R-PCQA on the established PCCQA and WPC [24] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Specifically, we use PCCQA to fit the model parame- ters and evaluate the fitting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Then, we evaluate on WPC dataset as cross check to verify the performance of R-PCQA and its generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Error Analysis The proposed PCCQA consists of three parts, part 1: lossless-G-lossy-A, part 2: lossy-G-lossless-A and part 3: lossy-G-lossy-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The proposed R-PCQA is fitted on the lossless-G-lossy-A and lossy-G-lossless-A parts, and we use the remaining lossy-G-lossy-A part to evaluate the perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Especially, we note the former two parts as the training (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' 2: Variation of Qs as function of average MOS for each compression condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The top row is under lossless-G-lossy-A, and the bottom row is under lossy-G-lossless-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' (a) (d) is for V-PCC, (b) (e) is for G-PCC, and (c) (f) is for AVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' set and the latter part as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The mean, standard deviation and 95% quantile of fitting errors MOS − MOSf on the testing set are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The correlation performance on the testing set is shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' TABLE IV: Mean, standard deviation and 95% quantile of the fitting errors on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Mean Standard deviation 95% quantile V-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='7561 G-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='0969 AVS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='1356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2531 We can see from Table IV and Table V that the proposed model can not only fit the dataset accurately, but also conforms to the characteristics of human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Combination Analysis The correlation performance of four combination schemes of the attribute model and geometry model on the testing set is shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' TABLE V: Correlation performance of four combination schemes on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' PLCC SROCC RMSE PLCC SROCC RMSE Linear Combination Multiplicative Combination V-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5070 G-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2100 AVS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='1650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='1691 GA Combination AG Combination V-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='8554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5075 G-PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='4046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2644 AVS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='1978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='9854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='2127 We can see from Table V that: i) the linear combination is determined due to its slightly better performance and sim- pler calculation for two relationship model mixing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' ii) the combination schemes hardly affect the performance, which indicates that the obtained relationship models for attribute and geometry are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Due to the removal of mutual masking, it is not necessary to consider the interaction of attribute and geometry components in the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Cross-dataset Evaluation After the model is established on the proposed dataset, we evaluate its generalization performance on another independent dataset, the V-PCC part of WPC [24], which contains 400 distorted samples derived from 16 reference point clouds with 25 different quantization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The results are shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' TABLE VI: Cross-dataset performance on WPC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' PLCC SROCC PLCC SROCC M-p2po (FR) [3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='58 H-PSNRyuv (FR) [33] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='23 M-p2pl (FR) [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='59 PCQM (FR) [12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='75 H-p2po (FR) [3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='46 GraphSIM (FR) [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='75 H-p2pl (FR) [34] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='48 MPED (FR) [35] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='59 PSNRyuv (FR) [33] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='47 PCM RR (RR) [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='38 R-PCQA (RR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='88 We can see from Table VI that: i) for the compression distortions, the proposed R-PCQA exhibits the state-of-the- art performance which only needs the assistance of two compression parameters, even compared with the existing FR metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' ii) the model parameters derived from PCCQA still exhibit robust performance on another independent dataset, which demonstrates the generalization ability of the proposed RR metric R-PCQA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' iii) the massive increase in points after reconstruction may interfere with the measurement of point- wise FR metrics, resulting in the poor performance of some FR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' CONCLUSION In this paper, we analyze the compression distortions of point clouds under separate attribute compression and geom- etry compression to avoid their mutual masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' Then by fitting the relationship between the quantization steps and the 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SO 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 100 200 300 s5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SO) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 100 200 300 Os5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SOI 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 Average 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 20 40 60 80 Qs5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 so 4 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 100 200 300 Os5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SOI 4 M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 Average 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 5 10 15 20 Qs5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 SOI 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 Average 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='5 1 0 5 10 15 20 Qsperceptual quality, we propose a RR PCQA model, called R- PCQA, for evaluating V-PCC, G-PCC and AVS distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The experiment results have demonstrated that the proposed R-PCQA exhibits reliable and robust performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' ACKNOWLEDGEMENT This paper is supported in part by National Key R&D Program of China (2018YFE0206700), National Natural Sci- ence Foundation of China (61971282, U20A20185).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content=' The cor- responding author is Yiling Xu (e-mail: yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'} +page_content='xu@sjtu.' 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+arXiv:2301.01022v1 [math.AP] 3 Jan 2023 +DECAY OF SOLUTIONS OF ISENTROPIC GAS DYNAMICS +FOR LARGE DATA +NAOKI TSUGE +Abstract. In this paper, we are concerned with the Cauchy problem for +isentropic gas dynamics. Through the contribution of many researchers such +as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay of solutions +was established. They treated with initial data with the small total variation. +On the other hand, the decay for large initial data has been open for half a +century. Our goal is to provide a new method to analyze this problem. We +prove the existence of a global attractor, which yields a decay of solutions +for large data. To construct approximate solutions, we introduce a modified +Godunov scheme. +1. Introduction +The present paper is concerned with isentropic gas dynamics + + + +ρt + mx = 0, +mt + +�m2 +ρ + p(ρ) +� +x += 0, +x ∈ R, +t ∈ R+, +(1.1) +where ρ, m and p are the density, the momentum and the pressure of the gas, +respectively. If ρ > 0, v = m/ρ represents the velocity of the gas. For a barotropic +gas, p(ρ) = ργ/γ, where γ ∈ (1, 5/3] is the adiabatic exponent for usual gases. +We consider the initial value problem (1.1) with the initial data +(ρ, m)|t=0 = (ρ0(x), m0(x)), +(1.2) +where +(ρ0(x), m0(x)) = (¯ρ, 0) outside a finite interval +(1.3) +and ¯ρ is a positive constant. The above problem (1.1)–(1.2) can be written in the +following form +� +ut + f(u)x = 0, +x ∈ R, +t ∈ R+, +u|t=0 = u0(x), +(1.4) +by using u = t(ρ, m), f(u) = t +� +m, m2 +ρ + p(ρ) +� +. +We recollect the known results of the above problem. DiPerna [5] proved the +global existence of solutions to (1.1)–(1.2) by the vanishing viscosity method and +2020 Mathematics Subject Classification. Primary 35L03, 35L65, 35Q31, 76N10, 76N15; Sec- +ondary 35A01, 35B35, 35B50, 35L60, 76H05, 76M20. +Key words and phrases. The Compressible Euler Equation, isentropic gas dyanmics, the com- +pensated compactness, the Godunov scheme, global attractor, decay estimates. +N. Tsuge’s research is partially supported by Grant-in-Aid for Scientific Research (C) +17K05315, Japan. +1 + +2 +NAOKI TSUGE +a compensated compactness argument. DiPerna first applied the method to (1.1) +for the special case where γ = 1 + 2/n and n is an odd integer. We notice that this +result can treat with the arbitrary L∞ data. Subsequently, Ding, Chen and Luo [6] +and Chen [1] and [2] extended his analysis to any γ in (1, 5/3]. +On the other hand, the existence of solutions to conservation laws including +(1.1) was established by Glimm [8]. Glimm treatd the Cauchy problem with initial +data having small total variation. +The theory of decay for genuinely nonlinear +2×2 systems of conservation laws was constructed by Glimm-Lax [9]. Glimm-Lax +showed that if initial data are constant outside a finite interval and have locally +bounded total variation and small oscillation, then the tonal variation of the solution +of [8] decays to zero at the rate t−1/2. The Glimm-Lax theory had been further +developed by DiPerna and Liu: general conservation laws with a convex entropy +function [4], general conservation laws with small initial data in total variation [10]. +However, the decay for large initial data has been open for half a century. Our +goal in the present paper is to provide a new method to analyze this problem and +investigate the decay structure of (1.1). We first introduce a modified Godunov +scheme to construct approximate solutions. We next prove the existence of a global +attractor, which yields decay estimates of solutions for large initial data. +To state our main theorem, we define the Riemann invariants w, z, which play +important roles in this paper, as +Definition 1.1. +w(u) := m +ρ + ρθ +θ = v + ρθ +θ , +z(u) := m +ρ − ρθ +θ = v − ρθ +θ +� +θ = γ − 1 +2 +� +. +These Riemann invariants satisfy the following. +Remark 1.1. +|w| ≥ |z|, w ≥ 0, when v ≥ 0. +|w| ≤ |z|, z ≤ 0, when v ≤ 0. +v = w + z +2 +, ρ = +�θ(w − z) +2 +�1/θ +, m = ρv. +(1.5) +From the above, the lower bound of z and the upper bound of w yield the bound of +ρ and |v|. +We next introduce the mechanical energy as η∗(u) = 1 +2 +m2 +ρ + +1 +γ(γ − 1)ργ and set +J(u) = η∗(u) − (¯ρ)γ−1 +γ − 1 ρ + (¯ρ)γ +γ . +(1.6) +Remark 1.2. From the convexity of η∗, we have +J(u) = η∗(u) − η∗(¯ρ, 0) − ∂η∗ +∂ρ (¯ρ, 0)(ρ − ¯ρ) − ∂η∗ +∂m (¯ρ, 0)m ≥ 0. +(1.7) +From the conservation of mass and the energy inequality, we have +0 ≤ +� x +−∞ +J(u(y, t))dy ≤ +� ∞ +−∞ +J(u(x, t))dx ≤ +� ∞ +−∞ +J(u0(x))dx. +(1.8) +Moreover, we define the entropy weak solution. + +THE COMPRESSIBLE EULER EQUATIONS +3 +Definition 1.2. A measurable function u(x, t) is called a global entropy weak so- +lution of the Cauchy problems (1.4) if +� ∞ +−∞ +� ∞ +0 +uφt + f(u)φxdxdt + +� ∞ +−∞ +u0(x)φ(x, 0)dx = 0 +holds for any test function φ ∈ C1 +0(R × R+) and +� ∞ +−∞ +� ∞ +0 +η(u)ψt + q(u)ψx + +� ∞ +−∞ +η(u0(x))ψ(x, 0)dx ≥ 0 +holds for any non-negative test function ψ ∈ C1 +0(R × R+), where (η, q) is a pair of +convex entropy–entropy flux of (1.1). +Finally, we define ˜z, ˜w by +˜z(x, t) = z(x, t) − +� x +−∞ +J(u(y, t))dy, +˜w(x, t) = w(x, t) − +� x +−∞ +J(u(y, t))dy. +(1.9) +Then, our main theorem is as follows. +Theorem 1.1. We assume that +ρ0(x) ≥ 0 +a.e. x ∈ R, +ρ0 ∈ L∞(R), +m0 +ρ0 +∈ L∞(R). +(1.10) +Then, there exists a global entropy weak solution of the Cauchy problems (1.4). +Moreover, for any positive constant ε, there exist positive constants t0 such that the +solution satisfies +− (¯ρ)θ +θ +− E0 − ε ≤ ˜z(x, t), +˜w(x, t) ≤ (¯ρ)θ +θ ++ ε, +ρ(x, t) ≥ 0, +a.e. (x, t) ∈ R × [t0, ∞), +(1.11) +where +E0 = +� ∞ +−∞ +J(u0(x))dx. +(1.12) +Remark 1.3. We remark some points for the above theorem. +In view of (1.3), we find that −(¯ρ)θ +θ −E0 ≥ ess infx (˜z(x, 0)) , ess supx ( ˜w(x, 0)) ≥ +(¯ρ)θ +θ . Therefore, (1.11) means the decay estimate of ess infx (˜z(x, t)) and ess infx ( ˜w(x, t)). +We similarly obtain −(¯ρ)θ +θ +≥ ess infx (z(x, 0)) , ess supx (w(x, 0)) ≥ (¯ρ)θ +θ . +If +−(¯ρ)θ +θ +− E0 − ε > ess infx (z(x, 0)) and ess supx (w(x, 0)) > (¯ρ)θ +θ ++ E0 + ε, (1.11) +yields the decay estimate of ess infx (z(x, t)) and ess supx (w(x, t)). In fact, we find + +4 +NAOKI TSUGE +that +ess inf +x (z(x, t0)) − ess inf +x (z(x, 0)) += ess inf +x +� +˜z(x, t0) + +� x +−∞ +J(u(y, t))dy +� +− ess inf +x (z(x, 0)) +≥ −(¯ρ)θ +θ +− E0 − ε − ess inf +x (z(x, 0)) > 0, +ess sup +x (w(x, t0)) − ess sup +x (w(x, 0)) += ess sup +x +� +˜w(x, t0) + +� x +−∞ +J(u(y, t))dy +� +− ess sup +x (w(x, 0)) +≤ (¯ρ)θ +θ ++ ε + E0 − ess sup +x (w(x, 0)) < 0. +1.1. Outline of the proof (formal argument). The proof of main theorem is +a little complicated. Therefore, before proceeding to the subject, let us grasp the +point of the main estimate by a formal argument. Although (1.1) has a discontin- +uous solution in general, we assume that solutions are smooth and the density is +nonnegative in this section. +We consider the physical region ρ ≥ 0 (i.e., w ≥ z.). Recalling Remark 1.1, it +suffices to derive the lower bound of z(u) and the upper bound of w(u) to obtain +the bound of u. To do this, we diagonalize (1.1). If solutions are smooth, we deduce +from (1.1) +zt + λ1zx = 0, +wt + λ2wx = 0, +(1.13) +where λ1 and λ2 are the characteristic speeds defined as follows +λ1 = v − ρθ, +λ2 = v + ρθ. +(1.14) +From the conservation of mass (1.1)1 and the conservation of energy (η∗)t + +(q∗)x = 0, we obtain +˜zt + λ1˜zx = g1(u), +˜wt + λ2 ˜wx = g2(u), +(1.15) +where +q∗(u) = m +�1 +2 +m2 +ρ2 + ργ−1 +γ − 1 +� +and +g1(u) = − (¯ρ)γ +γ λ1 + +1 +γ(γ − 1)ργ+θ + 1 +γ ργv + 1 +2ρθ+1v2 − (¯ρ)γ−1 +γ − 1 ρθ+1, +g2(u) = − (¯ρ)γ +γ λ2 − +1 +γ(γ − 1)ργ+θ + 1 +γ ργv − 1 +2ρθ+1v2 + (¯ρ)γ−1 +γ − 1 ρθ+1. +(1.16) +To prove Theorem 1.1, we prepare the following proposition. + +THE COMPRESSIBLE EULER EQUATIONS +5 +Proposition 1.2. + + + + + + + + + + + +• +g1(u(x, t)) > 0, when ˜z(x, t) < −(¯ρ)θ +θ +− E0, +• +g1(u(x, t)) = 0, when ˜z(x, t) = −(¯ρ)θ +θ +− E0, +� x +−∞ J(u(y, t))dy = E0 +and ρ(x, t) = ¯ρ, +(1.17) + + + + + + + + + + + +• +g2(u(x, t)) < 0, when ˜w(x, t) > (¯ρ)θ +θ , +• +g2(u(x, t)) = 0, when ˜w(x, t) = (¯ρ)θ +θ , +� x +−∞ J(u(y, t))dy = 0 +and ρ(x, t) = ¯ρ. +(1.18) +Proof. We first investigate (1.17). +When ˜z(x, t) ≤ −(¯ρ)θ +θ +− E0, from (1.8), we +observe that +z(x, t) = ˜z(x, t) + +� x +−∞ +J(u(y, t))dy ≤ −(¯ρ)θ +θ . +(1.19) +Then, we deduce that +g1(u)= +1 +γ(γ − 1)ργ+θ + 1 +γ ργv + 1 +2ρθ+1v2 − (¯ρ)γ +γ λ1 − (¯ρ)γ−1 +γ − 1 ρθ+1 += +1 +γ(γ − 1)ργ+θ + 1 +γ ργv + 1 +2ρθ+1v2 − (¯ρ)γ +γ +� +z + 3 − γ +γ − 1ρθ +� +− (¯ρ)γ−1 +γ − 1 ρθ+1 += +1 +γ(γ − 1)ργ+θ + 1 +γ ργv + 1 +2ρθ+1v2 − (¯ρ)γ +γ +� +z + 3 − γ +γ − 1ρθ +� +− (¯ρ)γ−1 +γ − 1 ρθ+1 += +1 +γ(γ − 1)ργ+θ + 1 +γ ργ +� +z + ρθ +θ +� ++ 1 +2ρθ+1 +� +z + ρθ +θ +�2 +− (¯ρ)γ +γ +� +z + 3 − γ +γ − 1ρθ +� +− (¯ρ)γ−1 +γ − 1 ρθ+1 +=ρθ+1 +2 +� +z − 1 +γ +(¯ρ)γ +ρθ+1 + 3γ − 1 +γ(γ − 1)ρθ +�2 ++ +γ + 1 +2γ2(γ − 1)ργ+θ +− +1 +γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1 +γ2 +(¯ρ)γ ρθ − +1 +2γ2 (¯ρ)2γ +1 +ρθ+1 . +(1.20) +When ρ ≤ ¯ρ, since (1.20) attains the minimum at z = −(¯ρ)θ +θ , we deduce from +Appendix A +g1(u) ≥ 5γ − 3 +γ(γ − 1)2 ργ+θ − 2(3γ − 1) +γ(γ − 1)2 (¯ρ)θ ργ + +3 − γ +(γ − 1)2 (¯ρ)γ−1 ρθ+1 − +3 − γ +γ(γ − 1) (¯ρ)γ ρθ ++ +2 +γ(γ − 1) (¯ρ)γ+θ +≥0, +(1.21) +where the equal sign of the second inequality can be used only when ρ = ¯ρ. + +6 +NAOKI TSUGE +When ρ > ¯ρ, since (1.20) attains the minimum at z = 1 +γ +(¯ρ)γ +ρθ+1 − 3γ − 1 +γ(γ − 1)ρθ, we +deduce from Appendix A +g1(u) ≥ +γ + 1 +2γ2(γ − 1)ργ+θ − +1 +γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1 +γ2 +(¯ρ)γ ρθ − +1 +2γ2 (¯ρ)2γ +1 +ρθ+1 +≥0, +(1.22) +where the equal sign of the second inequality can be used only when ρ = ¯ρ. +We next investigate (1.18). When ˜w(x, t) ≥ (¯ρ)θ +θ , from (1.8), we observe that +w(x, t) = ˜w(x, t) + +� x +−∞ +J(u(y, t))dy ≥ (¯ρ)θ +θ . +(1.23) +Then, we deduce that +g2(u) = − (¯ρ)γ +γ λ2 − +1 +γ(γ − 1)ργ+θ + 1 +γ ργv − 1 +2ρθ+1v2 + (¯ρ)γ−1 +γ − 1 ρθ+1 += − (¯ρ)γ +γ +� +w − 3 − γ +γ − 1ρθ +� +− +1 +γ(γ − 1)ργ+θ + 1 +γ ργv − 1 +2ρθ+1v2 + (¯ρ)γ−1 +γ − 1 ρθ+1 += − (¯ρ)γ +γ +� +w − 3 − γ +γ − 1ρθ +� +− +1 +γ(γ − 1)ργ+θ + 1 +γ ργ +� +w − ρθ +θ +� +− 1 +2ρθ+1 +� +w − ρθ +θ +�2 ++ (¯ρ)γ−1 +γ − 1 ρθ+1 += − ρθ+1 +2 +� +w + 1 +γ +(¯ρ)γ +ρθ+1 − 3γ − 1 +γ(γ − 1)ρθ +�2 +− +γ + 1 +2γ2(γ − 1)ργ+θ ++ +1 +γ − 1 (¯ρ)γ−1 ρθ+1 − γ + 1 +γ2 +(¯ρ)γ ρθ + +1 +2γ2 (¯ρ)2γ +1 +ρθ+1 . +From (1.21)–(1.22), we conclude that +g2(u) ≤ 0, +(1.24) +where the equal sign can be used only when ρ = ¯ρ. +□ +Proof of Theorem (1.11) +From (1.17)–(1.18), for any positive constant ε, there exists a positive constant +δ such that +g1(u) > 2δ, when ˜z ≤ −(¯ρ)θ +θ +− E0 − ε +2 and 0 ≤ ρ, +g2(u) < −2δ, when (¯ρ)θ +θ ++ ε +2 ≤ ˜w and 0 ≤ ρ. +(1.25) +We introduce ˆz, ˆw as follows. +ˆz(x, t) = ˜z(x, t) − δt, +ˆw(x, t) = ˜w(x, t) + δt. +(1.26) +We deduce from (1.15) that +ˆzt + λ1ˆzx = g1(u) − δ, +ˆwt + λ2 ˆwx = g2(u) + δ. +(1.27) + +THE COMPRESSIBLE EULER EQUATIONS +7 +We set +M0 = max +� +(¯ρ)θ +θ , −ess inf +x (˜z(x, 0)) + E0, ess inf +x ( ˜w(x, 0)) +� +. +(1.28) +Then, we notice that +−M0 − E0 ≤ ˆz(x, 0), +ˆw(x, 0) ≤ M0. +Let us prove that +ˆSinv = {(ˆz, ˆw) ∈ R2; ρ ≥ 0, ˆz ≥ −M0 − E0, ˆw ≤ M0} +is an invariant region for the Cauchy problem of (1.27) on +0 ≤ t ≤ max +� +0, M0 − (¯ρ)θ +θ +− ε +δ +� +=: t0. +We notice that this yields (1.11) on 0 ≤ t ≤ t0. +To achieve this, assuming +−M0 − E0 < ˆz(x, 0), +ˆw(x, 0) < M0 +and there exist x∗ ∈ R, 0 < t∗ ≤ t0 such that the following (1.29) or (1.30) holds, +− M0 − E0 < ˆz(x, t), ˆw(x, t) < M0, +x ∈ R, 0 ≤ t < t∗ +and +ˆz(x∗, t∗) = −M0 − E0, ˆw(x∗, t∗) ≤ M0, +(1.29) +− M0 − E0 < ˆz(x, t), ˆw(x, t) < M0, +x ∈ R, 0 ≤ t < t∗ +and +ˆz(x∗, t∗) ≥ −M0 − E0, ˆw(x∗, t∗) = M0, +(1.30) +we will deduce a contradiction. +To do this, we prove +g1(u(x∗, t∗)) − δ > 0, when (1.29) holds, +(1.31) +g2(u(x∗, t∗)) + δ < 0, when (1.30) holds. +(1.32) +Let us consider (1.31). When (1.29) and 0 ≤ t∗ ≤ t0, we notice that +˜z(x∗, t∗) ≤ −(¯ρ)θ +θ +− E0 − ε. +Therefore, from (1.25), we prove (1.31). Since ˆz attains the minimum at (x∗, t∗), +we can deduce from (1.27)1 a contradiction. We can similarly prove (1.32). +We notice that (˜z(x, t0), ˜w(x, t0)) is contained in +˜Sinv = +� +(˜z, ˜w) ∈ R2; ρ ≥ 0, ˜z ≥ −(¯ρ)θ +θ +− ε − E0, ˜w ≤ (¯ρ)θ +θ ++ ε +� +. +Then, we can similarly prove that ˜Sinv is an invariant region for the Cauchy problem +of (1.15). Therefore, we conclude (1.11). +Although the above argument is formal, it is essential. In fact, we shall implicitly +use this argument in the proof of Theorem 3.1. We must next justify the above +argument. +To do this, we introduce a modified Godunov scheme in Section 2. +Recently, the various difference schemes are developed in [11]–[18], which consist +of known functions. On the other hand, the present approximate solutions include +unknown functions in the form of (1.9) with constants ˜z, ˜w (see (2.15)). + +8 +NAOKI TSUGE +2. Construction of Approximate Solutions +In this section, we construct approximate solutions. Let T be any fixed positive +constant. In the strip 0 ≤ t ≤ T , we denote the approximate solutions by u∆(x, t) = +(ρ∆(x, t), m∆(x, t)). We denote the space mesh lengths by ∆x. Using E0 in (1.12) +and M0 in (1.28), we take time mesh length ∆t such that +∆x +∆t = 2(M0 + E0). +(2.1) +In addition, we set +(j, n) ∈ 2Z × Z≥0, +where Z≥0 = {0, 1, 2, 3, . . .}. For simplicity, we use the following terminology +xj = j∆x, tn = n∆t, tn.5 = +� +n + 1 +2 +� +∆t, tn− = n∆t − 0, tn+ = n∆t + 0. (2.2) +First we set u∆(x, t0−) = u0(x). +Then, for j ∈ 2Z, we define E0 +j (u) by +E0 +j (u) = +1 +2∆x +� xj+1 +xj−1 +u∆(x, t0−)dx. +Next, we assume that u∆(x, t) is defined for t < tn. +Then, for j ∈ 2Z, we define En +j (u) by +En +j (u) = +1 +2∆x +� xj+1 +xj−1 +u∆(x, tn−)dx. +To determine un +j = (ρn +j , mn +j ) for j ∈ 2Z, we define symbols In +j and Ln. Let the +approximation of +� x +−∞ J(u(y, t))dy be +In +j := +� xj +−∞ +J(En(x; u))dx, +where +En(x; u) = En +j (u) +x ∈ [xj−1, xj+1) +(2.3) +and J is defined in (1.6). +Let D = (x(t), t) denote a discontinuity in u∆(x, t), [η∗] and [q∗] denote the +jump of η∗(u∆(x, t)) and q∗(u∆(x, t)) across D from left to right, respectively, +[η∗] = η∗(u∆(x(t) + 0, t)) − η∗(u∆(x(t) − 0, t)), +[q∗] = q∗(u∆(x(t) + 0, t)) − q∗(u∆(x(t) − 0, t)), +where q∗(u) is the flux of η∗(u) defined by +q∗(u) = m +�1 +2 +m2 +ρ2 + ργ−1 +γ − 1 +� +. +Next, to measure the error in the entropy condition and the gap of the energy at +tn±, we introduce a functional. To do this, we deduce from the Taylor expansion + +THE COMPRESSIBLE EULER EQUATIONS +9 +that +η∗ +� +u∆(x, tn−) +� +− η∗ +� +En +j (u) +� +=∇η∗(En +j (u)) +� +u∆(x, tn−) − En +j (u) +� ++ +� 1 +0 +(1 − τ) · t � +u∆(x, tn−) − En +j (u) +� +× ∇2η∗ +� +En +j (u) + τ +� +u∆(x, tn−) − En +j (u) +�� +dτ +× +� +u∆(x, tn−) − En +j (u) +� +=∇η∗(En +j (u)) +� +u∆(x, tn−) − En +j (u) +� ++ Rn +j (x), +(2.4) +where +Rn +j (x) = +� 1 +0 +(1 − τ) · t � +u∆(x, tn−) − En +j (u) +� +∇2η∗ +� +En +j (u) + τ +� +u∆(x, tn−) − En +j (u) +�� +× +� +u∆(x, tn−) − En +j (u) +� +dτ. +Then, we define a functional Ln as +Ln = +� tn +0 +� +x∈R +σ[η∗] − [q∗]dt + +n +� +k=0 +� ∞ +−∞ +� +η∗(u∆(x, tk−0)) − η∗(Ek(x; u)) +� +dx ++ +n +� +k=0 +� +j∈2Z +1 +2∆x +� xj+1 +xj−1 +� x +xj−1 +Rk +j (y)dydx, +(2.5) +where the summention in � +x∈R is taken over all discontinuities in u∆(x, t) at a +fixed time t over x ∈ R, σ is the propagating speed of the discontinuities. +Moreover, we set +M1 = M0 − δ∆t, +Mn+1 = + + + + + + + +Mn − δ∆t, +when Mn + Ln ≥ (¯ρ)θ +θ ++ ε, +Mn, +when Mn + Ln < (¯ρ)θ +θ ++ ε, +(2.6) +where δ is defined in (1.25). We notice that Mn ≥ (¯ρ)θ +θ ++ ε − δ∆t. +Using In +j , Ln and Mn, we define un +j as follows. +We choose µ such that 1 < µ < 1/(2θ). If +En +j (ρ) := +1 +2∆x +� xj+1 +xj−1 +ρ∆(x, tn−)dx < (∆x)µ, +(2.7) +we define un +j by un +j = (0, 0); otherwise, setting +zn +j := max +� +z(En +j (u)), −Mn − E0 − Ln + In +j +� +, +wn +j := min +� +w(En +j (u)), Mn + Ln + In +j +� +, +(2.8) +we define un +j by +un +j := (ρn +j , mn +j ) := (ρn +j , ρn +j vn +j ) := +��θ(wn +j − zn +j ) +2 +�1/θ +, +�θ(wn +j − zn +j ) +2 +�1/θ wn +j + zn +j +2 +� +. +Remark 2.1. We find +−Mn − E0 − Ln + In +j ≤ z(un +j ), +w(un +j ) ≤ Mn + Ln + In +j . +(2.9) + +10 +NAOKI TSUGE +This implies that we cut off the parts where z(En +j (u)) < −Mn − E0 − Ln + In +j +and w(En +j (u)) > Mn + Ln + In +j in defining z(un +j ) and w(un +j ). Observing (3.2), the +order of these cut parts is o(∆x). The order is so small that we can deduce the +compactness and convergence of our approximate solutions. +2.1. Construction of Approximate Solutions in the Cell. We then assume +that approximate solutions u∆(x, t) are defined in domains D1 : t < tn +(n ∈ N) +and D2 : x < xj−1 +(j ∈ 2Z), tn ≤ t < tn+1. +By using un +j defined in D1 +and u∆(x, t) defined in D2, we construct the approximate solutions in the cell +tn ≤ t < tn+1 +(n ∈ N), +xj−1 ≤ x < xj+1 +(j ∈ 2Z). +We first solve a Riemann problem with initial data (un +j−1, un +j+1). Call constants +uL(= un +j−1), uM, uR(= un +j+1) the left, middle and right states, respectively. Then +the following four cases occur. +• Case 1 A 1-rarefaction wave and a 2-shock arise. +• Case 2 A 1-shock and a 2-rarefaction wave arise. +• Case 3 A 1-rarefaction wave and a 2-rarefaction arise. +• Case 4 A 1-shock and a 2-shock arise. +We then construct approximate solutions u∆(x, t) by perturbing the above Riemann +solutions. +Let α be a constant satisfying 1/2 < α < 1. We choose a positive value β small +enough. +In this step, we consider Case 1 in particular. The constructions of Cases 2–4 +are similar to that of Case 1. We consider only the case in which uM is away from +the vacuum. The other case (i.e., the case where uM is near the vacuum) is a little +technical. Therefore, we postpone this case to Appendix C. +Consider the case where a 1-rarefaction wave and a 2-shock arise as a Riemann +solution with initial data (un +j , un +j+1). Assume that uL, uM and uM, uR are connected +by a 1-rarefaction and a 2-shock curve, respectively. +Step 1. +In order to approximate a 1-rarefaction wave by a piecewise constant rarefaction +fan, we introduce the integer +p := max {[[(zM − zL)/(∆x)α]] + 1, 2} , +where zL = z(uL), zM = z(uM) and [[x]] is the greatest integer not greater than x. +Notice that +p = O((∆x)−α). +(2.10) +Define +z∗ +1 := zL, z∗ +p := zM, w∗ +i := wL (i = 1, . . . , p), +and +z∗ +i := zL + (i − 1)(∆x)α (i = 1, . . . , p − 1). +We next introduce the rays x = (j + 1/2)∆x + λ1(z∗ +i , z∗ +i+1, wL)(t − n∆t) separating +finite constant states (z∗ +i , w∗ +i ) (i = 1, . . . , p), where +λ1(z∗ +i , z∗ +i+1, wL) := v(z∗ +i , wL) − S(ρ(z∗ +i+1, wL), ρ(z∗ +i , wL)), + +THE COMPRESSIBLE EULER EQUATIONS +11 +ρ∗ +i := ρ(z∗ +i , wL) := +�θ(wL − z∗ +i ) +2 +�1/θ +, +v∗ +i := v(z∗ +i , wL) := wL + z∗ +i +2 +and +S(ρ, ρ0) := + + + + + +� +ρ(p(ρ) − p(ρ0)) +ρ0(ρ − ρ0) +, +if ρ ̸= ρ0, +� +p′(ρ0), +if ρ = ρ0. +(2.11) +We call this approximated 1-rarefaction wave a 1-rarefaction fan. +Step 2. +In this step, we replace the above constant states with functions of x and t as +follows: +In view of (1.9), we construct u∆ +1 (x, t). +We first determine the approximation of ˜z, ˜w in (1.9) as follows. +˜z∆ +1 =zL − +� xj−1 +−∞ +J(u∆ +n,0(x))dx, ˜w∆ +1 = wL − +� xj−1 +−∞ +J(u∆ +n,0(x))dx, +where u∆ +n,0(x) is a piecewise constant function defined by +u∆ +n,0(x) = un +j , +x ∈ [xj−1, xj+1) +(j ∈ 2Z). +(2.12) +We set +ˇz∆ +1 (x, t) = ˜z∆ +1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + +� x +x∆ +1 +J(uL)dy ++ {g1(x, t; uL) + V (uL)} (t − tn), +ˇw∆ +1 (x, t) = ˜w∆ +1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + +� x +x∆ +1 +J(uL)dy ++ {g2(x, t; uL) + V (uL)} (t − tn), +(2.13) +where g1 and g2 are defined in (1.16), x∆ +1 = xj−1 and +V (u) = q∗(u) − (¯ρ)γ−1 +γ − 1 m. +(2.14) +From (2.13), we determine ˇu∆ +1 (x, t) by the relation (1.5), that is, +ˇu∆ +1 (x, t) = (ˇρ∆ +1 (x, t), ˇm∆ +1 (x, t)) = (ˇρ∆ +1 (x, t), ˇρ∆ +1 (x, t)ˇv∆ +1 (x, t)), +where +ˇρ∆ +1 (x, t) = +� +θ +� +ˇw∆ +1 (x, t) − ˇz∆ +1 (x, t) +� +2 +� 1 +θ +, +ˇv∆ +1 (x, t) = ˇw∆ +1 (x, t) + ˇz∆ +1 (x, t) +2 +. + +12 +NAOKI TSUGE +Using ˇu∆ +1 (x, t), we next define u∆ +1 (x, t) as follows. +z∆ +1 (x, t) = ˜z∆ +1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + +� x +x∆ +1 +J(ˇu∆ +1 (y, t))dy ++ +� +g1(x, t; ˇu∆ +1 ) + V (uL) +� +(t − tn), +w∆ +1 (x, t) = ˜w∆ +1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + +� x +x∆ +1 +J(ˇu∆ +1 (y, t))dy ++ +� +g2(x, t; ˇu∆ +1 ) + V (uL) +� +(t − tn). +(2.15) +From (2.15), we determine u∆ +1 (x, t) by the relation (1.5). +Remark 2.2. +(i) We notice that approximate solutions z∆ +1 , w∆ +1 and ˜z∆ +1 , ˜w∆ +1 correspond to +z, w and ˜z, ˜w in (1.9), respectively. +(ii) For tn < t < tn+1, our approximate solutions will satisfy +� xj−1 +−∞ +J(u∆(x, tn+1−))dx + +� tn+1 +tn +� +x≤xj−1 +(σ[η∗] − [q∗])dt += +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)∆t + o(∆x). +(2.16) +In (2.15), we thus employ the right hand side of (2.16) instead of the left +hand side. +(iii) Our construction of approximate solutions uses the iteration method twice +(see (2.13) and (2.15)) to deduce (3.12). +First, by the implicit function theorem, we determine a propagation speed σ2 +and u2 = (ρ2, m2) such that +(1.a) z2 := z(u2) = z∗ +2 +(1.b) the speed σ2, the left state u∆ +1 (x2, tn.5) and the right state u2 satisfy the +Rankine–Hugoniot conditions, i.e., +f(u2) − f(u∆ +1 (x∆ +2 (tn.5), tn.5)) = σ2(u2 − u∆ +1 (x∆ +2 (tn.5), tn.5)), +where x∆ +2 (t) = xj + σ2(t − tn). Then we fill up by u∆ +1 (x) the sector where tn ≤ t < +tn+1, xj−1 ≤ x < x∆ +2 (t) (see Figure 1). +Assume that uk, u∆ +k (x, t), a propagation speed σk and x∆ +k (t) are defined. Then +we similarly determine σk+1 and uk+1 = (ρk+1, mk+1) such that +(k.a) zk+1 := z(uk+1) = z∗ +k+1, +(k.b) σk < σk+1, +(k.c) the speed σk+1, the left state u∆ +k (x∆ +k+1(tn.5), tn.5) and the right state uk+1 +satisfy the Rankine–Hugoniot conditions, +where x∆ +k+1(t) = xj + σk+1(t − tn). Then we fill up by u∆ +k (x, t) the sector where +tn ≤ t < tn+1, x∆ +k (t) ≤ x < x∆ +k+1(t) (see Figure 1). +We construct u∆ +k+1(x, t) as follows. + +THE COMPRESSIBLE EULER EQUATIONS +13 +Figure 1. The approximate solution in the case where a 1- +rarefaction and a 2-shock arise in the cell. +We first determine +˜z∆ +k+1 =zk+1 − +� xj−1 +−∞ +J(u∆ +n,0(x))dx − V (uL)∆t +2 − +k +� +l=1 +� x∆ +l+1(tn.5) +x∆ +l (tn.5) +J(u∆ +l (x, tn.5))dx, +˜w∆ +k+1 =wk+1 − +� xj−1 +−∞ +J(u∆ +n,0(x))dx − V (uL)∆t +2 − +k +� +l=1 +� x∆ +l+1(tn.5) +x∆ +l (tn.5) +J(u∆ +l (x, tn.5))dx, +where x∆ +1 (t) = xj−1, x∆ +l (t) = xj +σl(t−tn) +(l = 2, 3, . . . , k+1) and tn.5 is defined +in (2.2). +We next define ˇu∆ +k+1 as follows. +ˇz∆ +k+1(x, t) =˜z∆ +k+1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +k +� +l=1 +� x∆ +l+1(t) +x∆ +l (t) +J(u∆ +l (x, t))dx ++ +� x +x∆ +k+1(t) +J(uk+1)dy + g1(x, t; uk+1)(t − tn.5) + +� t +tn.5 +� +xj−1≤y≤x +(σ[η∗] − [q∗])ds, +ˇw∆ +k+1(x, t) = ˜w∆ +k+1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +k +� +l=1 +� x∆ +l+1(t) +x∆ +l (t) +J(u∆ +l (x, t))dx ++ +� x +x∆ +k+1(t) +J(uk+1)dy + g2(x, t; uk+1)(t − tn.5) + +� t +tn.5 +� +xj−1≤y≤x +(σ[η∗] − [q∗])ds. +From the above, we determine ˇu∆ +k+1(x, t) by the relation (1.5). +Finally, using ˇu∆ +k+1(x, t), we define u∆ +k+1(x, t) as follows. + +()m(cf)mg(f)()(°)(°)十-JQSQ3b +QQ +G14 +NAOKI TSUGE +z∆ +k+1(x, t) =˜z∆ +k+1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +k +� +l=1 +� x∆ +l+1(t) +x∆ +l (t) +J(u∆ +l (x, t))dx ++ +� x +x∆ +k+1(t) +J(ˇu∆ +k+1(y, t))dy + g1(x, t; ˇu∆ +k+1)(t − tn.5) ++ +� t +tn.5 +� +xj−1≤y≤x +(σ[η∗] − [q∗])ds, +w∆ +k+1(x, t)= ˜w∆ +k+1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +k +� +l=1 +� x∆ +l+1(t) +x∆ +l (t) +J(u∆ +l (x, t))dx ++ +� x +x∆ +k+1(t) +J(ˇu∆ +k+1(y, t))dy + g2(x, t; ˇu∆ +k+1)(t − tn.5) ++ +� t +tn.5 +� +xj−1≤y≤x +(σ[η∗] − [q∗])ds. +(2.17) +From (2.17), we determine u∆ +k+1(x, t) by the relation (1.5). +By induction, we define ui, u∆ +i (x, t) and σi (i = 1, . . . , p − 1). +Finally, we +determine a propagation speed σp and up = (ρp, mp) such that +(p.a) zp := z(up) = z∗ +p, +(p.b) the speed σp, and the left state u∆ +p−1(x∆ +p (tn.5), tn.5) and the right state up +satisfy the Rankine–Hugoniot conditions, +where x∆ +p (t) = xj + σp(t − tn). We then fill up by u∆ +p−1(x, t) and up the sector +where tn ≤ t < tn+1, x∆ +p−1(t) ≤ x < x∆ +p (t) and the line tn ≤ t < tn+1, x = x∆ +p (t), +respectively. +Given uL and zM with zL ≤ zM, we denote this piecewise functions of x and t +1-rarefaction wave by R∆ +1 (uL, zM, x, t). +On the other hand, we construct u∆ +R(x, t) as follows. +We first set +˜z∆ +R = zR − +� xj+1 +−∞ +J(u∆ +n,0(x))dx, ˜w∆ +R = wR − +� xj+1 +−∞ +J(u∆ +n,0(x))dx. +We next construct ˇu∆ +R +ˇz∆ +R (x, t)=˜z∆ +R + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) ++ +� x +xj+1 +J(uR)dy + g1(x, t; uR)(t − tn), +ˇw∆ +R (x, t)= ˜w∆ +R + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) ++ +� x +xj+1 +J(uR)dy + g2(x, t; uR)(t − tn). +From the above, we determine ˇu∆ +R(x, t) by the relation (1.5). + +THE COMPRESSIBLE EULER EQUATIONS +15 +Using ˇu∆ +R(x, t), we define u∆ +R(x, t) as follows. +z∆ +R (x, t) =˜z∆ +R + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) ++ +� x +xj+1 +J(ˇuR(y, t))dy + g1(x, t; ˇuR)(t − tn), +w∆ +R (x, t) = ˜w∆ +R + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) ++ +� x +xj+1 +J(ˇuR(y, t))dy + g2(x, t; ˇuR)(t − tn). +(2.18) +From (2.18), we determine u∆ +R(x, t) by the relation (1.5). +Now we fix u∆ +R(x, t) and u∆ +p−1(x, t). Let σs be the propagation speed of the 2- +shock connecting uM and uR. Choosing σ⋄ +p near to σp, σ⋄ +s near to σs and u⋄ +M near to +uM, we fill up by u∆ +M(x, t) the gap between x = xj+σ⋄ +p(t−tn) and x = xj+σ⋄ +s(t−tn), +such that +(M.a) σp−1 < σ⋄ +p < σ⋄ +s, +(M.b) the speed σ⋄ +p, the left and right states u∆ +p−1(x⋄ +p, tn.5), u∆ +M(x⋄ +p, tn.5) satisfy +the Rankine–Hugoniot conditions, +(M.c) the speed σ⋄ +s, the left and right states u∆ +M(x⋄ +s, tn.5), u∆ +R(x⋄ +s, tn.5) satisfy the +Rankine–Hugoniot conditions, +where x⋄ +p := xj + σ⋄ +p∆/2, x⋄ +s := xj + σ⋄ +s∆/2 and u∆ +M(x, t) defined as follows. +We first set +˜z∆ +M =z⋄ +M − +� xj+1 +−∞ +J(u∆ +n,0 (x))dx − V (uR)∆t +2 − +� x∆ +R (tn.5) +xj+1 +J(u∆ +R(x, tn.5))dx, +˜w∆ +M =w⋄ +M − +� xj+1 +−∞ +J(u∆ +n,0 (x))dx − V (uR)∆t +2 − +� x∆ +R (tn.5) +xj+1 +J(u∆ +R(x, tn.5))dx, +where x∆ +R(t) = xj + σ⋄ +s(t − tn). +We construct ˇu∆ +M +ˇz∆ +M(x, t) =˜z∆ +M + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) + +� x∆ +R (t) +xj+1 +J(u∆ +R(x, t))dy ++ +� x +x∆ +R (t) +J(uM)dy + g1(x, t; uM)(t − tn.5) +− +� t +tn.5 +� +x≤y≤xj+1 +(σ[η∗] − [q∗])ds, +ˇw∆ +M(x, t) = ˜w∆ +M + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) + +� x∆ +R (t) +xj+1 +J(u∆ +R(x, t))dy ++ +� x +x∆ +R (t) +J(uM)dy + g2(x, t; uM)(t − tn.5) +− +� t +tn.5 +� +x≤y≤xj+1 +(σ[η∗] − [q∗])ds. + +16 +NAOKI TSUGE +From the above, we determine ˇu∆ +M(x, t) by the relation (1.5). +Using ˇu∆ +M(x, t), we next define u∆ +M(x, t) as follows. +z∆ +M(x, t) =˜z∆ +M + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) + +� x∆ +R (t) +xj+1 +J(u∆ +R(x, t))dy ++ +� x +x∆ +R (t) +J(ˇu∆ +M(y, t))dy + g1(x, t; ˇu∆ +M)(t − tn.5) +− +� t +tn.5 +� +x≤y≤xj+1 +(σ[η∗] − [q∗])ds, +w∆ +M(x, t) = ˜w∆ +M + +� xj+1 +−∞ +J(u∆ +n,0(x))dx + V (uR)(t − tn) + +� x∆ +R (t) +xj+1 +J(u∆ +R(x, t))dy ++ +� x +x∆ +R (t) +J(ˇu∆ +M(y, t))dy + g2(x, t; ˇu∆ +M)(t − tn.5) +− +� t +tn.5 +� +x≤y≤xj+1 +(σ[η∗] − [q∗])ds. +(2.19) +From (2.19), we determine u∆ +M(x, t) by the relation (1.5). +We denote this approximate Riemann solution, which consists of (2.17), (2.18), +(2.18) , by u∆(x, t). The validity of the above construction is demonstrated in [11, +Appendix A]. +Remark 2.3. u∆(x, t) satisfies the Rankine–Hugoniot conditions at the middle +time of the cell, t = tn.5. +Remark 2.4. The approximate solution u∆(x, t) is piecewise smooth in each of the +divided parts of the cell. Then, in the divided part, u∆(x, t) satisfies +(u∆)t + f(u∆)x − g(x, u∆) = o(1). +To deduce that Ln is uniformly bounded, we prove the following lemma. +Lemma 2.1. +0 ≤ +n +� +k=0 +� ∞ +−∞ +� +η∗(u∆(x, tk−0)) − η∗(Ek(x; u)) +� +dx + +� tn +0 +� +x∈R +(σ[η∗] − [q∗])dt +(2.20) += +n +� +k=0 +� +j∈2Z +� xj+1 +xj−1 +Rn +j (x)dx + +� tn +0 +� +x∈R +(σ[η∗] − [q∗])dt + o(∆x) +(2.21) += +� ∞ +−∞ +� +J +� +u∆(x, t0−) +� +− J +� +u∆(x, tn+) +�� +dx + o(∆x) +(2.22) +≤ +� ∞ +−∞ +J(u0(x))dx + o(∆x). +(2.23) +where o(∆x) depends only on M0, E0 and T . +Ln ≤ C, +(2.24) +where C depends only on initial data. + +THE COMPRESSIBLE EULER EQUATIONS +17 +Proof. We recall that our approximate solutions are constructed in [0, T ] for any +fixed positive constant T . From (1.3) and the finite propagation, we find that our +approximate solutions are (¯ρ, 0) outside a finite interval. +First, from the Jensen inequality and the entropy condition, we obtain (2.20). +Second, from (2.4), we have (2.21). +Finally, we consider (2.22). From the similar argument to [11, (6.10)], taking +J(u) as η in [11], we have +n +� +k=0 +� ∞ +−∞ +� +η∗(u∆(x, tk−0)) − η∗(u∆ +n,0(x)) +� +dx + +� tn +0 +� +x∈R +(σ[η∗] − [q∗])dt += +� ∞ +−∞ +� +J +� +u∆(x, t0−) +� +− J +� +u∆(x, tn+) +�� +dx + o(∆x). +On the other hand, (2.9) and Theorem 3.2, we find that un +j = En +j (u) + o(∆x). +Recalling (2.3) and (2.12), we have +n +� +k=0 +� ∞ +−∞ +� +η∗(u∆(x, tk−0)) − η∗(u∆ +n,0(x)) +� +dx += +n +� +k=0 +� ∞ +−∞ +� +η∗(u∆(x, tk−0)) − η∗(Ek(x; u)) +� +dx + o(∆x). +Finally, observing Rn +j (x) ≥ 0, from (2.5) and (2.23), we have (2.24). +□ +3. The L∞ estimate of the approximate solutions +The aim in this section is to deduce from (2.9) the following theorem: +Theorem 3.1. For j ∈ 2Z≥0, n ∈ Z≥0 and xj−1 ≤ x < xj+1, +z∆(x, tn+1−) ≥ − Mn+1 − E0 − Ln + +� x +−∞ +J(u∆(y, tn+1−))dy − o(∆x), +w∆(x, tn+1−)≤Mn+1 + Ln + +� x +−∞ +J(u∆(y, tn+1−))dy ++ +� tn+1 +tn +� +x Mn + Ln + In +j , +(3.9) +the following holds. +1 +2∆x +� xj+1 +xj−1 +� x +xj−1 +Γn +j (y)dydx ≤ +� xj+1 +xj−1 +Rn +j (x)dx + o(∆x). +Proof. From Theorem 3.1, we have z(En +j (u)) ≥ −Mn − Ln + In +j − O(∆x). From +(3.9), we find En +j (v) ≥ In +j − O (∆x) (recall the definition of En +j (v) in (3.3) ). If +En +j (v) < 0, since In +j − O (∆x) ≤ En +j (v) ≤ 0, we have +−En +j (v) ≤ O (∆x) . +(3.10) +We first treat with B1 in (3.8). If En +j (v) ≥ 0, we deduce from Theorem 3.1 and +(3.9) that +B1 ≤ +� xj+1 +xj−1 +En +j (v)ρ∆(x, tn−) +� +w(x, tn−) − wn +j +� +dx = o(∆x). +If En +j (v) < 0, from (3.10), we have B1 = o(∆x). +We next consider B2. If En +j (v) ≥ 0, we find that B2 ≤ 0. If En +j (v) < 0, from +(3.10), we have B2 = o(∆x). +□ +From Lemma 3.4, we can complete the proof of (3.2). +3.2. Proof of Theorem 3.1. We next prove Theorem 3.1. +Estimates of w∆(x, t) along R∆ +1 in Case 1 In this step, we estimate w∆(x, t) +along R∆ +1 in Case 1 of Section 2. We recall that u∆ along R∆ +1 consists of u∆ +k +(k = +1, 2, 3, . . ., p−1). In this case, w∆(x, t) has the following properties, which is proved +in [11, Appendix A]: +w∆ +k+1(x∆ +k+1(tn.5), tn.5) =wk+1 = w∆ +k (x∆ +k+1(tn.5), tn.5) + O((∆x)3α−(γ−1)β) +(k = 1, . . . , p − 2), +(3.11) +where tn.5 is defined in (2.2). +We first consider ˜w∆ +1 . We recall that +˜w∆ +1 = wL − +� xj−1 +−∞ +J(u∆ +n,0(x))dx. +From (2.9), we have ˜w∆ +1 ≤ Mn + Ln. +Since +ˇu∆ +1 (x, t) = u∆ +1 (x, t) + O((∆x)2), +(3.12) + +22 +NAOKI TSUGE +we have +w∆ +1 (x, t)= ˜w∆ +1 + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +� x +x∆ +1 +J(ˇu∆ +1 (y, t))dy ++ g2(x, t; ˇu∆)(t − tn) +≤Mn + Ln + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) + +� x +x∆ +1 +J(u∆ +1 (y, t))dy ++ g2(x, t; u∆)(t − tn) + o(∆x). +On the other hand, from the construction of our approximate solutions, we ob- +serve that w∆ +1 (x, t) = w∆ +1 (x, tn−) + O(∆x) +(xj−1 ≤ x < xj+1, tn ≤ t < tn+1). +Separating three cases, we prove (3.1)2. +(i) If w∆ +1 (x, tn−) < Mn + Ln + In +j − +√ +∆x, we obtain (3.1)2. +(ii) If w∆ +1 (x, tn−) ≥ Mn + Ln + In +j − +√ +∆x and Mn + Ln ≥ (¯ρ)θ +θ ++ ε, we +observe that w∆ +1 (x, t) ≥ w∆ +1 (x, tn−) − O( +√ +∆x) ≥ Mn + Ln − O( +√ +∆x) ≥ +(¯ρ)θ +θ ++ ε − O( +√ +∆x) ≥ (¯ρ)θ +θ ++ ε/2, by choosing ∆x small enough. From +(1.25), we obtain g2(x, t; u∆ +1 ) < −2δ. From (2.16), we obtain (3.1)2. +(iii) If w∆ +1 (x, tn−) ≥ Mn +Ln +In +j − +√ +∆x and Mn +Ln < (¯ρ)θ +θ ++ε, from (2.6), +we find that (¯ρ)θ +θ ++ ε − δ∆t ≤ Mn + Ln. Therefore, we have w∆ +1 (x, t) ≥ +w∆ +1 (x, tn−)−O( +√ +∆x) ≥ Mn+Ln−O( +√ +∆x) ≥ (¯ρ)θ +θ +ε−O( +√ +∆x) ≥ (¯ρ)θ +θ + +ε/2, by choosing ∆x small enough. From (1.25), we obtain g2(x, t; u∆ +1 ) < +−2δ < 0. Recalling (2.6), from (2.16), we obtain (3.1)2. +Next, we assume that +w∆ +k (x, t)≤Mn + Ln + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)(t − tn) ++ +� x +xj−1 +J(u∆(y, t))dy + +� t +tn.5 +� +xj−1≤y 1, we conclude (3.1)2. +4. Proof of Theorem 1.1 +Our approximate solutions satisfy the following propositions holds (these proofs +are similar to [11]–[13].). +Proposition 4.1. The measure sequence +η∗(u∆)t + q(u∆)x +lies in a compact subset of H−1 +loc (Ω) for all weak entropy pair (η∗, q), where Ω ⊂ +[0, 1] × [0, 1] is any bounded and open set. +Proposition 4.2. Assume that the approximate solutions u∆ are bounded and +satisfy Proposition 4.1. Then there is a convergent subsequence u∆n(x, t) in the +approximate solutions u∆(x, t) such that +u∆n(x, t) → u(x, t) +a.e., +as n → ∞. +The function u(x, t) is a global entropy solution of the Cauchy problem (1.4). +Moreover, from Theorem 3.1, the above solution satisfies (1.11). Therefore, we +can prove Theorem 1.1. +Appendix A. Proof of (1.21) and (1.22) +A.1. Proof of (1.21). First, when 0 ≤ ρ ≤ ¯ρ, we will prove +5γ − 3 +γ(γ − 1)2 ργ+θ − 2(3γ − 1) +γ(γ − 1)2 (¯ρ)θ ργ + +3 − γ +(γ − 1)2 (¯ρ)γ−1 ρθ+1 − +3 − γ +γ(γ − 1) (¯ρ)γ ρθ ++ +2 +γ(γ − 1) (¯ρ)γ+θ ≥ 0. +To this, setting t = ρ/¯ρ, we consider +f(t) =(5γ − 3)t3θ+1 − 2(3γ − 1)t2θ+1 + γ(3 − γ)tθ+1 − (3 − γ)(γ − 1)tθ ++ 2(γ − 1), +0 ≤ t ≤ 1. + +24 +NAOKI TSUGE +Separating 3 steps, we will deduce that f(t) ≥ 0, +0 ≤ t ≤ 1. +Step 1 +First, we consider the neighborhood of t = 0. We set X = tθ. Solving two +inequalities +(5γ − 3)t3θ+1 − 2(3γ − 1)t2θ+1 + γ(3 − γ)tθ+1 += tθ+1 � +(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ) +� +≥ 0 +and +−(3 − γ)(γ − 1)tθ + 2γ(γ − 1) = −(3 − γ)(γ − 1)X + 2(γ − 1) ≥ 0, +we have 0 ≤ X ≤ ξ, where ξ is the smaller solution of (5γ − 3)X2 − 2(3γ − 1)X + +γ(3 − γ) = 0. We notice that f(t) ≥ 0 in the interval 0 ≤ X ≤ ξ. +Step 2 +Next, we consider the neighborhood of t = 1. We find that f(1) = f ′(1) = 0. +On the other hand, from 0 ≤ t ≤ 1, γ > 1, we have +4f ′′(t) =3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1 ++ γ(γ − 1)(γ + 1)(3 − γ)tθ−1 + (γ − 1)2(3 − γ)2tθ−2 +≥3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1 ++ (5γ − 3)(γ − 1)(3 − γ)tθ−1 +≥3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1 ++ (3γ − 1)(γ − 1)(3 − γ)tθ−1 +=(3γ − 1)(γ − 1)tθ−1 � +3(5γ − 3)X2 − 8γX + 3 − γ +� +. +We thus find that f(t) ≥ 0 in the interval η ≤ X ≤ 1, where η is the larger solution +of 3(5γ − 3)X2 − 8γX + 3 − γ = 0. +Step 3 +Since ξ < η, from Step 1,2, it suffices to prove f(t) ≥ 0 in the interval ξ ≤ X ≤ η. +Observing that (5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ) ≤ 0 in this interval, we have +f(t) =tθ+1 � +(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ) +� +− (3 − γ)(γ − 1)X + 2(γ − 1) +≥X +� +(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ) +� +− (3 − γ)(γ − 1)X + 2(γ − 1) +=(5γ − 3)X3 − 2(3γ − 1)X2 + (3 − γ)X + 2(γ − 1) =: g(X). +Let α, β (α < β) be tow solutions of g′(X) = 0. Then, we find that 0 < α < ξ < +η < β < 1. Moreover, we deduce that g(η) > 0. Therefore, we can complete the +proof. +A.2. Proof of (1.22). Our goal in this appendix is to prove +γ + 1 +2γ2(γ − 1)ργ+θ − +1 +γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1 +γ2 +(¯ρ)γ ρθ − +1 +2γ2 (¯ρ)2γ +1 +ρθ+1 ≥ 0, +where ρ ≥ ¯ρ. To do this, setting t = ρ/¯ρ, we prove +g(t) = +γ + 1 +2γ2(γ − 1)t2γ − +1 +γ − 1tγ+1 + γ + 1 +γ2 tγ − +1 +2γ2 ≥ 0, +t ≥ 1. +First, we observe that g(1) = g′(1) = g′′(1) = 0. In addition, we find that g′′′(t) ≥ +0, +t ≥ 1. We thus conclude that g(t) ≥ 0. + +THE COMPRESSIBLE EULER EQUATIONS +25 +Appendix B. Proof of Lemma 3.3 +Proof. Due to space limitations, we denote tn− by T in this section. +Set +ρ∆ +† (x, T ) := ˆρ(x, T ) {A(x, T )} +2 +γ−1 , +m∆ +† (x, T ) := ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 , +En+1 +j +(ρ∆ +† ) := +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx, +En+1 +j +(m∆ +† ) := +1 +2∆x +� xj+1 +xj−1 +ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 dx. +Then, we find that +w(ˆu(x, T )) ≤ 1 + o(∆x). +(B.1) +Let us prove +w(En+1 +j +(ρ∆ +† ), En+1 +j +(m∆ +† )) ≤ ¯Aj(T ) + o(∆x), +where +¯Aj(T ) = +1 +2∆x +� xj+1 +xj−1 +A(x, T )dx +and +w(En+1 +j +(ρ∆ +† ), En+1 +j +(m∆ +† )) += En+1 +j +(m∆ +† )/En+1 +j +(ρ∆ +† ) + {En+1 +j +(ρ∆ +† )}θ/θ += +1 +2∆x +� xj+1 +xj−1 +ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 dx + +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ+1 +/θ +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +. +(B.2) +Step 1. +We find +En+1 +j +(ρ∆ +† ) = +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 {A(x, T )}−1 dx += +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 dx ++ +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 × +� +{A(x, T )}−1 − +� ¯Aj(T ) +�−1� +dx += +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 dx +− +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx + o(∆x), +where r(x, T ) = A(x, T ) − ¯Aj(T ). Recalling (3.6), we notice that r(x, T ) = O(∆x). + +26 +NAOKI TSUGE +Substituting the above equation for (B.2), we obtain +w(En+1 +j +(ρ∆ +† ), En+1 +j +(m∆ +† )) += +1 +2∆x +� xj+1 +xj−1 +ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 dx + +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ+1 +/θ +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 dx ++ +1 +2∆x +� xj+1 +xj−1 +ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 dx + +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ+1 +/θ +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�2 +× +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx + o(∆x). +(B.3) +Set +ω := +2 +γ + 1 +1 +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ +× +1 +2∆x +� xj+1 +xj−1 +ˆm(x, T ) {A(x, T )} +γ+1 +γ−1 dx + +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ+1 +/θ +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +. +(B.4) +Then assume that the following holds. +(En+1 +j +(ρ∆ +† ))θ+1 ≤ +1 +2∆x +� xj+1 +xj−1 +(ˆρ(x, T ))θ+1 {A(x, T )} +γ+1 +γ−1 dx +− γ + 1 +2 +ω +� ¯Aj(T ) +�−1 +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ +× +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx +− +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +1 +2∆x +� xj+1 +xj−1 +r(x, T )dx +� ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx. +(B.5) + +THE COMPRESSIBLE EULER EQUATIONS +27 +This estimate shall be proved in step 2–4. Then, substituting (B.5) for (B.3), +we deduce from (B.1) that +w(En+1 +j +(¯ρ), En+1 +j +(m∆ +† )) ≤ +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 +� +ˆv(x, T ) + {ˆρ(x, T )}θ +θ +� +dx +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +γ+1 +γ−1 dx ++ o(∆x) +≤ ¯Aj(T ) + o(∆x). +Therefore we must prove (B.5). Separating three steps, we derive this estimate. +Step 2. +From (3.5), we notice that +|ω| ≤ C(∆x)−θδ−ε, +where C depends only on M. +In this step, we consider the first equation of (B.3): +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ+1 +. +Since θδ < 1/2, we first find +En+1 +j +(ρ∆ +† ) = +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 {A(x, T )}−ω dx += +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +− ω +� ¯Aj(T ) +�−ω−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 r(x, T )dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +:=I0 − I1 + I2. +We next estimate I1 as follows: +I1 = ω +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx. + +28 +NAOKI TSUGE +Therefore, we have +En+1 +j +(ρ∆ +† ) = +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 {A(x, T )}−ω dx += +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +− ω +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx. +From the above, we deduce that +(En+1 +j +(ρ∆ +† ))θ+1 = +� +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +−ω +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx +�θ+1 ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx += +� +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +�θ+1 ++ (θ + 1) +� +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +�θ +× −ω +� ¯Aj(T ) +�−1 +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx += +� +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +�θ+1 +− γ + 1 +2 +ω +� ¯Aj(T ) +�−1 +� +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +�θ +× +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 r(x, T )dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx. +(B.6) + +THE COMPRESSIBLE EULER EQUATIONS +29 +Step 3 +Applying the Jensen inequality to the first term of the right-hand of (B.6), we have +� +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +�θ+1 += + + + + + +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx + + + + + +θ+1 +× +� +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx +�θ+1 += + + + + + +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω+ +2 +γ−1 dx +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx + + + + + +θ+1 +× +� +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx +� +× +� +� ¯Aj(T ) +� γ+1 +2 +ω + γ + 1 +2 +ω +� ¯Aj(T ) +� γ+1 +2 +ω−1 +1 +2∆x +� xj+1 +xj−1 +r(x, T )dx + o(∆x) +� += + + + + + +� ¯Aj(T ) +�−ω +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )}ω− γ+1 +γ−1 ω+ +2 +γ−1 {A(x, T )} +γ+1 +γ−1 ω dx +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx + + + + + +θ+1 +× +� +1 +2∆x +� xj+1 +xj−1 +{A(x, T )} +γ+1 +γ−1 ω dx +� +� ¯Aj(T ) +� γ+1 +2 +ω ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx +≤ +1 +2∆x +� xj+1 +xj−1 +(ˆρ(x, T ))θ+1 {A(x, T )} +γ+1 +γ−1 dx ++ o(∆x) +1 +2∆x +� xj+1 +xj−1 +ˆρ(x, T ) {A(x, T )} +2 +γ−1 dx. +(B.7) +From (B.6) and (B.7), we obtain (B.5) and complete the proof of lemma 3.3. +□ +Appendix C. Construction and L∞ estimates of approximate +solutions near the vacuum in Case 1 +In this step, we consider the case where ρM ≤ (∆x)β, which means that uM +is near the vacuum. Since we cannot use the implicit function theorem, we must +construct u∆(x, t) in a different way. +Case 1 A 1-rarefaction wave and a 2-shock arise. + +30 +NAOKI TSUGE +In this case, we notice that ρR ≤ (∆x)β, zR ≥ −Mn − Ln + In +j and wR ≤ +Mn + Ln + In +j . +Case 1.1 ρL > (∆x)β +We denote u(1) +L +a state satisfying w(u(1) +L ) = w(uL) and ρ(1) +L += (∆x)β. Let u(2) +L +be a state connected to u∆ +1 (xj−1, tn+1−) on the right by R∆ +1 (uL, z(1) +L , x, tn+1−). We +set +(z(3) +L , w(3) +L ) = +� +(z(2) +L , w(2) +L ), +if z(2) +L +≥ Dn +j , +(Dn +j , w(2) +L ), +if z(2) +L +< Dn +j , +where +Dn +j = − Mn+1 − Ln + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)∆t + +� xj+1 +xj−1 +(¯ρ)γ +γ dx ++ +� xj+λ1(u(2) +L )∆t +xj−1 +η(R∆ +1 (uL, z(1) +L , x, tn+1−))dx. +Then, we define u∆(x, t) as follows. +u∆(x, t) = + + + + + + + + + + + + + + + + + + + + + + + +R∆ +1 (uL, z(1) +L , x, t), +if xj−1 ≤ x ≤ xj + λ1(u(2) +L )(t − tn) +and tn ≤ t < tn+1, +uRw(x, t), +if xj + λ1(u(2) +L )(t − tn)< x ≤ xj + λ2(uM, uR)(t − tn) +and tn ≤ t < tn+1, +u∆ +R(x, t) defined in (2.18), +if xj + λ2(uM, uR)(t − tn)< x ≤ xj+1 +and tn ≤ t < tn+1, +where (a) λ2(uM, uR) is a propagation speed of 2-shock wave; (b) uRw(x, t) is a +rarefaction wave connecting u(3) +L and u(4) +L ; (c) u(4) +L is defined by z(4) +L += max{z(3) +L , zM}, +w(4) +L += w(3) +L . +Rarefaction wave +Figure 2. Case 1.1: The approximate solution u∆ in the cell. + +()m(cf)mg(f)(°)(°)十-JQSQ3THE COMPRESSIBLE EULER EQUATIONS +31 +Case 1.2 ρL ≤ (∆x)β +We set (z(5) +L , w(5) +L ) = (max{zL, Dn +j }, min{wL, U n +j }), where +U n +j =Mn+1 + Ln + +� xj−1 +−∞ +J(u∆ +n,0(x))dx + V (uL)∆t. +Then, we define u∆(x, t) as follows. +u∆(x, t) = + + + + + + + + + + + + + + + + + + + + + + + +u∆ +1 (x, t) defined in (2.15), +if xj−1 ≤ x ≤ xj + λ1(uL)(t − tn) +and tn ≤ t < tn+1, +uRw(x, t), +if xj + λ1(uL)(t − tn)< x ≤ xj + λ2(uM, uR)(t − tn) +and tn ≤ t < tn+1, +u∆ +R(x, t) defined in (2.18), +if xj + λ2(uM, uR)(t − tn)< x ≤ xj+1 +and tn ≤ t < tn+1, +where (a) uRw(x, t) is a rarefaction wave connecting u(5) +L +and u(6) +L ; (b) u(6) +L +is defined +by z(6) +L += max{z(5) +L , zM}, w(6) +L += w(5) +L . +Remark C.1. We notice that ρ∆(x, t) = O((∆x)β) in (1.ii), (1.iii) and (2.i)– +(2.iii). Therefore, the followings hold in these areas. +Although (1.ii) and (2.ii) are solutions of homogeneous isentropic gas dynamics +(i.e., g(x, t, u)) = 0), they is also a solution of (1.4) approximately +(u∆)t + f(u∆)x − g(x, u∆) = −g(x, u∆) = O((∆x)β). +In addition, discontinuities separating (1.i)–(1.iii) and (2.i)–(2.iii) satisfy [11, +Lemma 5.3]. +C.1. L∞ estimates of approximate solutions. We consider Case 1.1 in partic- +ular. It suffices to treat with uRw(x, t) in the region where xj + λ1(u(2) +L )(t − tn) < +x ≤ xj + λ2(uM, uR)(t − tn) and tn ≤ t < tn+1. The other cases are similar to +Theorem 3.1. +In this case, since ρ∆(x, t) = O((∆x)β), we have +η∗(u∆(x, t)) = O((∆x)β). +(C.1) +Moreover, we notice that +w∆(x, tn+1−) = w(2) +L += w(R∆ +1 (uL, z(1) +L , xj + λ1(u(2) +L )∆t, tn+1−)). +Applying Theorem 3.1 to R∆ +1 (uL, z(1) +L , x, tn+1−), we drive +w∆(x, tn+1−)≤Mn+1 + Ln + +� xj+λ1(u(2) +L )∆t +−∞ +J(u∆(y, tn+1−))dy ++ +� tn+1 +tn +� +y ˜τ0(u) +and +˜τ0(u) − +∑ +(u,v)∈Eu +C +′(v) ≤ −1 +2 +Overall, we have +P(C +′,S) − P(C,S) +(34) += ⎛ +⎝ ∑ +u∈V0−1 +⎛ +⎝ +∑ +(u,v)∈Eu +C +′(v)⎞ +⎠ − ˜τ0(u)⎞ +⎠ + ⎛ +⎝ ∑ +u∈V1−0 +˜τ0(u) − ⎛ +⎝ +∑ +(u,v)∈Eu +C +′(v)⎞ +⎠ +⎞ +⎠ +≤ − ∑ +u∈V0−1 +1 +2 − ∑ +u∈V1−0 +1 +2 += −∣V0−1∣ + ∣V1−0∣ +2 +≤ −1 +2 +This concludes the proof. +Overall, we have shown that the potential gap between any two configurations is 4m−n. Further- +more, the overall system configuration decreases by at least 1/2 after each time step. It follows that +for a (SN, IT)-SyDS, starting from an arbitrary initial configuration, the system dynamic converges +in at most 8m − 2n time steps, irrespective of the underlying network structure. Subsequently, the +convergence time result for SN-SyACG follows. +Theorem 5.4. For SN-SyACG, starting from any initial action profile, the best-response dynamic +converges to a Nash equilibrium or a 2-cycle in O(m) time steps. +With simple modifications of Lemma 5.3 (i.e., consider u as a neighbor of itself), we can also show +that the same convergence time can be extended to SE-SyACG. +Theorem 5.5. For a SE-SyACG, starting from any initial action profile, the best-response dynamic +converges to a Nash equilibrium or a 2-cycle in O(m) time steps. +Corollary 5.6. For both SE-SyACG and SN-SyACG, starting from any initial action profile, the best- +response dynamic converges to a Nash equilibrium or a 2-cycle in O(n) time steps if the graph is +degree bounded. +49 + diff --git a/ENE1T4oBgHgl3EQfEQP0/content/tmp_files/load_file.txt b/ENE1T4oBgHgl3EQfEQP0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc0e4860ffb0e3f1e813764f8e07c80d2d165eed --- /dev/null +++ b/ENE1T4oBgHgl3EQfEQP0/content/tmp_files/load_file.txt @@ -0,0 +1,1423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf,len=1422 +page_content='Networked Anti-Coordination Games Meet Graphical Dynamical Systems: Equilibria and Convergence Zirou Qiu,1,2 Chen Chen,2 Madhav V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Marathe,1,2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Ravi,2,3 Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Rosenkrantz,2,3 Richard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Stearns,2,3 Anil Vullikanti1,2 1 1Computer Science Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 2Biocomplexity Institute and Initiative, University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 3Computer Science Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', University at Albany – SUNY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Evolutionary anti-coordination games on networks capture real-world strategic situa- tions such as traffic routing and market competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In such games, agents maximize their utility by choosing actions that differ from their neighbors’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Two important problems concerning evolutionary games are the existence of a pure Nash equilibrium (NE) and the convergence time of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this work, we study these two problems for anti-coordination games under sequential and synchronous update schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each up- date scheme, we examine two decision modes based on whether an agent considers its own previous action (self essential) or not (self non-essential) in choosing its next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Us- ing a relationship between games and dynamical systems, we show that for both update schemes, finding an NE can be done efficiently under the self non-essential mode but is computationally intractable under the self essential mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To cope with this hardness, we identify special cases for which an NE can be obtained efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For convergence time, we show that the best-response dynamics converges in a polynomial number of steps in the synchronous scheme for both modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' for the sequential scheme, the convergence time is polynomial only under the self non-essential mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Through experiments, we empiri- cally examine the convergence time and the equilibria for both synthetic and real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='02889v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='GT] 7 Jan 2023 1 Introduction Evolutionary anti-coordination (AC) games have been widely used to model real-world strategic sit- uations such as product marketing [25], balanced traffic routing [16], and social competition [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In the networked version of such a game, vertices are agents (players), and edges are interactions between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' At each time step, agents make new decisions based on the decisions of their neighbors [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, under the best-response dynamics of an anti-coordination game with binary actions, each agent maximizes its utility at the current time step by choosing a particular action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', 0 or 1) if and only if a sufficient number of its neighbors chose the opposite action at the previous step [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' There are two main types of update schemes for evolutionary games, where agents either choose actions synchronously or sequentially [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The decision mode where each agent only considers its neighbors’ actions in making its decisions is common in the game theory literature [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Nevertheless, in real-world situations where agents compete for resources, it is natural for an agent to also consider its own previous action before choosing a new action [17, 36, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For example, drivers on a highway can be seen as agents in an evolutionary anti- coordination game, where people switch lanes to avoid traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, an agent’s choice of lanes in the next time step is influenced by both its current lane and the lanes of neighboring cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Such considerations motivate us to investigate two decision modes: (i) self essential (SE), where each agent considers both its previous action and the actions of its neighbors, and (ii) self non-essential (SN), where each agent only considers the actions of its neighbors1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Pure Nash equilibria (NE) are a central concept in game theory [3, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In evolutionary games, another key notion is the time it takes for the best-response dynamics to reach an NE or a limit cycle [13, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Nevertheless, researchers have given limited attention to efficiently finding an NE for anti-coordination games under the SE and SN modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, to our knowledge, whether the best- response dynamics of anti-coordination games has a polynomial convergence time remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this work, we close the gap with a systematic study of the following two problems for the synchronous and sequential games under both SE and SN modes: (i) Equilibrium existence/finding (EQE/EQF): Does the game have an NE, and if so, can we find one efficiently?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) Convergence (Conv): Starting from an action profile, how fast does the best-response dynamics converge2 to a cycle of length at most 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The best-response dynamics of an evolutionary anti-coordination game can be specified using a threshold framework [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This naturally allows us to model such a game as a graphical dynamical 1Our usage of the word “essential” here is based on the term “essential variable” used in the context of Boolean (and other) functions to indicate the dependence of a function on a variable [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 2We use convergence to mean that the dynamics reaches a limit cycle of length at most 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', an NE is a limit cycle of length 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For games considered in this work, it is known that the length of each limit cycle is at most 2, except for SE sequential AC games, where there can be limit cycles of exponential length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 2 system [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In such a system, at each time step, each vertex uses its local function to compute its state in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To model anti-coordination games, the domain is Boolean, and the local functions are inverted-threshold functions whereby each vertex u is assigned state 1 for the next step if and only if enough neighboring vertices of u are currently in state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Graphical dynamical systems are commonly used to model the propagation of contagions and decision-making processes over networks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Here, we use dynamical systems with inverted-threshold functions as a theoretical framework to study EQE/EQF and Conv problems for evolutionary networked anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Main Contributions Finding an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We demonstrate an interesting contrast in the complexity of EQE/EQF between the SE and SN modes for anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, we show that EQE/EQF is NP- hard under the SE mode for both synchronous and sequential games, even on bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we show that the corresponding counting problem is #P-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, we can find an NE efficiently under the SN mode for synchronous and sequential games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We also identify special cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the underlying graph is a DAG, or even-cycle free) of EQE/EQF for the SE mode where an NE can be efficiently found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We show that starting from an arbitrary action profile, the best-response dynamics of synchronous anti-coordination games under either SE or SN mode converge in O(m) time steps, where m is the number of edges in the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we establish a similar O(m) bound on the convergence time for sequential anti-coordination games under the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We do not consider the convergence problem for the sequential games under the SE mode since such systems can have exponentially long cycles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We study the contrast in the empirical convergence time for both modes under different classes of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we perform simulations to explore how convergence time changes with network density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We also investigate the number of equilibria for problem instances of reasonable sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Problems SN-SyACG SE-SyACG SN-SACG SE-SACG EQE / EQF Trivial / P NP-hard Trivial / P NP-hard Conv O(m) O(m) O(m) NA Table 1: Overview of key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' All results, except those marked with “Trivial” and “NA”, are established in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' SyACG (SACG) denotes synchronous (sequential) anti-coordination game, and SE (SN) stands for the self essential (self non-essential) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The number of edges is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The entry “Trivial” for EQE denotes that the corresponding game always has a NE [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' “NA” denotes that the problem is not applicable since there can be exponentially long cycles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For Conv, O(m) is the number of steps for the best-response dynamics to reach a limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 3 2 Related Work The existence of NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The self non-essential sequential anti-coordination games are potential games and always have an NE [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This argument follows from [28], which guarantees the existence of NE at the maximum potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, Monderer and Shapley [28] show that general potential games converge in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that this result does not imply a polynomial-time algorithm in finding an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [21] study a special form of anti-coordination games where each agent chooses the decision that is the opposite of the majority of neighboring decisions, and show that in such a game, an NE can be found efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Vanelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [37] examine synchronous games with both coordinating and anti-coordinating agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' They present several special cases on the threshold distributions for the existence of NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Auletta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [5] define the class of generalized discrete preference games and show that such games always have an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' They also show that every ordinal potential game with binary actions can be reduced to a discrete preference game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Goles and Martinez [18] prove that for synchronous coordination games, the length of any limit cycle is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Many researchers have studied the existence of NE in other games of different forms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', [35, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Limiting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Adam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [2] show that the length of a limit cycle in a synchronous anti- coordination game is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' However, they did not bound the convergence time to reach a limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Ramazi, Riehl, and Cao [31] investigate the convergence time for asynchronous self non-essential anti-coordination games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' they establish that an NE will be reached in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We note that the asynchronous dynamic they consider is different from the sequential dynamic studied in our work, and their convergence result does not imply ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Goles and Martinez [18] establish that for coordination games, the dynamics converges in a poly- nomial number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ([8]) study phase space properties of sequential dynamical systems (which includes modeling sequential AC games);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' the results imply that the length of a limit cycle in a self essential sequential anti-coordination game can be exponential in the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The convergence of the best-response dynamics for games of other types has also been studied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', [19, 6, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Minority games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Anti-coordination games are closely related to minority games [12, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Many ver- sions of minority games have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For example, Challet and Marsili [11] study the dynamics of minority games where agents make decisions based on the action profile history of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [27] examine the action distribution of minority games over different network topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Several other forms of minority games are studied in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A more thorough discussion of related work is given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To our knowledge, the complexity of EQF for SACG/SyACG has not been established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' nor has a polynomial bound on the 4 convergence time of these games been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We resolve these problems in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 3 Preliminaries and Problem Definition A networked game operates on a graph G where vertices are agents and edges represent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' At each discrete time step, an agent chooses a binary action from the set {0,1} based on neighbors’ actions, and receives a payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Under the best-response dynamics, each agent chooses an action that yields a maximum payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For the best-response dynamics of an anti-coordination (AC) game, each agent v has a nonnegative threshold τ1(v), and v chooses action 1 if and only if the number of neighbors (plus possibly v itself) that chose action 0 is at least τ1(v) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now define the problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Equilibrium existence/finding (EQE/EQF): Given an anti-coordination game, does it have an NE, and if so, can we find it efficiently?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Convergence (Conv): Given an anti-coordination game and an initial action profile, how fast does the best-response dynamics converge to a limit cycle of length at most 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We examine two decision modes, based on whether or not each agent considers its own previous action in making a new decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Under the self non-essential (SN) mode, each agent only considers neighbors’ actions, whereas, under the self essential (SE) mode, each agent considers both its own previous action and its neighbors’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We further consider two types of update schemes: (i) synchronous (SyACG): agents choose actions simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) sequential (SACG): at each time step, agents update their actions in a predefined order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we examine four classes of AC games based on update scheme (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', synchronous or sequential) and decision mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', SN or SE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Graphical dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We use dynamical systems as a mathematical framework to study anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We follow the notation used in [9, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A synchronous dynamical system (SyDS) over the Boolean state domain B = {0,1} is a pair S = (GS,F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' GS = (V,E) is the underlying graph, with n = ∣V ∣ and m = ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We assume GS is connected and undirected, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The set F = {f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',fn} consists of functions with fi being the local function of vertex vi ∈ V,1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The output of fi gives the state value of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In a SyDS, vertices update states simultaneously at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A sequential dynamical system (SDS) S′ is a tuple (GS′,F′,Π) where GS′ and F′ are defined the same as for the above synchronous systems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In addition, Π is a permutation of V that determines the sequential order in which vertices update their states at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, each time step of S′ consists of n substeps, in each of which one vertex updates its state using the current vertex states in S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 5 Update rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To model anti-coordination dynamics, we consider inverted-threshold local functions for the state-update of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Formally, each vertex vi has a fixed integer threshold τ1(vi) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Starting from an initial configuration, all vertices update states at each time step using their local functions, and the next state of vi is 1 iff the number of neighbors (plus possibly vi itself) with state-0 at the previous time step (for a synchronous scheme) or at the current time step (for a sequential scheme) is at least τ1(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, analogous to anti-coordination games, we consider the same two decision modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', SN and SE) for dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the schemes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', SyDS or SDS) and decision modes, we have four classes of systems that map to the four classes of anti-coordination games described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We use the notation (SN/SE, IT)-SDS/SyDS to denote different classes of systems (IT stands for inverted-threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' An example of SN mode is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Limit cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A configuration C of a system S is a vector C = (C(v1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',C(vn)) where C(vi) ∈ B is the state of vertex vi under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The dynamics of S from an initial configuration C can be represented by a time-ordered sequence of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A configuration C′ is the successor of C if S evolves from C to C′ in one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C′ be the successor of C, and C′′ be the successor of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If C = C′, that is, no vertices undergo state changes from C, then C is a fixed point of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' if C ≠ C′, but C = C′′, then C ⇌ C′ forms a 2-cycle of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' under the synchronous scheme under the sequential scheme Figure 1: Example dynamics of SDS and SyDS under the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, C is an initial configuration, and C′ is its successor under either the synchronous or the sequential (with vertex update order (v1,v2,v3,v4)) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' State-1 vertices are highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The dynamics of an AC game is captured by the dynamics of the underlying dynamical system S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, an agent v’s action at time t corresponds to v’s state at time t in S, and v’s decision threshold is described by τ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Moreover, the evolution of the action profile for the game coincides with the transition of configurations under S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, We have: Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A fixed point and a limit cycle of S correspond respectively to an NE and a limit cycle of the action profile for the underlying anti-coordination game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the convergence time of S precisely characterizes the convergence time of the corresponding game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given this connection, proofs of our results for anti-coordination games are given in the context 6 of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4 Equilibrium Existence and Finding For self essential (SE) anti-coordination games, we establish that EQE (and therefore EQF) is NP- hard, even on bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the corresponding counting problem is #P-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In contrast, for self non-essential (SN) anti-coordination games, we can find an NE in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We remark that the simple difference between the two modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', whether each agent considers its own state or not) yields a major difference in the complexity of finding an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We further discuss the reasons for this contrast in a later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, to cope with the hardness under the SE mode, we identify special classes where an NE (if it exists) can be found efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first observe that if a SyDS S and an SDS S′ have the same underlying graph and the same local functions, then they have the same set of fixed points, regardless of the vertex permutation Π of S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A SyDS and an SDS with the same underlying graph and the same local functions have the same set of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since a fixed point of a dynamical system corresponds to an NE of the underlying anti-coordination game, it follows that the complexities of EQE/EQF are the same for SN-SyACG and SN-SACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The same observation holds for SE-SyACG and SE-SACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1 Intractability for the self essential mode We establish that EQE (and therefore EQF) is hard for the anti-coordination games under the SE mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', SE-SyACG and SE-SACG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, we present a reduction from 3SAT to EQE for the SE-SyACG (modeled as a SyDS), and by Observation 2, the hardness is carried over to the SE-SACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the reduction is parsimonious, which further implies that #EQE is #P-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We present detailed proofs in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both SE-SyACG and SE-SACG, EQE is NP-complete, and the counting problem #EQE is #P-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' These results hold even when the graph is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a 3SAT formula f, we construct a (SE, IT)-SyDS S such that there is a one- to-one correspondence between satisfying assignments of f and fixed points of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The construction creates a positive and a negative literal vertex for each variable in f, and a clause vertex for each clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Each clause vertex is adjacent to the corresponding literal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We construct two carefully 7 designed gadgets to ensure that (i) in any fixed point of S, a positive and a negative literal vertex for the same variable have complementary values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) there is a one-to-one correspondence between satisfying assignments and fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 Finding NE under the self non-essential mode It is known that a SN-SACG always has an NE, and an NE can be found in finite time [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, EQE is trivially true for SN-SACG, and by Observation 2, EQE is also true for SN-SyACG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' however, the complexity of the search problem EQF is not implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this section, we look beyond the existence problem and show that an NE for an SN game (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', SN-SyACG and SN-SACG) can be found in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, we show that starting an (SN, IT)-SDS S′ (modeling a SN-SACG) from any configuration C, a fixed point of S′ is always reached in at most 3m steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since each step of S′ can be carried out in O(m) time, a fixed point of S′ can be found in O(m2) time (Theorem 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for a (SN, IT)-SyDS S (modeling a SN-SyACG), we can transform it into a corresponding SDS S′, and then find a fixed point of S′ (obtained as described above), which is also a fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We provide more details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S′ be an (SN, IT)-SDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Recall that for each vertex u ∈ V (GS′), τ1(u) is the minimum number of state-0 neighbors of u such that fu evaluates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For the SN mode, τ0(u) = d(u) + 1 − τ1(u) is the minimum number of state-1 neighbors of u such that fu evaluates to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a configuration C of S′, the potentials P of vertices, edges, and C are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Vertex potential: The potential of u ∈ V (GS′) under C is P(C,u) = τ0(u) if C(u) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' P(C,u) = τ1(u) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Edge potential: The potential of e = (u,v) ∈ E(GS′) under C is P(C,e) = 1 if C(u) = C(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' P(C,e) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Configuration potential: The potential of C is the sum of the vertex potentials and edge potentials over all vertices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' That is, P(C,S′) = ∑u∈V (GS′) P(C,u) + ∑e∈E(GS′) P(C,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now establish that under any configuration C, the potential is lower and upper bounded by polynomials in n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A detailed proof is in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For any configuration C of S′, we have 0 ≤ P(C,S′) ≤ 3m (1) Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' One can easily verify that the configuration potential over any configuration is at least 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the upper bound, we argue that ∑ u∈V (GS) P(C,u) ≤ ∑ u∈V (GS) max{τ0(u),τ1(u)} ≤ ∑ u∈V (GS) d(u) = 2m (2) and that the upper bound for the edge potential is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The upper bound of 3m for the configuration potential follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4 establishes that the configuration potential gap between any two configurations is at most 3m for S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Decrease of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Next, we argue that whenever a substep of S′ changes the state of a vertex, the configuration potential decreases by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, the system reaches a fixed point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', no vertices further update their states) in at most 3m total steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The detailed proof appears in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose that in a substep of S′ for a vertex u, the evaluation of fu results in a state change of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C denote the configuration before the substep, and ˆC the configuration after the substep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then, P( ˆC,S′) − P(C,S′) ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since u is the only vertex that undergoes a state change, the overall configuration potential is affected by only the change of u’s potential and the potentials of edges incident to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', suppose u’s state changes from 0 to 1 in the transition from C to ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We can show that the change in the configuration potential is of the form P( ˆC,S′) − P(C,S′) = τ1(u) + d1(u) − τ0(u) − d0(u) (3) where d0(u) and d1(u) are the numbers of u’s neighbors in state-0 and state-1 in C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since ˆC(u) = 1, it follows that d0(u) ≥ τ1(u) and d1(u) ≤ τ0(u)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, P( ˆC,S′)−P(C,S′) ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' From the above two Lemmas, starting from an arbitrary configuration of (SN, IT)-SDS S′, a fixed point is reached in 3m steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we can find a fixed point by starting with an arbitrary inititial configuration, and simulating the operation of S′ for 3m steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Each step consists of n substeps, each of which evaluates one of the local functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, each step can be simulated in O(m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, a fixed point of S′ can be found in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, given a (SN, IT)-SyDS S, we can first convert S into an SDS S′ by assigning a random vertex permutation, and then find a fixed point of S′, which is also a fixed point S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, given that a fixed point of a dynamical system is an NE of the underlying self non-essential game, we have: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both SN-SACG and SN-SyACG, we can find an NE in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 9 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A key reason for the drastic contrast in the complexity of EQE/EQF between the SN and SE modes is the difference in the behavior of threshold-1 vertices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', vertices with τ1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In the SE mode, a threshold-1 vertex u cannot be in state 0 under any fixed point because u will change to state 1 in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (Since u counts its own state, there is at least one 0 input to fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, the gadgets used in the hardness proof for the SE mode critically depend on this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In contrast, such a constraint does not hold for threshold-1 vertices under the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Hence, our hardness proof does not carry over to the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3 Efficient algorithms for special classes Given the hardness of EQE/EQF for SE anti-coordination games, we identify several sufficient con- ditions under which an NE can be obtained efficiently (if one exists), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both SE-SyACG and SE-SACG, there is a O(m + n) time algorithm for EQE/ EQF for any of the following restricted cases: (i) The underlying graph is a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) The underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (iii) The underlying graph is a directed acyclic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (iv) The threshold for each u satisfies τ1(u) ∈ {1,du + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A detailed proof appears in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Here, we provide a sketch for (i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', complete graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let P = {V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',Vk} be the partition of the vertex set V where each block of P consists of a maximal set of vertices with the same value of τ1, and the blocks of P are indexed so that for each i, 1 ≤ i < k, the members of Vi have a lower value of τ1 than the members of Vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose that configuration C is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let q be the highest block index such that the number of 0’s in C is at least the τ1 value of the vertices in Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since the underlying graph is a complete graph, for each vertex u, C(u) = 1 iff u ∈ Vi for some i where i ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus there are only k + 1 candidate configurations that can possibly be a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our algorithm constructs each of these candidates, and checks if it is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 5 Convergence We have shown in the previous section that for SN-SACG, starting from any action profile, the best- response dynamics converges to an NE in O(m) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In contrast, it is known that the best-response for SE-SACG could have exponentially long limit cycles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The dynamics of an SDS and its corresponding SyDS can be drastically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the O(m) convergence time for a SACG (SDS) established in the previous section does not imply an O(m) convergence time for a SyACG (SyDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In fact, synchronous anti-coordination games are not potential games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the approach we used for SACG does not carry over to the analysis of 10 SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Also, both SN-SyACG and SE-SyACG can have length-2 limit cycles, and there are instances of SE-SyACG that do not have an NE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the underlying graph is an odd cycle and τ1 = 2 for all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1 Convergence of synchronous games Synchronous anti-coordination games are not potential games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the results on potential games by Monderer and Shapley ([28]) do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As shown in [2], the limit cycles of such a game are of length at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this section, we study the convergence time to either an NE or a 2-cycle for SN-SyACG and SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Using a potential function argument inspired by [18], in Theorem 10 we establish that for both SN-SyACG and SE-SyACG, starting from an arbitrary action profile, the best-response dynamics converges in O(m) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (A detailed proof of the theorem appears in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=') Here, we present a proof sketch for the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SN, IT)-SyDS corresponding to a given SN-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Due to the existence of length-2 limit cycles, our definitions of potentials at each time step account for not only the states of vertices at the current step but also that of the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SN mode, let τ0(u) = d(u) + 1 − τ1(u) be the minimum number of state-1 neighbors of vertex u such that fu evaluates to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We henceforth assume that no local function is a constant function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', 1 ≤ τ1(u) ≤ d(u), ∀u ∈ V (GS));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' a justification is given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, 1 ≤ τ0(u) ≤ d(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We define ˜τ0(u) = τ0(u) − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given an arbitrary configuration C of S, let C ′ be the successor of C, and C′′ the successor of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Vertex potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential of a vertex u under C is defined by P(C,u) = [C(u) + C ′(u)] ⋅ ˜τ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Edge potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential of an edge e = (u,v) under C is defined by P(C,e) = C(u) ⋅ C ′(v) + C(v) ⋅ C ′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential of a configuration C is defined by P(C,S) = ∑ e∈E(GS) P(C,e) − ∑ u∈V (GS) P(C,u) We now establish lower and upper bounds on the potential under an arbitrary configuration C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For any configuration C of a (SN, IT)-SyDS S, we have −4m + n ≤ P(C,S) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' One can easily verify that the maximum value of the sum of vertex potentials is 4m−n, which gives the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now address the upper bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each vertex u, let Eu be the 11 set of edges incident on u, and let σu = ∑u∈V (GS) C(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that for inverted-threshold function fu, C′(u) = 1 iff σu < τ0(u), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', iff σu ≤ τ0(u) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let βu = σu ⋅ C′(u) − C ′(u) ⋅ ˜τ0(u) − C(u) ⋅ ˜τ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The configuration potential can be restated as: P(C,S) = ∑u∈V (GS) βu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If C ′(u) = 0, then βu = −C(u) ⋅ ˜τ0(u), which is at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If C ′(u) = 1, then βu ≤ (τ0(u) − 1) − ˜τ0(u), which is at most −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Decrease of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We show that from any configuration C, the potential decreases by at least 1/2 every step until a fixed point or a 2-cycle is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, the dynamics converges in at most 8m − 2n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C be an arbitrary configuration of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let ∆(S) = P(C ′,S) − P(C,S) denote the change of configuration potential from C to C ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then ∆(S) = 0 if and only if C = C ′′, that is C is a fixed point or is part of a 2-cycle C ←→ C ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, if C ≠ C ′′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the dynamics has not converged), then, the configuration potential has decreased by at least 1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', ∆(S) ≤ −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof (sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We define the change of potential for an edge e = (u,v) as ∆(e) = P(C ′,e) − P(C,e), and the change of potential for a vertex u as ∆(u) = P(C ′,u) − P(C,u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then, ∆(S) = P(C ′,S) − P(C,S) = ∑ e∈E(GS) ∆(e) − ∑ u∈V (GS) ∆(u) (4) Rearranging terms from the definition of potentials, we get ∆(e) = C ′(u) ⋅ [C ′′(v) − C(v)] + C ′(v) ⋅ [C ′′(u) − C(u)] (5) and ∆(u) = [C ′′(u) − C(u)] ⋅ ˜τ0(u) (6) Now we argue that ∆(S) = 0 iff C = C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' First suppose that C = C ′′, so that for every vertex u, C(u) = C ′′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, ∆(e) = 0, ∀e ∈ E(GS) and ∆(u) = 0, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Now suppose that C ≠ C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let V0−1 denote the set of vertices u such that C(u) = 0 and C ′′(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, let V1−0 denote the set of vertices u such that C(u) = 1 and C ′′(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We establish the following two equalities: ∑ u∈V (GS) ∆(u) = ∑ u∈V0−1 ˜τ0(u) − ∑ u∈V1−0 ˜τ0(u) (7) 12 and ∑ e∈E(GS) = ⎛ ⎝ ∑ u∈V0−1 ∑ (u,v)∈Eu C ′(v)⎞ ⎠ − ⎛ ⎝ ∑ u∈V1−0 ∑ (u,v)∈Eu C ′(v)⎞ ⎠ (8) where Eu is the set of edges incident on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Recall that ∆(S) equals the change in edge potentials minus the change in vertex potentials, so by (i) and (ii) above, ∆(S) = ∑ u∈V0−1 ⎛ ⎝ ⎛ ⎝ ∑ (u,v)∈Eu C ′(v)⎞ ⎠ − ˜τ0(u)⎞ ⎠ + ∑ u∈V1−0 ⎛ ⎝˜τ0(u) − ⎛ ⎝ ∑ (u,v)∈Eu C ′(v)⎞ ⎠ ⎞ ⎠ (9) We argue that if u ∈ V0−1, then: ∑ (u,v)∈Eu C ′(v) ≤ τ0(u) − 1 = ˜τ0(u) − 1 2 (10) and thus (∑(u,v)∈Eu C ′(v)) − ˜τ0(u) ≤ −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Likewise, if u ∈ V1−0, then: ∑ (u,v)∈Eu C ′(v) ≥ τ0(u) = ˜τ0(u) + 1 2 (11) and thus ˜τ0(u) − ∑(u,v)∈Eu C ′(v) ≤ −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since V0−1 ∪ V1−0 ≠ ∅, we have ∆(S) ≤ − ∑ u∈V0−1 1 2 − ∑ u∈V1−0 1 2 ≤ −1 2 (12) and the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The above discussion establishes that starting from an arbitrary configuration of S, the dynamics stabilizes in O(m) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our convergence time results for the SN synchronous anti-coordination games follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In the Appendix, we also show that the above proof can be easily extended to SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SN-SyACG and SE-SyACG, starting from any initial action profile, the dynamics converges in O(m) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The polynomial-time convergence (to either an NE or a 2-cycle) for SE-SyACG does not contradict the results in the previous section where we showed that determining if a SE-SyACG has an NE is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, despite the fast convergence to a limit cycle, the limit cycle will often be a 2-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 13 6 Experimental Studies We conduct experiments to study the contrast between the empirical convergence time of synchronous anti-coordination games under the two modes (SN and SE) on networks with varying structures, shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, astroph, google+ and Deezer are real-world social networks [33, 23, 22], and synthetic networks from the classes Gnp, scale-free [7], and small-world [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we investigate the contrast in the number of Nash equilibria for small problem instances under the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' All experiments were conducted on Intel Xeon(R) Linux machines with 64GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Source code, documentation, and selected networks appear in the code supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1 Experimental results on convergence Convergence time on networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each network, we randomly generate 200 copies of threshold assignments, where τ1(u) is chosen uniformly at random in the range [1,d(u) + 1] for the SE mode, and in the range [1,d(u)] for the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This gives us 200 problem instances for each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Next, for each instance, we construct random initial configurations using various probabilities p of each vertex having state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, we vary p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='9, and for each p, we generate 500 random initial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This gives us 2,500 initial configurations for each instance and a total of 500,000 simulations of the dynamics on each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The average number of steps to converge are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that this average for all the examined networks is less than 20 for both modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the maximum number of steps (over all simulations) for the SE and SN modes are 53 and 24 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Network n m Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' steps (SE) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' steps (SN) Small-world 10,000 90,000 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='41 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='71 Scale-free 10,000 97,219 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='78 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='66 Gnp 10,000 99,562 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='09 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='71 astroph 17,903 196,972 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='31 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='72 google+ 23,613 39,182 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='05 Deezer 28,281 92,752 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='41 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='29 Table 2: Convergence for different networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Average number of time steps for the best-response dynamics to converge to a NE or a 2-cycle under the SE and SN modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Impact of network density on convergence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We study the contrast in the average con- vergence time between the two modes under different network densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, we simulate the dynamics on Gnp networks of size 10,000, with average degrees varying from 5 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The results appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The variances of the two modes are shown as shaded regions, with one stdev above and below the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, as the network density increases, we observe a close to linear increase 14 in the convergence time and the variance for the SE mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In contrast, the convergence time and the variance for the SN mode change marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 0 20 40 60 80 100 Average degree 0 10 20 30 40 50 Average number of time steps Self essential Self non-essential Figure 2: Impact of network density on the average number of steps for the SE and SN modes to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The underlying Gnp networks have 10,000 vertices with average degrees varying from 5 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The variances for the SE and the SN modes are shown in the beige and blue shaded regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To gain additional insight, we computed the average (over all pairs of consecutive steps) number n of vertices whose states change every 2 steps until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As suggested by Lemma 9 in section 5, a higher n implies a faster decrease in the system potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, the value of n for the SE mode and SN mode are 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='24 and 544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This provides one reason for the observed difference in the convergence time between the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the maximum convergence time over all simulations for the SE mode is 186, whereas that for the SN mode is only 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 Experimental results on NE existence As we have shown, determining whether a game has an NE is intractable for the SE mode but easy for the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Here, we compare the number of NE in small instances between the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, we construct 100 Gnp networks of size 20 with average degrees of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each Gnp network, we construct 200 different threshold assignments where τ1(u) is selected uniformly at random in range [1,d(u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This gives us a total of 20,000 instances for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, for each instance, we check all 220 possible configurations and count the number of NE among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The contrast in the number of NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For the SE mode, 710 instances have NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Among these 710 instances, 706 have exactly one NE each, and the remaining 4 instances have 2 NE each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In contrast, all the 20,000 instances for the SN mode have at least one NE, and the average number of NE per instance is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='61% of the SN instances have at most 9 NE, and 98% have at most 28 NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For 1 ≤ η ≤ 28, the number of instances with η NE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This shows a contrast in the number of NE for the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, most configurations that are NE under the SN mode 15 0 5 10 15 20 25 Number of Nash equilibria 0 1000 2000 Number of instances Self non-essential Self essential Figure 3: The distribution of the number of instances with at most 28 NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The underlying Gnp net- works are of size 20 with an average degree of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' are no longer NE under the SE mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 7 Conclusions and Future Work We studied the problem of finding Nash equilibria in evolutionary anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We also considered the convergence problem for such games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our results present a contrast between the self essential (SE) and self non-essential (SN) modes w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' the complexity of finding NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we rigorously established an upper bound on the convergence time for both modes by showing that the best-response dynamics reaches a limit cycle in a polynomial number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' One possible future direction is to tighten the bound on convergence time as the empirical convergence time is much smaller than the theoretical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Another direction is to study the problem of finding an approximate NE [15] for the SE anti-coordination games, since finding an exact NE is hard under the SE mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, it is also of interest to examine the existence of other forms of NE such as mixed-strategy equilibria in anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 16 References [1] Elie M Adam, Munther A Dahleh, and Asuman Ozdaglar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the behavior of threshold models over finite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pages 2672–2677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [2] Elie M Adam, Munther A Dahleh, and Asuman Ozdaglar.' metadata={'source': 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+page_content=' The evolution of social norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Economics, 7(1):359–387, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Technical Appendix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Symbols ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Sequential anti-coordination games ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SyACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Synchronous anti-coordination games ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Self essential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Self non-essential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SE-SACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Sequential anti-coordination games under self essential mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SN-SACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Sequential anti-coordination games under self non-essential mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SE-SyACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Synchronous anti-coordination games under self essential mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SN-SyACG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Synchronous anti-coordination games under self non-essential mode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SDS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Sequential dynamical system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='SyDS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Synchronous dynamical system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='IT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Inverted threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='(SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SDS Sequential dynamical system under self essential mode (SN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SDS Sequential dynamical system under self non-essential mode (SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SyDS Synchronous dynamical system under self essential mode (SN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SyDS Synchronous dynamical system under self non-essential mode S = (GS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='F) A SyDS S with underlying graph GS and set of local functions F S′ = (GS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Π) An SDS S′ where Π is the vertex update sequence n The number of vertices in GS m The number of edges in GS fv Local function of v τ1(v) For vertex v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' the threshold value on the number of 0’s for fv to equal 1 τ0(v) For vertex v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' the threshold value on the number of 1’s for fv to equal 0 N(v) The open neighborhood of v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the set of vertices adjacent to v N +(v) The closed neighborhood of v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the set of vertices adjacent to v and v itself d(v) The degree of v C A configuration of S C′ The successor of C C′′ The successor of C′ C(v) The state of v in configuration C of S Table 3: Symbols and Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that τ1(v) uniquely determines τ0(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, τ0(v) + τ1(v) = d(v) + 2 for the self essential mode, and τ0(v) + τ1(v) = d(v) + 1 for the self non-essential mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the mapping between anti-coordination games and discrete dynamical systems, we present all proofs in the context of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 20 A Detailed Discussion of Related Work It is known that the self non-essential sequential anti-coordination games are potential games, and by the argument from Monderer and Shapley [28], such a game always has an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, a potential game admits a potential function that increases when an agent chooses an action that yields a strictly higher payoff [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a potential game starting from an initial action profile, consider the sequence of profiles such that at each step, a player updates its action to obtain a strictly higher payoff (if possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Such a sequence is called an improvement path, and for potential games, all the improvement paths are of finite length (known as the finite improvement property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' More importantly, a maximal improvement path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', an improvement path that goes to a maximum of the potential) always ends at an equilibrium point [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that this result does not immediately imply that an NE can be reached in polynomial time, as the number of possible action profiles is exponential in the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we emphasize that anti-coordination games under synchronous update schemes are not potential games as limit cycles of length 2 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The book by Goles and Martinez [18] discussed the phase space properties of dynamical systems, and one can verify that such systems also model coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, they prove that for synchronous coordination games, the length of any limit cycle is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Their argument uses a mathematical tool called algebraic invariants, and they show that if we consider each limit cycle as a periodical sequence, then the length of such a sequence is either 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In the same work, Goles and Martinez proposed a Lyapunov function for synchronous coordination games and show that the best-response dynamics of the game converges to a limit cycle of length at most 2 in a polynomial number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For anti-coordination games, Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ([8]) study phase space properties of sequential dynamical systems (which includes modeling sequential AC games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Their results imply that when local functions are nad’s and nor’s (which are inverted-threshold functions), the length of a limit cycle in a self essential sequential anti-coordination game can be 2O(√n) where n is the number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Later, Adam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' [2] use a combinatorial approach and argue that the length of a limit cycle in a synchronous anti-coordination game is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' However, they did not bound the convergence time to reach a limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A more recent work by Ramazi, Riehl, and Cao [31] investigates the convergence time for asyn- chronous self non-essential anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In their asynchronous dynamics, agents are 21 updated at each time step in a random order, and for each agent, the number of steps between any two consecutive updates is guaranteed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on this scheme, they show that under the best-response dynamics, an equilibrium is always reached in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Additional Material for Section 4 In this section, we present the detailed proofs of the results given in section 4 of the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We start with a key observation: Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A SyDS and an SDS with the same underlying graph and the same set of local functions have the same set of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, SN-SyACG (SE-SyACG) and SN-SACG (SE-SACG) have the same complexity for EQE / EQF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1 Intractability for the self essential mode We establish that Equilibrium existence (EQE) is NP-hard for self essential synchronous anti- coordination games (SE-SyACG), and the problem remains hard on bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the corresponding counting problem is #P-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This immediately implies that Equilibrium find- ing (EQF) is also hard for SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, Let S be a (SE, IT)-SyDS that models a SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We show that determining the existence of a fixed point of S is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1, it follows that EQE / EQF is also hard for SE-SACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now proceed with the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose that a vertex v of a (SE, IT)-SyDS S has threshold τ1(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then in every fixed point (if any exists) C of S, C(v) = 1 and at least one neighbor of v has state 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose that a vertex v of a (SE, IT)-SyDS S has threshold τ1(v) = d(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then in every fixed point (if any exists) C of S, if C(v) = 1, then then all neighbors of v have state 0 in C, and if C(v) = 0, then at least two neighbors of v have state 1 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S be a (SE, IT)-SyDS whose underlying graph GS is a complete bipartite graph of the form: 22 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The bipartitions V (GS) = {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ∣A∣ = ∣B∣ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' τ1(v) = 3, ∀v ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The phase space of S has only 2 distinct fixed points for which either (i) vertices in A are in state 1, and vertices in B are in state 0, or (ii) vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Figure 4: An example (SE, IT)-SyDS S for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4, where GS is a complete bipartite graph with bipartitions {A,B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The thresholds τ1 of all vertices are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By the threshold values of vertices in A and in B, it is easy to see that the two configurations given in the Lemma are fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now argue that the system has no other fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C be a configuration of S that is different from the two fixed points above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C′ be the successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first consider the case where both A and B have at least one state-1 vertex under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v ∈ A be such a vertex in state 1 under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that since B has at least one state-1 vertex, the number of state-0 vertices in v’s closed neighborhood is at most ∣B∣ − 1 < τ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we have C′(v) = 0 and C is not a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Next, we consider the case where both A and B have at least one state-0 vertex under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v and w be two state-0 vertices in A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since τ1(u) = 3, ∀u ∈ V (GS), C(v) = 0 implies that there exists at least two state-1 vertex in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, C(w) = 0 implies that there exists at least two state-1 vertex in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By our previous argument, it follows that C is not a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now present the Theorem on the intractability of EQE for SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SE-SyACG, the Equilibrium existence( EQE) is NP-complete and the counting problem #EQE is #P-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the problem remains hard on bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' One can easily verify that the problem is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now establish the NP-hardness of EQE via a reduction from 3SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let f be the formula for an arbitrary 3SAT instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We construct a SyDS S such that S has a vertex for each literal, a vertex for each clause, and a collection of gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, S is constructed so that there is a one-to-one correspondence between fixed points of S and satisfying assignments to the given 3SAT formula f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Literal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each variable xi in f, there is a positive literal vertex yi and a negative literal vertex zi in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Both these vertices have threshold τ1(yi) = τ1(zi) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Under any configuration, we interpret positive literal vertex yi having state 0 as corresponding to variable xi being true, and negative literal vertex zi having value 0 as corresponding to variable xi being false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our Gadgets (shown later) ensure that in any fixed point of S, these two vertices have complementary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Clause vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each clause cj in f, there is a clause vertex wj, τ1(wj) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, wj is adjacent to each of the literal vertices that correspond to literals occurring in clause cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' There is no other edges incident on wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' From Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2, in every fixed point of S, wj has state 1, and at least one of the adjacent literal vertices has state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now construct the gadgets, which involve auxiliary vertices for the literal vertices, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each variable xi in f, there are three auxiliary vertices, namely ai, di and ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, (i) The vertex ai is adjacent to both xi’s positive literal vertex yi and negative literal vertex zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, τ1(ai) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' From Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2, in every fixed point of S, ai has state 1, and at least one of the two literal vertices yi and zi has state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) The vertex di and ei has threshold τ1(di) = 1 and τ1(ei) = 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, we introduce three auxiliary edges (ei, di), (ei, yi) and (ei,zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' From Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2, in every fixed point, di must has state 1, thus, its neighbor ei has state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then, from an application of Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3 to vertex ei, at least one of the two literal vertices yi and zi has state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The combined effect of the auxiliary vertices and edges described thus far is that under every fixed point of S, exactly one of yi and zi has state 0, and exactly one has state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Finally, for every literal vertex (positive or negative), denoted by v, we add an auxiliary structure based on the bipartite graph in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The structure involves five auxiliary vertices: g1 v, g2 v, h1 v, h2 v, and h3 v, each with threshold τ1 = 3 (which is the degree of each of these vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consider the 24 subsets Av = {v, h1 v, h2 v} and Bv = {g1 v, g2 v,g3 v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The structure has the following nine auxiliary edges: (v, h1 v), (v, h2 v), (v, h3 v), (g1 v, h1 v), (g1 v, h2 v), g1 v,h3 v), (g2 v,h1 v), (g2 v,h2 v), (g2 v,h3 v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' That is, the subgraph of S induced on Av ∪ Bv is a complete bipartite graph with bipartitions Av and Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4, that in any fixed point of S, either all the vertices in Av have state 1 and all the vertices in Bv have state 0, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' To see this, first suppose that in a given fixed point C, at least one of the vertices in Bv has state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then, from Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3, the three neighbors of this vertex, namely the three vertices in Av, all have state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, since the threshold of each vertex in Bv is 3, all three vertices in Bv have value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Now suppose that none of the vertices in Bv has value 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', they all have value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since the threshold of each vertex in Av is 3, all three vertices in Av have value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This completes the construction of S which clearly takes polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the resulting graph is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now claim that there is a one-to-one correspondence between fixed points of S and satisfying assignments of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (⇒) Let α be a satisfying assignment of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We construct a configuration Cα of S as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each variable under α, if α[xi] = true, then the positive literal vertex yi has the state Cα(yi) = 0, and the negative literal vertex zi has the state Cα(zi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if If α[xi] = false, then Cα(yi) = 1 and Cα(zi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, every clause vertex wj has state 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the states of auxiliary vertices for each xi, we set ai and di to state 1, and ei to state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, for each literal vertex v, the other two members of Av have the same value as v, and the three members of Bv have the complement of the value of v under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This completes the specification of Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By checking the number of state-0 vertices in the closed neighborhood of each vertex, it can be verified that Cα is a fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (⇐) Let let C be a fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let αC be the assignment to f where αC(xi) = 1 iff Cα[yi] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since C is a fixed point, every clause vertex is adjacent to at least one literal vertex with the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, αC is a satisfying assignment of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now have established that determining if S has a fixed point is NP-complete, even when the underlying graph is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, it can be verified that C = CαC, so the reduction is parsimonious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The NP-hardness of EQE and the #P-hardness of the counting version #EQE for 25 SE-SyACG immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SE-SACG, the Equilibrium existence( EQE) is NP-complete and the counting problem #EQE is #P-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the problem remains hard on bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A fixed point under the SE mode is generally not a fixed point under the SN mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the above hardness result does not imply a hardness of EQE for the SN anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 Finding NE under the self non-essential mode As pointed out in [37], a self non-essential sequential anti-coordination game (SN-SACG) always has a pure Nash equilibrium (NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1, a self non-essential synchronous anti-coordination game (SN-SyACG) also always has an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this section, we explore beyond the existence problem and tackle the problem of finding an NE in a for SN mode under arbitrary network topology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', EQF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, inspired by the potential function approach developed in [9], given a (SN, IT)-SDS S′ (modeling a SN-SACG), we show that starting from an arbitrary configuration, a fixed point of S′ is always reached in at most 3m−n steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since each step of S′ can be carried out in O(m) time, a fixed point of S′ is then found in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consequently, a fixed point of a (SN, IT)-SyDS S (modeling a SN-SyACG) can also be founded in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We note that our results does not follow from [9] since (i) we study anti-coordination games, and the results in [9] are for coordination games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) in the self non-essential mode, each vertex does not consider its own state while playing the game, whereas [9] focuses on the case where its own state is considered by each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If τ1(v) = 0 or τ1(v) = dv + 1 for some vertex v, then v is a vertex whose state is constant after at most one transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we can remove v from the graph and update the threshold of neighbors correspondingly without affecting system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Without loss of generality, we assume that there are no constant vertices, that is 1 ≤ τ1(v) ≤ d(v), ∀v ∈ V (GS′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 26 The potential functions and bounds Let S′ = (GS′,F′,Π) be a (SN, IT)-SDS that models a SN-SACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a configuration C, We now define the potentials of vertices, edges, and the system under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The vertex potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a vertex u ∈ V (GS′), the potential of u under C is defined as follows P(C, u) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ τ0(u) if C(u) = 0 τ1(u) if C(u) = 1 The edge potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given an edge e = (u, v) ∈ E(GS′), the potential of e under C is defined as follows P(C, e) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 1 if C(u) = C(v) 0 if C(u) ≠ C(v) The configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential of the system S′ under C is defined as the sum of the vertex potentials and edge potentials over all vertices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' P(C,S′) = ∑ u∈V (GS′) P(C,u) + ∑ e∈E(GS′) P(C,e) A lower bound on the configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first establish a lower bound of the P(C,S′) for any configuration C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For any configuration C of S′, we have P(C,S′) ≥ ⎛ ⎝ ∑ u∈V (GS′) min{τ0(u),τ1(u)}⎞ ⎠ + γ where γ is the minimum number of edges whose endpoints have the same color in GS′, over all 2-coloring of V (GS′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given any configuration C, since the potential of each vertex is either τ1(u) or τ0(u), it imme- diately follows that ∑u∈V (GS) min{τ0(u), τ1(u)} is the lower bound on the sum of potentials from all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the bound on the sum of edge potentials, remark that C corresponds to a 2-coloring of V (GS′) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', each state represents a color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, the minimum number of edges whose endpoints 27 have the same state under C is at most γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since each edge with the same-state endpoints has potential value 1, it follows that the sum of edge potential under any configuration C is at least γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' An upper bound on the configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Next, we present the upper bound of the configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given an arbitrary configuration C Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The configuration potential satisfies P(C,S′) ≤ 3m Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that under an arbitrary configuration of S′, the sum of vertex potential satisfies ∑ u∈V (GS′) P(C, u) ≤ ∑ u∈V (GS′) max{τ0(u),τ1(u)} ≤ ∑ u∈V (GS′) d(u) = 2m As for the sum of edge potential ∑e∈E(GS′) P(C,e), it is easy to see that the upper bound is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The upper bound of the overall configuration potential follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, We have shown that the gap in the configuration potential between an arbitrary initial configuration and a system’s converged configuration is at most 3m − ∑ v∈V (GS) (min{τ0(v),τ1(v)}) − γ ≤ 3m − n (13) Decrease of the potential after each update We establish that starting from an arbitrary initial configuration that is not a fixed point, the con- figuration potential of S′ decreases by at least 1 after each vertex switches its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given that the potential gap is at most 3m−n/2, it follows that the system reaches a fixed point in at most 3m−n/2 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a configuration C of S′ that is not a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let ˜C be a configuration that results from the state change of a single vertex u due to the dynamics of S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We then have P(C,S′) − P( ˜C,S′) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 28 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since u is the only vertex that undergoes the state change, the overall configuration potential is affected by only the change of u’s potential and the potentials of edges incident to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let d0(u) and d1(u) denote the number of u’s neighbors in state-0 and the number of u’s neighbors in state-1 under C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Without loss of generality, suppose u changes its state from C(u) = 0 to ˜C(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, the sums of potentials of u and edges incident to u under C and ˜C are P(C, u) + ∑ e=(u,v),v∈N(u) P(C,e) = τ0(u) + d0(u) and P( ˜C,u) + ∑ e=(u,v),v∈N(u) P( ˜C,e) = τ1(u) + d1(u) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since ˜C(u) = 1, it follows that d0(u) ≥ τ1(u) and d1(u) ≤ τ0(u) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, we have P(C,S′) − P( ˜C,S′) = P(C, u) + ∑ e=(u,v),v∈N(u) P(C,e) − ⎛ ⎝P( ˜C,u) + ∑ e=(u,v),v∈N(u) P( ˜C,e)⎞ ⎠ = τ0(u) + d0(u) − τ1(u) − d1(u) ≥ 1 This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In summary, we have shown that the potential gap between any two configurations is at most 3m−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, each change of vertex state due to the system dynamic decreases the overall potential by at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since in each time step (before reaching a fixed point), at least one vertex updates its state, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='9 immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For (SN, IT)-SDS, starting from an arbitrary initial configuration, the system dynamics reaches a fixed point in at most 3m − n time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In each time step, we compute the local functions of all vertices to determine the successor con- figuration, which takes O(m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, a fixed point of S′ can be obtained in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given that a fixed point of a (SN, IT)-SDS S′ is also a fixed point of its twin (SN, IT)-SyDS S, we also establish the tractability of finding an NE for (SN, IT)-SyDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our results for SN anti-coordination games follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 29 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both SN-SyACG and SN-SACG, we can find a pure Nash equilibrium of the game in O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3 Finding Nash equilibria under special cases We have established the intractability of EQE / EQF for self essential (SE) anti-coordination games, whereas a self non-essential (SN) game admits a polynomial time algorithm for finding an NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In this section, we identify several special classes of the problem instances such that an NE (if any) can be found in polynomial time for SE anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, We extend the results of some special classes to SN anti-coordination games such that an NE can be found in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the connection between anti-coordination games and dynamical systems, we present all proofs in the context of synchronous dynamical systems (and the results for sequential systems follow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Inclination for one action over another We consider the special case where agents are inclined to choose one action over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, during the game evolution, each agent u either (i) chooses the action 1 in the next time step if at least one agent in u’s open/closed neighborhood chose action 0 in the previous time step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (ii) chooses the action 0 in the next time step if at least one agent in u’s open/closed neighborhood chose action 1 in the previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that the case (i) above corresponds to τ1(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the case (ii) implies τ1(u) = d(v) for the SN game, and τ1(u) = d(v) + 1 for the SE game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We call the local function of a vertex u with τ1(u) = 1 a nand function, and of u with τ1(u) = d(v) / τ1(u) = d(v) + 1 a nor vertex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SE anti-coordination games and SN anti-coordination games, an NE can be found in O(m + n) time if the corresponding local functions of vertices are nand’s and nor’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first present the result for SE-SyACG, modeled by a (SE, IT)-SyDS S = (GS,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that under any fixed point of S, a nand vertex must be in state 1, and a nor vertex must be in state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This uniquely determines a configuration C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We can then compute C’s successor C′, and examine if C is a fixed point in O(m + n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We further establish the claim below 30 Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Such a configuration C is a fixed point if and only if each nor vertex is adjacent to at least one nand vertex and vise versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For sufficiency, suppose C is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If there exists a nor vertex v whose neighbors are all nor vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then the number of state-0 vertices in v’ closed neighborhood is d(v)+1 (since C is a fixed point, all nor vertices are in state 0) which equals to the threshold of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, C(v) = 0 ≠ C′(v) = 1 and C is not a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' An analogous argument applies to the case where v is a nand vertex whose neighbors are all nand vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, the number of state-0 vertices in v’s closed neighborhood is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, C(v) = 1 ≠ C′(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the necessity of the condition, observe that if a nor vertex v is adjacent to at least one nand vertex, the number of state-0 vertices in v’s closed neighborhood is at most d(v) < τ1(v), thus, C(v) = C′(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, if a nand vertex v is adjacent to at least one nor vertex, then the number of state-0 vertices in v’s closed neighborhood is at least τ1(v) = 1, thus, C(v) = C′(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof for SE-SyACG (and thus also SE-SACG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now establish the result for a SN-SyACG, modeled by a (SN, IT)-SyDS, ¯S = (G ¯ S,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we first construct a candidate configuration C and then modify C to make it a fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, for each vertex v ∈ V (G ¯ S), if v is a nand vertex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', τ1(v) = 1), we set C(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if v is a nor vertex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', τ1(v) = d(v)), set C(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the same argument in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1, it follows that C is a fixed point of S′ if and only if each nand vertex is adjacent to at least one nor vertex and visa versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If this condition does not hold, however, there must exist at least one nand (nor) vertex whose neighbors are all nand (nor) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, We further modify C as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' First, for each nand vertex v whose neighbors are all in state 1 under C, we set C(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let Vnand denote such a set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, for each nor vertex v whose neighbors are all in state 0 under C, we set C(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let Vnor denote this set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The pseudocode is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now argue that the resulting configuration C is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C′ be the successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' First, observe that for each nand vertex u ∈ V (GS)∖(Vnand ∪Vnor), we have C(u) = 1 (since we never change the state of vertices that are not in Vnand or Vnor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, u must be adjacent to at least one state-0 vertex, or else u will be in Vnand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, for each nor vertex u ∈ V (GS)∖(Vnand∪Vnor), we have that C(u) = 0 and u is adjacent to at least one state-1 vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It immediately follows that C(u) = C′(u), ∀u ∈ V (GS) ∖ (Vnand ∪ Vnor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now consider the states of vertices in Vnand or 31 Algorithm 1 EQF NAND NOR SN(S′) Input: A (SN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SyDS ¯S = (G ¯S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='F),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' where τ1(v) = 1 or d(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ∀v ∈ V (G ¯S) Output: A fixed point C of S′ 1: C ← an initial configuration of all 0’s 2: for v ∈ V (GS′) do 3: if τ1(v) = 1 then C(v) ← 1 ▷ v is a nand vertex 4: else if τ1(v) = d(v) then C(v) ← 0 ▷ v is a nor vertex 5: end for 6: for v ∈ V (GS′) do 7: if τ1(v) = 1 and C(w) = 1 for all neighbors w ∈ N(v) then C(v) ← 0 ▷ Vnand = Vnand ∪ {v} 8: if τ1(v) = d(v) and C(w) = 0 for all neighbors w ∈ N(v) then C(v) ← 1 ▷ Vnor = Vnor ∪ {v} 9: end for 10: return C Vnor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each vertex v ∈ Vnand, observe that neighbors of v must all be nand vertices, or else, v is adjacent to at least one state-0 vertex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', a nor vertex) which contradicts the fact that v ∈ Vnand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, we claim that Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Each neighbor of v is not in Vnand, ∀v ∈ Vnand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contradiction, suppose v′ is a neighbor of v who is also in Vnand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If v′ is added to Vnand before v, then v cannot also be in Vnand since v has v′ as a state-0 neighbor under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, if v is added to Vnand before v′, then v′ cannot also be in Vnand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 implies that v has no state-0 neighbor under C, thus, C(v) = C′(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By a similar argument, for each vertex v ∈ Vnor, neighbors of v must all be nor vertices, or else, v is adjacent to at least one state-1 vertex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', a nand vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Moreover, each neighbor of v is not in Vnor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that the number of state-0 neighbors of v is d(v), and C(v) = C′(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the correctness for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the time complexity, the for loop from line 2 to 5 takes O(n) time, and the for loop from line 6 to 9 takes O(m+n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the overall running time of Algorithm 1 is O(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The results for SN-SyACG and SN-SACG follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The underlying graph is a DAG Under a directed graph, at each time step, an agent in the self essential mode considers its own state and the state of in-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, each agent in the self non-essential mode only considers the states of its in-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both the SE anti-coordination game and the SN anti-coordination game, an NE 32 can be found in O(m + n) time if the underlying graph is a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first present the result for SE-SyACG modeled by a (SE, IT)-SyDS S = (GS,F′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let V ′ be the set of vertices with 0 indegree in GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that V ′ ≠ ∅, or else, GS contains directed cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first remark that S has no fixed points if there exists a vertex v ∈ V ′ that is not a constant vertex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', τ1(v) = 1, ∃v ∈ V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contrapositive, suppose τ1(v) = 1 for some v ∈ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given any configuration C where C(v) = 1, it follows that C′(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, if C(v) = 0, then C′(v) = 1 for such a vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, any configuration C of S cannot be a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose all vertices in V ′ are constant vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our algorithm constructs a potential fixed point C as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each vertex v ∈ V ′, if τ1(v) = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', v is a constant-1 vertex), we set C(v) = 1, and remove v from GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if τ1(v) = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', v is a constant-0 vertex), we set C(v) = 0, then decrease the threshold values of all v’s out-neighbors by 1 and remove v from GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Note that after removing v, we might set some other vertices in V (GS) ∖ V ′ to have 0 in-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, we add these new 0 in-degree vertices to V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The pseudocode is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If at any iteration of the algorithm, we found a vertex in V ′ to have threshold 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', a non-constant vertex), the algorithm terminates and concludes that S has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if all vertices in V ′ are constant vertices, it follows that the algorithm uniquely determines a fixed point C of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the running time, we may consider the algorithm as a breadth-first search process that takes O(m + n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof SE-SyACG (and thus SE-SACG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the SN anti-coordination games, note that a 0-indegree vertex v is always a constant vertex, irrespective of τ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, a SN anti-coordination game (either SN-SyACG or SN-SACG) has a unique equilibrium which is determined by the threshold of each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The underlying graph has no even cycles We first introduce three characterizations of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Definition 11 (Terminal vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let G be an undirected graph with no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A vertex v is a terminal vertex if v is not in any cycles, and v is on the path between two cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Definition 12 (Gate vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let G be an undirected graph with no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A vertex v is a 33 Algorithm 2 EQF DAG SE(S) Input: A (SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' IT)-SyDS S = (GS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='F),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' where GS is a DAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='Output: A fixed point C of S (if exists) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1: C ← an initial configuration of all 0’s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2: V ′ ← the set of vertices with 0 indegree in GS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3: for v ∈ V ′ do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='if τ1(v) = 1 then return Null ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='▷ The system has no fixed points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='else if τ1(v) = 0 then C(v) ← 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='else if τ1(v) = 2 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='C(v) ← 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='for w ∈ N(v) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='if τ1(w) ≠ 0 then τ1(w) ← τ1(w) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='V ′ ← V ′ ∖ {v} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='V ′′ ← the set of new vertices with 0 in-degree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='V ′ ← V ′ ∪ V ′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='15: end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='16: return C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='gate vertex if v is on at least one cycle and either (i) adjacent to a terminal vertex or (ii) adjacent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='to another vertex on another cycle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' or (iii) on at least two cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Intuitively, gate vertices for a cycle C act as “entrances” on C, such that a transversal from any other cycles to C must reach one of the gate vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Definition 13 (Tree vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let G be an undirected graph with no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A vertex v is a tree vertex if v is not on any cycles and v is not a terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given an undirected graph G with no even cycles, all cycles in G are edge-disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contrapositive, suppose there exists two different odd cycles, denoted by C and C′, such that they share a common path L = (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',vl) of length l ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that the set of vertices V (C)∪V (C′)∖V (L) form another cycle, denoted by C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let 2k +1 and 2k′ +1 be the length of C and C′, respectively, k, k′ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that the length of C′′ is 2k + 1 + 2k′ + 1 − 2l = 2(k + k′ + 1 − l) which is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proved by contraposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let G be an undirected graph with no even cycles, then there must exist a cycle C with at most one gate vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given any cycle C, let v be a gate vertex of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By the definition of gate vertices, remark that 34 Gate Tree Tree Terminal Terminal Gate Gate Gate Gate Gate Tree Figure 5: An example graph G where blue vertices are terminals, green vertices are gates, and red vertices are tree vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' there exists at least one path P from v to another cycle3, where P intersects with C only on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contradiction, suppose all cycles have at least two gate vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C1 be any cycle in G, and denoted by v1 a gate vertex of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We label all vertices in C1 as visited, and consider a depth-first search (DFS) process from v1 that traverses unvisited vertices until another gate vertex is found (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', a new cycle is reached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' During the DFS process, we label traversed vertices as visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v2 be the gate vertex that the DFS (from v1) encounters, and let C2 be a cycle that contains v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We label vertices in C2 also as visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since all cycles have at least 2 gate vertices, let v′ 2 be another gate vertex of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark that there must not exists a path that consists of only unvisited vertices (excepts the two endpoints) from v′ 2 to any visited vertices, or else, there exists a cycle that share common edges with C2 which contradicts G being even-cycle free (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that if we continue the DFS process from v′ 2 while traversing unvisited vertices, it must reach a new gate vertex v3 on a cycle C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v′ 3 be another gate vertex of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, there must not exists a path that consists of only unvisited vertices (except the two endpoints) from v′ 3 to any visited vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, given a newly visited gate vertex vk on a cycle Ck, k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v′ k be another gate vertex of Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By induction, there exists a path with only unvisited vertices (except v′ k itself who is visited) from v′ k to a cycle that is different from Ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' However, this contradicts G being finite (and thus the number of cycles is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The Lemma immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 3If v is on more than one cycle, then the path consists of only vertex v itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 35 Now consider a (SE, IT)-SyDS S = (GS,F) that models a SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SE, IT)-SyDS where the underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' A tree vertex v ∈ V (GS) has the same state over any fixed point C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first argue that if a vertex v has degree 1, then v has the same state under any fixed point C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, note that the threshold τ1(v) could be either 0, 1, 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If τ1(v) = 0 or τ1(v) = 3, then v is a constant vertex whose state is uniquely determined by τ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, by Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3, we know that C(v) = 1 if τ1(v) = 1 and C(v) = 0 if τ1(v) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that if there exists at least one tree vertex in GS, then at least one of the tree vertices has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let v be any tree vertex with degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the claim above, we know that the state of v under any fixed point C is predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, we can effectively remove v from the graph and update the threshold values of v’s neighbor accordingly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', decrease the neighbor’s threshold by one if C(v) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By recursion, it follows that the state of any tree vertex v is the same over any fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SE, IT)-SyDS where the underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C be a cycle in GS with at most one gate vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let u ∈ C be a non-gate vertex on C, then the state of u is the same under any fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='14, we know that such a cycle C with at most one gate vertex must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a non-gate vertex u on C, observe that any of u’s neighbors that are not on C must be tree vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='15, the states of all tree vertices are predetermined under a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we can consider the graph where all the tree vertices are removed, and the threshold values of their non-tree neighbors are updated accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, all non-gate vertices on C have degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If S has a fixed point, then at least one non-gate vertex on C does not have threshold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contradiction, suppose all the non-gate vertices on C have thresholds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since C is of odd length, it cannot be 2-colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, under any configuration C, there must exist an edge (u,v) on 36 C where the state of u and v are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, at least one of them is a non-gate vertex with threshold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', let u be such a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since u has degree 2, if C(u) = C(v) = 1, then the number of state-0 vertices in the closed neighborhood of u is at most 1, which is less than τ1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, C(u) = 1 ≠ C′(u) = 0 where C′ is the successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, if C(u) = C(v) = 0, we have C(u) = 0 ≠ C′(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, it follows that S does not have a fixed point if all the non-gate vertices on C have thresholds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1 implies that if S has a fixed point, then at least one non-gate vertex u on C has a threshold not equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now argue that the state of u remains the same under any fixed point C of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, if τ1(u) = 0 or τ1(u) = 4, then u is a constant vertex whose state is predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if τ1(u) = 1, then C(u) = 1, and if τ1(u) = 3 = d(v) + 1, then C(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that the state of u is the same under any fixed point and is predetermined by τ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we may remove u from the graph and update the thresholds of u’s neighbors accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark that after removing u, all non-gate vertices on C become tree vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='15, the state of all non-gate vertices on C are predetermined, and are the same over any fixed point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For a SE anti-coordination game, an NE can be found in O(m + n) time if the underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SE, IT)-SyDS that models a SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose S has a fixed point, denoted by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='15, we can determine the state of all the tree vertices under C based on their threshold values, and remove all the tree vertices from GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C1 be a cycle that consists of only one gate vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='14, we can also determine the states of all non-gate vertices on C1 and effectively eliminate the cycle C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let GS1 be the resulting graph which is still even-cycle free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If GS1 contains other cycles, then there must exists another cycle that consists of only one gate vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By recursively determining the state of non-gate vertices on single-gate cycles and the states of tree vertices, it follows that we can determine the states of all vertices under C in time O(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since a fixed point of S corresponds to an NE of the underlying SE-SyACG, we conclude that an NE can be found in O(m + n) time for a SE-SyACG (and thus for SE-SACG) when the underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' With the same argument, we also establish the same result for SN anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 37 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For a SN anti-coordination game, an NE can be found in O(m + n) time if the underlying graph has no even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The underlying graph is complete Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For a SE anti-coordination game, an NE can be found in O(n) time if the underlying graph is a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SE, IT)-SyDS that models a SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We partition the set of vertices based on their thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, let k be the number of distinct thresholds τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let P = {V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',Vk} be a partition of the vertex set V (GS), such that τ1(u) = τ1(v), ∀u,v ∈ Vi, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, τ1(u) < τ1(v), ∀u ∈ Vi, v ∈ Vj, 1 ≤ i < j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Remark that since GS is a complete graph, the closed neighborhoods of all vertices are the same, which is V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, given any fixed point C of S, if C(v) = 1 for some v ∈ Vj, then C(u) = 1,∀u ∈ Vi,i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Our algorithm consists of at most k − 1 iterations, where in each iteration j, 1 ≤ j ≤ k − 1, we construct a configuration C such that C(v) = 1,∀v ∈ Vi,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',j, and C(v) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' After each iteration of the algorithm, we check if the resulting C is a fixed point, and return C if so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if all the resulting k − 1 configurations are not fixed points, we conclude that S does not have a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The pseudocode is shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Algorithm 3 EQF Complete SE(S) Input: A (SE, IT)-SyDS/SDS S = (GS,F), where GS is a complete graph Output: A fixed point C of S 1: C ← an initial configuration of all 0’s 2: P ← {V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',Vk} be a partition of V (GS) such that τu < τv iff u ∈ Vi,v ∈ Vj,1 ≤ i < j ≤ k 3: τ1(Vi) ← the threshold value of vertices in Vi ∈ P,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=',k 4: a0 ← ∣V (GS)∣ ▷ The number of state-0 vertices in C 5: for j = 1 to k − 1 do 6: for v ∈ Vj do 7: C(v) ← 1 8: end for 9: a0 ← a0 − ∣Vj∣ 10: if τ1(Vj) ≤ a0 < τ1(Vj+1) then ▷ Determine if C is a fixed point return C 11: end if 12: end for 13: return Null ▷ The system has no fixed point We establish the correctness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' First, we claim that 38 Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The system S has a fixed point if and only if there exists a bipartition {V ′,V ′′} of V (GS), such that max{τ1(v) ∶ v ∈ V ′} ≤ ∣V ′′∣ < min{τ1(v) ∶ v ∈ V ′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, such a fixed point C has vertices in V ′ in state 1, and vertices in V ′′ in state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose S has a fixed point C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C′ be the successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let V ′ = {v ∶ C(v) = 1,v ∈ V (GS)} and V ′′ = {v ∶ C(v) = 0, v ∈ V (GS)} be a bipartition of the vertex set based on the state of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We argue that max{τ1(v) ∶ v ∈ V ′} < min{τ1(v) ∶ v ∈ V ′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For contrapositive, assume that there exists a vertex v ∈ V ′ such that τ1(v) ≥ τ1(w) for some w ∈ V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since C is a fixed point and C(v) = 1, we have ∣V ′′∣ ≥ τ1(v) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the number of state-0 vertices in C is at least τ1(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' However, it follows that ∣V ′′∣ ≥ τ1(v) ≥ τ1(w), thus, C(w) = 0 ≠ C′(w) = 1 and C is not a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we have max{τ1(v) ∶ v ∈ V ′} ≤ ∣V ′′∣ < min{τ1(v) ∶ v ∈ V ′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' As for the necessity of the claim, note that if such a bipartition {V ′,V ′′} exists, we can construct a fixed point C by assigning C(v) = 1, ∀v ∈ V ′ and C(v) = 0 ∀v ∈ V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that ∣V ′′∣ ≥ max{τ1(v) ∶ v ∈ V ′} ≥ τ1(v), ∀v ∈ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, C(v) = C′(v) = 1, ∀v ∈ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, we have ∣V ′′∣ ≤ min{τ1(v) ∶ v ∈ V ′′} ≤ τ1(v), ∀v ∈ V ′′ and C(v) = C′(v) = 0, ∀v ∈ V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now argue that the for loop of Algorithm 3 from line 5 to 12 essentially discovers if such a bipartition {V ′, V ′′} exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, at the jth iteration, 1 ≤ j ≤ k − 1, denoted by a0 the number of state-0 vertices in C (after the updates from line 6 to 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let τ1(Vj) be the threshold value of vertices in Vj ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that the algorithm effectively construct a configuration C by setting vertices in V ′ = ⋃j i=1 Vi to state 1, and vertices in V ′′ = ⋃k i=j+1 Vi to state 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The inequality τ1(Vj) ≤ a0 < τ1(Vj+1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the condition at line 10) is satisfied if and only if max{τ1(v) ∶ v ∈ V ′} ≤ ∣V ′′∣ < min{τ1(v) ∶ v ∈ V ′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that a0 = ∣V ′′∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose τ1(Vj) ≤ a0 < τ1(Vj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since τ1(Vj) ≥ τ1(v), ∀v ∈ V ′ = ⋃j i=1 Vi, it follows that a0 = ∣V ′′∣ ≥ max{τ1(v) ∶ v ∈ V ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, given that τ1(Vj+1) ≤ τ1(v), ∀v ∈ V ′′ = ⋃k j=i+1 Vj, we have ∣V ′′∣ ≤ min{τ1(v) ∶ v ∈ V ′′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For the other direction, by an analogous argument, it is easy to see that if max{τ1(v) ∶ v ∈ V ′} ≤ ∣V ′′∣ < min{τ1(v) ∶ v ∈ V ′′}, then τ1(Vi) ≤ a0 < τ1(Vi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The correctness of the algorithm immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, the partition P can be constructed in O(n) time via the bucket sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, the 39 for loop at line 7 is called at most O(n) time, and the operations from line 9 to line 12 take constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Therefore, the overall running time is O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Since a fixed point of S corresponds to an NE of the underlying game, we conclude that an NE can be found in O(n) time for a SE-SyACG (and therefore for SE-SACG) when the underlying graph is a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' With the same argument, we can show that the above result for complete graphs carries over to SN anti-coordination games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For a SN anti-coordination game, an NE can be found in O(n) time if the underlying graph is a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4 ILP for Finding Nash equilibria in SE mode In this section, we present an integer linear program formulation that finds a Nash equilibrium (if one exists) for SE anti-coordination games under networks of reasonable sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SE, IT)-SyDS that models a SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The proposed ILP constructs a configuration COPT of S that maximizes the number of vertices whose states remain unchanged in the successor of COPT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, COPT is a fixed point if and only if S has a fixed point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the states of all vertices remain unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if S does not have fixed points, our ILP ensures that COPT is (i) a 2-cycle, and (ii) the number of vertices whose states remain unchanged (under the 2-cycle) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The formulation of ILP is presented in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, each u ∈ V (GS) is associated with 4 variables: (i) xu, (ii) yu, (iii) au, and (iv) bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' In particular, (i) xu is the state of u under the constructed configuration COPT , (ii) yu is the state of u under the successor of COPT , denoted by C′ OPT , (iii) au = min{xu, yu}, and (iv) bu = max{xu,yu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 40 max ∑ u∈V (GS) au − bu (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' ∑ v∈N +(u) 1 − xv ≥ τ1(u) ⋅ yu ∀u ∈ V (GS) (14b) ∑ v∈N +(u) xv ≥ τ0(u) ⋅ (1 − yu) ∀u ∈ V (GS) (14c) ∑ v∈N +(u) 1 − yv ≥ τ1(u) ⋅ xu ∀u ∈ V (GS) (14d) ∑ v∈N +(u) yv ≥ τ0(u) ⋅ (1 − xu) ∀u ∈ V (GS) (14e) au ≤ xu ∀u ∈ V (GS) (14f) au ≤ yu ∀u ∈ V (GS) (14g) bu ≥ xu ∀u ∈ V (GS) (14h) bu ≥ yu ∀u ∈ V (GS) (14i) au,bu,xu,yu ∈ {0,1} ∀u ∈ V (GS) (14j) Let x∗ u and y∗ u, ∀u ∈ V (GS), be optimal state assignments for the variable xu and yu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let COPT be the corresponding configuration for which COPT (u) = x∗ u, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C′ OPT be the configuration where C′ OPT (u) = y∗ u, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now establish the correctness of the integer linear program (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The resulting configuration COPT is a fixed point of S if and only if S has at least one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, if S has no fixed points, then COPT ←→ C′ OPT is a 2-cycle of S where the number of vertices v such that COPT (v) = C′ OPT (v) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consider a feasible assignment of variables xu and yu, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C and C′ be the corresponding configurations of S, where C(u) = xu and C′(u) = yu, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first establish the following claim Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The configuration C is a successor of C′, and C′ is also a successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Consider the constraint 14b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that ∑v∈N +(u) 1 − xv is the number of state-0 vertices in u’s closed neighborhood under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If yu = 0, then clearly constraint 14b is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' On the other hand, if yu = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', C(u) = 1), constraint guarantees that the number of state-0 vertices in u’s closed 41 neighborhood is at least τ1(u), which aligns with the inverted-thershold dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now consider constraint 14c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that ∑v∈N +(u) xv is the number of state-1 vertices in u’s closed neighborhood under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If yu = 1, then constraint 14c is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Alternatively, if yu = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', C′(u) = 0), then constraint 14c enforces that the number of state-1 vertices in u’s closed neighborhood under C is at least τ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, constraint 14b and 14c together ensure that C′ is a successor of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By analogous arguments, it follows that constraint 14d and 14e ensure that C is a successor of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Next, we argue that maximizing the objective 14a equivalently maximizes the number of vertices u that satisfies C(u) = C′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The objective 14a is maximized if and only if the number of vertices u that satisfies C(u) = C′(u) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let au and bu be feasible assignments of variables au and bu for u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It is easy to see that constraints 14f and 14g ensures that au ≤ min{xu,yu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Similarly, constraints 14h and 14i ensures that bu ≥ max{xu, yu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Observe that when xu ≠ yu (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', C(u) ≠ C′(u)), we must have au = 0 and bu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, the resulting objective a∗ u − b∗ u = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Alternatively, the objective for vertex u is the maximum when au − bu = 0, that is au = bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose the objective 14a is maximized, it follows that the number of vertices u where xu = yu (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', C(u) = C′(u)) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Conversely, if the number of vertices u where xu = yu is maximized, then clearly the objective is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, let x∗ u and y∗ u, ∀u ∈ V (GS), be optimal state assignments for the variable xu and yu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let COPT and C′ OPT be the corresponding configuration for which COPT (u) = x∗ u, ∀u ∈ V (GS), and C′ OPT (u) = y∗ u, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1, C′ OPT is a successor of COPT , and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If COPT is a fixed point of S, then clearly S has a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Conversely, if S has a fixed point, that is, the state of all vertices remains unchanged in the successor of the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2, it follows that COPT (u) = C′ OPT (u), ∀u ∈ V (G) and COPT is a fixed point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lastly, suppose S does not have a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1, COPT ←→ C′ OPT is a 2-cycle of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2 implies that the number of vertices u such that COPT (u) = C′ OPT (u) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' This concludes the correctness of the ILP (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 42 Additional Material for Section 5 In this section, we present the detailed proofs of the convergence result for synchronous anti-coordination games, given in section 5 of the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Based on the connection between the games and dynamical systems, all proofs are given in the context of synchronous dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first present the convergence result for (SN, IT)-SyDS (modeling SN-SyACG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Then with simple modifications of the proof, the result for (SE, IT)-SyDS (modeling SE-SyACG) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let S = (GS,F) be a (SN, IT)-SyDS that models a SN-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Recall that for a vertex v, τ0(v) is the minimum number of state-1 neighbors of v for fv to equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', we assume there are no constant vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' That is, 1 ≤ τ1(u), τ0(u) ≤ d(u) for all u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential functions and bounds Let C be an arbitrary configuration of S whose successor is C ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now define the potentials of vertices, edges, and the system under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For each vertex u ∈ V (GS), let ˜τ0(u) = τ0(u) − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Vertex potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given a vertex u ∈ V (GS), the potential of u under C, is defined as follows: P(C, u) = C(u) ⋅ ˜τ0(u) + C ′(u) ⋅ ˜τ0(u) Edge potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Given an edge e = (u, v) ∈ E(GS), the potential of e under C is defined as follows: P(C, e) = C(u) ⋅ C ′(v) + C(v) ⋅ C ′(u) The configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The potential of the system S under C is defined as the subtraction of the sum of vertex potentials from the sum of edge potentials: P(C,S) = ∑ e∈E(GS) P(C,e) − ∑ u∈V (GS) P(C,u) A lower bound on the configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We first present a lower bound on the configuration potential under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 43 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The configuration potential satisfies P(C,S) ≥ −4m + n Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' First observe that the sum of edge potentials is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further, the sum of potentials of vertices is upper bounded by ∑ u∈V (GS) P(C, u) = ∑ u∈V (GS) C(u) ⋅ ˜τ0(u) + C ′(u) ⋅ ˜τ0(u) ≤ 2 ∑ u∈V (GS) ˜τ0(u) (15) ≤ 2 ∑ u∈V (GS) (d(u) − 1 2) = 4m − n The lower bound of the configuration potential immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' An upper bound on the configuration potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now present an upper bound on the configuration potentials under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' The configuration potential P(C,S) is upper bounded by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let Eu = {(u, v) ∶ (u, v) ∈ E(GS)} be the set of edges incident to a vertex u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We can restate the sum of edge potentials as follows ∑ e∈E(GS) P(C, e) = ∑ (u,v)∈E(GS) (C(u) ⋅ C ′(v) + C(v) ⋅ C ′(u)) (16) = ∑ u∈V (GS) ⎛ ⎝C ′(u) ⋅ ∑ (u,v)∈Eu C(v)⎞ ⎠ We can further expend the configuration potential into the form P(C,S) = ∑ u∈V (GS) ⎛ ⎝ ⎛ ⎝C ′(u) ∑ (u,v)∈Eu C(v)⎞ ⎠ − C(u) ⋅ ˜τ0(u) − C ′(u) ⋅ ˜τ0(u)⎞ ⎠ (17) Note that ∑(u,v)∈Eu C(v) is exactly the number of state-1 neighbors of u under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' If C ′(u) = 0, then ⎛ ⎝C ′(u) ∑ e=(u,v)∈Eu C(v)⎞ ⎠ − C(u) ⋅ ˜τ0(u) − C ′(u) ⋅ ˜τ0(u) = −C(u) ⋅ ˜τ0(u) ≤ 0 (18) 44 Conversely, if C ′(u) = 1, then by the inverted-threshold dynamics, the number of state-1 neighbor of u under C is less than τ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, we have ∑ (u,v)∈Eu C(v) ≤ τ0(u) − 1 < ˜τ0(u) (19) and ⎛ ⎝C ′(u) ∑ (u,v)∈Eu C(v)⎞ ⎠ − C ′(u) ⋅ ˜τ0(u) − C(u) ⋅ ˜τ0(u) (20) = ⎛ ⎝ ∑ (u,v)∈Eu C(v)⎞ ⎠ − ˜τ0(u) − C(u) ⋅ ˜τ0(u) ≤ 0 It follows that the overall configuration potential satisfies P(C,S) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we have established that the gap in the configuration potential value between two config- urations is at most 4m − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Decrease of the configuration potential before convergence In this section, we show that from a configuration C, the configuration potential decrease by at least 1 after every two time-steps, until a fixed point or a 2-cycle is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let C an arbitrary configuration of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We have P(C ′,S) = P(C,S) if and only if C = C ′′, that is C is a fixed point or is on a 2-cycle C ←→ C ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Furthermore, if C ≠ C ′′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', the dynamic has not converged), then P(C ′,S) − P(C,S) ≤ −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We defined the change of potential of an edge e = (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v) ∈ E(GS) from C to C ′ as ∆(e) = P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e) − P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e) (21) and the change of potential of a vertex u as ∆(u) = P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='u) − P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='u) (22) 45 Subsequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' the change of the configuration potential from C to C ′ is as follows P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='S) − P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='S) (23) = ∑ e∈E(GS) P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' e) − ∑ u∈V (GS) P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='u) − ∑ e∈E(GS) P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e) + ∑ u∈V (GS) P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='u) = ∑ e∈E(GS) ∆(e) − ∑ u∈V (GS) ∆(u) We now expand ∆(e) for e = (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' v) and ∆(u) as follows ∆(e) = (C ′(u) ⋅ C ′′(v) + C ′(v) ⋅ C ′′(u)) − (C(u) ⋅ C ′(v) + C ′(u) ⋅ C(v)) (24) = C ′(u) ⋅ C ′′(v) + C ′(v) ⋅ C ′′(u) − C(u) ⋅ C ′(v) − C ′(u) ⋅ C(v) = C ′(u) ⋅ (C ′′(v) − C(v)) + C ′(v) ⋅ (C ′′(u) − C(u)) ∆(u) = (C ′(u)˜τ0(u) + C ′′(u)˜τ0(u)) − (C(u)˜τ0(u) + C ′(u)˜τ0(u)) (25) = (C ′′(u) − C(u)) ⋅ ˜τ0(u) We argue that the change in configuration potential is 0 if and only if C = C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (⇐) Suppose C = C ′′, it follows that C(u) = C ′′(u), ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, ∆(e) = 0, ∀e ∈ E(GS) and ∆(u) = 0, ∀u ∈ V (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we conclude that the change of potential P(C ′,S) − P(C,S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' (⇒) For contrapositive, suppose C ≠ C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now show that the configuration potential decreases by at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It is easy to see that given a vertex u such that C(u) = C ′′(u), we have ∆(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, we consider the set of vertices whose states in C are different from the states in C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let V0−1 = {u ∶ C(u) = 0, C ′′(u) = 1, u ∈ V (GS)} be the set of vertices whose states are 0’s under C and are 1’s under C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Analogously, denoted by V1−0 = {u ∶ C(u) = 1, C ′′(u) = 0, u ∈ V (GS)} the set of vertices whose states are 1’s under C and are 0’s under C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We remark that since C ≠ C ′′, ∣V0−1∣ + ∣V1−0∣ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now consider the change of potential for a vertex u whose state in C is different from the state in C ′′, under the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 46 Case 1: The vertex u ∈ V0−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, the change of vertex potential for u is ∆(u) = (C ′′(u) − C(u)) ⋅ ˜τ0(u) = ˜τ0(u) (26) Case 2: The vertex u ∈ V1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It then follows that ∆(u) = (C ′′(u) − C(u)) ⋅ ˜τ0(u) = −˜τ0(u) (27) Overall, we have ∑ u∈V (GS) ∆(u) = ∑ u∈V0−1 ˜τ0(u) − ∑ u∈V1−0 ˜τ0(u) (28) As for the change of edge potentials, observe that ∆(e) = 0 for e = (u,v) if C(u) = C ′′(u) and C(v) = C ′′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' We now consider ∆(e), e = (u,v) where either u or v (or both) undergoes state change from C to C ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Case 1: The state of one vertex altered, and the state of the other remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', suppose the vertex u ∈ V0−1, and C(v) = C ′′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that ∆(e) = C ′(u) ⋅ (C ′′(v) − C(v)) + C ′(v) ⋅ (C ′′(u) − C(u)) (29) = C ′(v) On the other hand, if u ∈ V1−0, we then have ∆(e) = −C ′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Analogously, if the state of v changed and the state of u remained the same, then ∆(e) = C ′(u) when v ∈ V0−1, and ∆(e) = −C ′(u) when v ∈ V1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Case 2: The the states of u, v both changed in the same direction from C to C ′′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', both from 0 to 1 or from 1 to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Specifically, if u,v ∈ V0−1, we then have ∆(e) = C ′(u) ⋅ (C ′′(v) − C(v)) + C ′(v) ⋅ (C ′′(u) − C(u)) (30) = C ′(v) + C ′(u) Conversely, if u, v ∈ V1−0, then ∆(e) = −C ′(v) − C ′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Case 3: The states of u, v changed in different directions from C to C ′′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', one from 0 to 1 and 47 the other from 1 to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', suppose u ∈ V0−1 and v ∈ V1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, we have ∆(e) = C ′(v) − C ′(u) (31) Similarly, if u ∈ V1−0 and v ∈ V0−1, then ∆(e) = C ′(u) − C ′(v) Overall, observe that for any vertex u ∈ V0−1, the change of edge potential ∆(e) for each incident edge e = (u, v) has a positive term C ′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Conversely, if u ∈ V1−0, then ∆(e) has a negative term −C ′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Let Eu = {(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' v) ∶ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' v) ∈ E(GS)} be the set of edges incident to u ∈ V (GS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' we can rewire the sum of the change in edge potentials as ∑ e∈E(GS) ∆(e) = ⎛ ⎝ ∑ u∈V0−1 ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v)⎞ ⎠ − ⎛ ⎝ ∑ u∈V1−0 ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v)⎞ ⎠ (32) In combined with Equation 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' we characterize the change in the configuration potential from C to C ′ as follows P(C ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='S) − P(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='S) (33) = ∑ e∈E(GS) ∆(e) − ∑ u∈V (GS) ∆(u) = ⎛ ⎝ ∑ u∈V0−1 ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v) − ∑ u∈V1−0 ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v)⎞ ⎠ − ⎛ ⎝ ∑ u∈V0−1 ˜τ0(u) − ∑ u∈V1−0 ˜τ0(u)⎞ ⎠ = ⎛ ⎝ ∑ u∈V0−1 ⎛ ⎝ ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v)⎞ ⎠ − ˜τ0(u)⎞ ⎠ + ⎛ ⎝ ∑ u∈V1−0 ˜τ0(u) − ⎛ ⎝ ∑ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v)⎞ ⎠ ⎞ ⎠ Remark that the term ∑(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='v)∈Eu C ′(v) is the number of state-1 neighbors of a vertex u under C ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Suppose u ∈ V0−1, that is, C ′′(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' By the dynamics of inverted-threshold model, it follows that the number of state-1 neighbors of u under C ′ is less than τ0(u) = ˜τ0(u) + 1/2, thus, ∑ (u,v)∈Eu C ′(v) ≤ τ0(u) − 1 < ˜τ0(u) and ⎛ ⎝ ∑ (u,v)∈Eu C ′(v)⎞ ⎠ − ˜τ0(u) ≤ −1 2 Conversely, if u ∈ V1−0 which implies that C ′′(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that the number of state-1 neighbors 48 of u is at least τ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Thus, ∑ (u,v)∈Eu C ′(v) ≥ τ0(u) > ˜τ0(u) and ˜τ0(u) − ∑ (u,v)∈Eu C ′(v) ≤ −1 2 Overall, we have P(C ′,S) − P(C,S) (34) = ⎛ ⎝ ∑ u∈V0−1 ⎛ ⎝ ∑ (u,v)∈Eu C ′(v)⎞ ⎠ − ˜τ0(u)⎞ ⎠ + ⎛ ⎝ ∑ u∈V1−0 ˜τ0(u) − ⎛ ⎝ ∑ (u,v)∈Eu C ′(v)⎞ ⎠ ⎞ ⎠ ≤ − ∑ u∈V0−1 1 2 − ∑ u∈V1−0 1 2 = −∣V0−1∣ + ∣V1−0∣ 2 ≤ −1 2 This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Overall, we have shown that the potential gap between any two configurations is 4m−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Further- more, the overall system configuration decreases by at least 1/2 after each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' It follows that for a (SN, IT)-SyDS, starting from an arbitrary initial configuration, the system dynamic converges in at most 8m − 2n time steps, irrespective of the underlying network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Subsequently, the convergence time result for SN-SyACG follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For SN-SyACG, starting from any initial action profile, the best-response dynamic converges to a Nash equilibrium or a 2-cycle in O(m) time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' With simple modifications of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=', consider u as a neighbor of itself), we can also show that the same convergence time can be extended to SE-SyACG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For a SE-SyACG, starting from any initial action profile, the best-response dynamic converges to a Nash equilibrium or a 2-cycle in O(m) time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' For both SE-SyACG and SN-SyACG, starting from any initial action profile, the best- response dynamic converges to a Nash equilibrium or a 2-cycle in O(n) time steps if the graph is degree bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} +page_content=' 49' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfEQP0/content/2301.02889v1.pdf'} diff --git a/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf b/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6b3a64f1d3acfb8150c406c3c4d3519acc252e16 --- /dev/null +++ b/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2f615023a06e3f3e9ffc84b1811be49cbc895535f2f576f0ec7be2d13624d499 +size 408379 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b/N9E3T4oBgHgl3EQfZQq9/content/tmp_files/2301.04496v1.pdf.txt @@ -0,0 +1,508 @@ +CONTENTS +CONTENTS +Photoinduced pairing in Mott insulators +Satoshi Ejima1,2⋆ and Holger Fehske1,3 +1 Institut für Physik, Universität Greifswald, 17489 Greifswald, Germany +2 Institut für Softwaretechnologie, Abteilung High-Performance Computing, Deutsches Zentrum +für Luft- und Raumfahrt (DLR), 22529 Hamburg, Germany +3 Erlangen National High Performance Computing Center, Friedrich-Alexander-Universität +Erlangen-Nürnberg, 91058 Erlangen, Germany +* satoshi.ejima@dlr.de +January 12, 2023 +International Conference on Strongly Correlated Electron Systems +(SCES 2022) +Amsterdam, 24-29 July 2022 +doi:10.21468/SciPostPhysProc.? +Abstract +Utilizing time-evolution techniques in (infinite) matrix-product-state representation, we study +the non-equilibrium dynamics of driven Mott insulators and demonstrate photoinduced η +pairing directly in the thermodynamic limit. Analyzing the time evolution of the correspond- +ing pairing correlations, we determine the optimal laser pump parameters for which long- +range η-pairing becomes dominant after pulse irradiation. The time-dependent photoemis- +sion spectra for this optimal pump parameter set show clear signatures of the photoinduced +insulator-to-metal phase transition related to the formation of η pairs. +Contents +1 +Introduction +2 +2 +Model +2 +3 +Pairing correlations +3 +4 +Non-equilibrium dynamics +5 +5 +Conclusions +6 +References +7 +1 +arXiv:2301.04496v1 [cond-mat.str-el] 11 Jan 2023 + +SCES +Amsterdam +2022 +金2 +MODEL +1 +Introduction +η pairing, proposed first by C. N. Yang in 1989 [1], gives rise to a pairing-density-wave-like off- +diagonal long-range order in the Hubbard model. While it can be used to construct exact eigen- +states of this model, η pairing is absent in the Hubbard model’s ground state, and therefore has +attracted only specific attention, mostly from mathematical point of view. Recently, however, it +was pointed out that the η-pairing state will be enforced by pulse irradiation [2]. The respective +enhancement of pairing correlations emerged in time-dependent exact diagonalisations: Calcu- +lating all eigenstates as well as pairing correlations for a small cluster and taking the selection +rule of η pairs into account, Kaneko et al. showed that this photoinduced state is related to the +η-pairing state [2]. +Meanwhile, as a result of on-going developments in (time-depenent) density-matrix renormal- +isation group [(t-)DMRG] technique [3,4], optically driven systems in (quasi-)one-dimension can +be simulated directly in the thermodynamic limit. In doing so, static correlation functions such +as η-pair correlations can be computed by means of the infinite time-evolving block decimation +(iTEBD) technique [5], taking advantage of translational invariance in the infinite matrix-product- +state (iMPS) representation. Building window sites with so-called infinite boundary conditions +(IBC) in the uniform update scheme [6] enables us to simulate non-equilibrium dynamics of ex- +cited (quasi-)one-dimensional (1D) systems by a laser electric field [7]. +On this basis, in this study, we reexamine the time-evolution of photoinduced η-pairing, mainly +to confirm or put in question previous small cluster results. Thereby we emphasize the impor- +tance of using optimal pump pulse parameters. Furthermore, we reconsider the relation between +the η-pairing correlations and the optical spectrum in the small-amplitude regime after pulse ir- +radiation. Finally we prove the photoinduced insulator-to-metal phase transition by simulating +time-dependent photoemission spectra of driven Mott insulators. +2 +Model +Let us consider the 1D half-filled Hubbard model, +ˆH = −th +� +j,σ +� +ˆc† +j,σˆcj+1,σ + H.c. +� ++ U +� +j +� +ˆnj,↑ − 1/2 +�� +ˆnj,↓ − 1/2 +� +, +(1) +where th is the nearest-neighbor transfer amplitude and U gives the on-site part of the Coulomb +interaction. In Eq. (1), ˆc† +j,σ (ˆcj,σ) creates (annihilates) a spin-σ (=↑,↓) electron at Wannier lattice +site j, and ˆnj,σ = ˆc† +j,σˆcj,σ. In the repulsive case (U > 0) the model realizes a Mott insulating +ground state with a finite charge gap ∆. +Exact eigenstates of the Hubbard model can be constructed by means of the operators ˆη+ = +� +j(−1)j ˆ∆† +j, +ˆη− = ( ˆη+)†, and ˆηz = 1 +2 +� +j(ˆnj,↑ + ˆnj,↓ − 1), where ˆ∆† +j = ˆc† +j,↓ˆc† +j,↑ denotes the singlet pair-creation +operator [1]. These so-called η operators fulfill SU(2) commutation relations [ ˆη+, ˆη−] = 2 ˆηz and +[ ˆηz, ˆη±] = ± ˆη±. Apparently, the Hubbard Hamiltonian (1) commutes with ˆη2 = 1 +2( ˆη+ ˆη−+ ˆη− ˆη+)+( ˆηz)2, +i.e., 〈η2〉 is a conserved quantity. Long-ranged pairing correlations 〈 ˆη+ +j ˆη− +ℓ 〉 develop when the ex- +pectation value 〈 ˆη2〉 becomes finite, but such η-pairing states cannot be the ground state of the +Hubbard model [1]. Pulse irradiation can establish η-paired states in Mott insulators however [2]. +To address this issue, we apply a pump pulse with amplitude A0, frequency ωp and width σp, +2 + +3 +PAIRING CORRELATIONS +0 +5 +10 +15 +20 +0 +0.5 +1.0 +1.5 +t · th +˜P(q = π, t) +2nd(t) +U/th = 8 +Figure 1: Typical time-evolution process of ˜P(q = π, t) and 2nd(t) for the photoin- +duced η-pairing states in the strong-coupling regime of the driven Hubbard model with +U/th = 8 and pump parameters A0=0.4, ωp/th = 7.0, σp = 2t−1 +h +and t0 = 10t−1 +h . +The iTEBD data are obtained for bond dimension χ = 1200, ensuring a truncation error +smaller than 10−5. For the iTEBD calculations, we employ a second-order Suzuki-Trotter +decomposition with time step 0.1t−1 +h +(0.01t−1 +h ). +centered at time t0(> 0): +A(t) = A0e−(t−t0)2/(2σ2 +p) cos +� +ωp(t − t0) +� +. +(2) +The external time-dependent electric field A(t) changes the hopping amplitude by a Peierls phase [8]: +thˆc† +j,σˆcj+1,σ → theiA(t)ˆc† +j,σˆcj+1,σ, i.e., ˆH → ˆH(t). As a result, the system being initially in the ground +state, is driven out of equilibrium, |ψ(0)〉 → |ψ(t)〉. +3 +Pairing correlations +The η-pairing state can be detected evaluating the time evolution of the pair-correlation function +P(r, t) = 1 +L +� +j +〈ψ(t)| ˆ∆† +j+r ˆ∆j + H.c.)|ψ(t)〉 +(3) +and its Fourier transform ˜P(q, t) = +� +r eiqr P(r, t). As found in Refs. [2,9] for small clusters, ˜P(π, t) +is enhanced after pulse irradiation, indicating the formation of an η-pairing state. By means of +iTEBD this is confirmed directly in the thermodynamic limit which is demonstrated in Fig. 1 for +a pump with A0 = 0.4, ωp/th = 7.0 and σp = 2t−1 +h +centered at t0 = 10t−1 +h . +˜P(π, t) shows +a clear response to pulse irradiation and is strengthened as the system progresses in time until +saturation is reached. Obviously, the nonlocal contributions have a stronger impact on ˜P(π, t) +than the double occupancy nd(t) = (1/L) +� +j〈ψ(t)|ˆnj,↑ˆnj,↓|ψ(t)〉 [note that P(r = 0, t) = 2nd(t), +where nd(0) > 0 for the finite U values considered]. +The enhancement process of η-pairing can be described as follows [2]: The initial state before +pulse irradiation is the ground state of the Hubbard chain with |η = 0,ηz = 0〉, which is consis- +tent with the numerical finding: ˜P(0, t = 0) ≃ 0 (see Fig. 1). Turning on the pump pulse, the +3 + +3 +PAIRING CORRELATIONS +0 +0.4 +0.8 +1.2 +4 +6 +8 +10 +12 +A0 +ωp/th +0 +0.4 +0.8 +1.2 +2 +4 +6 +8 +10 +12 +0 +0.4 +0.8 +1.2 +ω/th, +ωp/th +Im χJJ(ω) +0 +2 +4 +6 +8 +10 +� +P(π, t)/A2 +0 +A0 = 0.04 +A0 = 0.06 +A0 = 0.08 +A0 = 0.10 +Figure 2: (a): Contour plots of ˜P(q = π, t) in the ωp-A0 plane at t = 15t−1 +h . Again +U/th = 8, and the pump is parametrized by σp = 2t−1 +h +at t0 = 10t−1 +h . (b): ˜P(π, t) at +t = 15t−1 +h +in the small-A0 area enclosed by the dashed square in panel (a). Dividing by +A2 +0, data can be rescaled to Imχ(ω) (black line), where Imχ(ω) is the imaginary part of +the optical spectrum χJJ(ω). +Hamiltonian does not commute with the η-operators anymore, +[ ˆH(t),η+] = [ ˆH,η+]cos[A(t)] + +� +k +F(k, t)ˆc† +π−k,↓ˆc† +k,↑ , +(4) +where F(k, t) = 4th sin[A(t)]sin k. This alters the initial state to a state with a finite expectation +value 〈 ˆη2〉. Even though the commutation relation is recovered for t ≫ t0, i.e., [ ˆH(t), ˆη+] → [ ˆH, ˆη+] +[since A(t) → 0], |ψ(t)〉 now includes components of |η > 0,ηz = 0〉 leading to the enhancement +of ˜P(π, t), see Fig. 1 for t > t0. +Note that the enhancement of η pairing after pulse irradiation depends, however, strongly +on the pump pulse parameters. The optimal parameter set for inducing η-pairing states can be +determined examining the A0 and ωp dependences of ˜P(π, t) by iTEBD. Figure 2(a) shows the +contour plot of ˜P(π, t) after pulse irradiation (t = 15t−1 +h ). We find a single maximum around +A0 ≈ 0.4 and ωp/th ≈ 7.0 (marked by the “×" symbol), instead of the stripe structure observed in +the finite-system (L = 14) exact diagonalisation (ED) simulations [2]. +Another notable results of previous ED calculations [2] was that the peak structure of ˜P(π, t) as +a function of ωp for small A0 is essentially the same as those of the ground-state optical spectrum, +χJJ(ω > 0) = − 1 +L 〈ψ0|ˆJ +1 +E0 − ˆH + ħhω + iηL +ˆJ|ψ0〉, +(5) +where |ψ0〉 is the ground state having energy E0 and Lorentzian width ηL. In (5), the Hubbard- +model charge-current operator is ˆJ = ith +� +j,σ(ˆc† +j,σˆcj+1,σ − ˆc† +j+1,σˆcj,σ). +Figure 2(b) compares the iTEBD data, obtained for ˜P(π, t) at various small A0 and t = 15t−1 +h , +with the t-DMRG results for χJJ(ω) (using ηL/th = 0.2), in dependence on ωp respectively ω. +Most notably, ˜P(π, t) divided by A2 +0 scales to the imaginary part of the optical spectrum Imχ(ω). +This can be understood as follows: The hopping term including the Peierls phase can be divided +4 + +4 +NON-EQUILIBRIUM DYNAMICS +−π +π −π +π −π +π +0 +0.2 +A(ω; t) +t = 5t−1 +h +t = 10t−1 +h +t = 15t−1 +h +8 +−8 +−4 +0 +4 +0 +ω/th +k +(a) t = 5t−1 +h +0 +k +(b) t = 10t−1 +h +0 +k +(c) t = 15t−1 +h +0.1 +(d) +Figure 3: Snapshots of the photoemission spectra A(k,ω; t) indicating photoinduced η- +pairing during the pump at times t = 5t−1 +h +(a), 10t−1 +h +(b) and 15t−1 +h +(c). The pump is +parametrized by A0 = 0.4, ωp/th = 7.0 [see ‘×’-symbol in Fig. 2(a)], and σp = 2t−1 +h +at +t0 = 10t−1 +h . The transient integrated density of states A(ω; t) obtained from the data of +panels (a)-(c) is depicted in panel (d). All data are obtained by the (i)TEBD technique +with IBC for the 1D half-filled Hubbard model with U/th = 8. Note that the time cutoff in +the simulation of time-dependent correlation functions is T = 5t−1 +h , i.e., the integration +in Eq. 7 extends only over the interval −T ≤ τ1,τ2 ≤ T. As a compromise between time +and frequency resolutions we have chosen a probe pulse width σpr = 2t−1 +h . +into kinetic and current operators as +−th +� +j,σ +� +eiA(t)ˆc† +j,σˆcj+1,σ + H.c. +� += ˆK cos[A(t)] + ˆJ sin[A(t)], +(6) +where ˆK = −th +� +j,σ(ˆc† +j,σˆcj+1,σ +H.c.). For small A0 and large t, the second term in Eq. (6) can be +approximated by ˆJA0, yielding a significant contribution of A2 +0 to the pair correlations. Needless +to say that the finite-size effects are eliminated by simulating the pair correlations directly in +thermodynamic limit by iTEBD, leading to the single-peak structure in Fig. 2(b), in strong contrast +to the multiple-peak structure observed in the ED calculations [2]. +4 +Non-equilibrium dynamics +We now analyze the non-equilibrium photoemission spectra A(k,ω; t) = +� +σ=↑,↓ Aσ(k,ω; t) for +the optimal pump parameter set marked by the “×"-symbol in Fig. 2(a). To explore the system +dynamics in a non-equilibrium situation, time-dependent spectral functions of the form [10] +Aσ(k,ω; t) = +� +r +e−ikr +� ∞ +−∞ +� ∞ +−∞ +dτ1dτ2 f (τ1,τ2;ω) · Cσ(r,τ1,τ2; t) +(7) +are of interest. Here, the non-equilibrium two-point correlator +Cσ(r,τ1,τ2; t) = 〈φ(t)|ˆc† +j+r,σ(τ1; t)ˆcj,σ(τ2; t)|φ(t)〉 +(8) +5 + +5 +CONCLUSIONS +is defined relative to t, and +f (τ1,τ2;ω) = eiω(τ1−τ2)g(τ1)g(τ2), g(τ) = exp[−τ2/2σ2 +pr]/ +� +2πσpr +(9) +specify the shape of the probe pulse, e.g., in a time-dependent photoemission spectroscopy exper- +iment. How numerically simulate two-time-dependent quantities such as Cσ(r,τ1,τ2; t) has been +explained in detail in Ref. [7] [see paragraphs below Eq. (1)]. +Figure 3 displays our (i)TEBD results for the 1D half-filled Hubbard model in the strong- +coupling regime (U/th = 8). Before pump irradiation the state is a Mott insulator with a noticable +single-particle gap, see Fig. 3(a) for t = 5t−1 +h . In the midst of the pump (t = 10t−1 +h ), an extra +dispersion above Fermi energy (ω > EF) appears and persists afterwards [Fig. 3(c)]. +Evaluating the integrated density of states +A(ω; t) = 1 +L +� +k +A(k,ω; t), +(10) +we see more clearly how the spectral weight is shifted from ω < EF to ω > EF due to the photoin- +duced η-pairing. Figure 3(d) gives A(ω; t) for the photoinduced η-pairing state. Obviously, the +spectral weight for ω > EF increases distinctly over time, indicating a photoinduced phase tran- +sition from a Mott insulator to a metallic η-pairing state. This photoinduced insulator-to-metal +transition should be observed in time- and angle-resolved photoemission spectroscopy, when the +pure Hubbard model is realized experimentally, e.g., in optical lattices. We note that the pho- +toinduced phase transition cannot be observed by simulating the time-dependent photoemission +spectra with not-optimized pump-pulse parameters, see Ref. [7]. +5 +Conclusions +To summarize, combining tensor-network algorithms with infinite time-evolving block decimation +techniques, we revisited the problem of photoinducing η-pairing states in the one-dimensional +Hubbard model at half band filling. This allowed us to prove the enhancement of the pairing +correlations directly in the thermodynamic limit. We also determined the optimal pump-pulse +parameter set that maximizes the η-pairing tendency. An η-pairing related Mott insulator to metal +transition could be extracted from the time-dependent photoemission spectrum. +We wish to stress that the numerical approach presented here can be applied to simulate the +non-equilibrium dynamics of any (quasi-)one-dimensional translational-invariant system in entire +ranges of interacting and driving parameters. For example, the photoinduced metallization of +excitonic insulators was demonstrated quite recently in accordance with time- and angle-resolved +photoemission spectroscopy experiments on Ta2NiSe5 [14,15]. +Acknowledgements +The iTEBD simulations were performed using the ITensor library [16]. +Funding information +S.E. was supported by Deutsche Forschungsgemeinschaft through project +EJ 7/2-1. +6 + +REFERENCES +REFERENCES +References +[1] C. N. Yang, η pairing and off-diagonal long-range order in a Hubbard model, Phys. Rev. Lett. +63, 2144 (1989), doi:10.1103/PhysRevLett.63.2144. +[2] T. Kaneko, T. Shirakawa, S. Sorella and S. Yunoki, Photoinduced η pairing in the Hubbard +model, Phys. Rev. Lett. 122, 077002 (2019), doi:10.1103/PhysRevLett.122.077002. +[3] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. +69, 2863 (1992), doi:10.1103/PhysRevLett.69.2863. +[4] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, +Ann. Phys. 326(1), 96 (2011), doi:10.1016/j.aop.2010.09.012. +[5] G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, +Phys. Rev. Lett. 98, 070201 (2007), doi:10.1103/PhysRevLett.98.070201. +[6] V. Zauner, M. Ganahl, H. G. Evertz and T. Nishino, +Time evolution within a comoving +window: scaling of signal fronts and magnetization plateaus after a local quench in quan- +tum spin chains, J. Phys.: Condens. Matter 27(42), 425602 (2015), doi:10.1088/0953- +8984/27/42/425602. +[7] S. Ejima, F. Lange and H. Fehske, Nonequilibrium dynamics in pumped mott insulators, Phys. +Rev. Research 4, L012012 (2022), doi:10.1103/PhysRevResearch.4.L012012. +[8] R. Peierls, Zur Theorie des Diamagnetismus von Leitungselektronen, Z. Phys. 80(11), 763 +(1933), doi:10.1007/BF01342591. +[9] S. +Ejima, +T. +Kaneko, +F. +Lange, +S. +Yunoki +and +H. +Fehske, +Photoinduced +η- +pairing in one-dimensional Mott insulators, +JPS Conf. Proc. 30, +011184 (2020), +doi:10.7566/JPSCP.30.011184. +[10] J. K. Freericks, H. R. Krishnamurthy and T. Pruschke, Theoretical description of time-resolved +photoemission spectroscopy: Application to pump-probe experiments, +Phys. Rev. Lett. 102, +136401 (2009), doi:10.1103/PhysRevLett.102.136401. +[11] G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. +Lett. 91, 147902 (2003), doi:10.1103/PhysRevLett.91.147902. +[12] F. Lange, S. Ejima and H. Fehske, +Finite-temperature dynamic structure factor of +the spin-1 XXZ chain with single-ion anisotropy, +Phys. Rev. B 97, 060403 (2018), +doi:10.1103/PhysRevB.97.060403. +[13] S. Ejima, F. Lange and H. Fehske, +Finite-temperature photoemission in the extended +Falicov-Kimball model: +a case study for Ta2NiSe5, +SciPost Phys. 10(3), 077 (2021), +doi:10.21468/scipostphys.10.3.077. +[14] S. Ejima, F. Lange and H. Fehske, Photoinduced metallization of excitonic insulators, Phys. +Rev. B 105, 245126 (2022), doi:10.1103/PhysRevB.105.245126. +7 + +REFERENCES +REFERENCES +[15] K. Okazaki, Y. Ogawa, T. Suzuki, T. Yamamoto, T. Someya, S. Michimae, M. Watanabe, Y. Lu, +M. Nohara, H. Takagi, N. Katayama, H. Sawa et al., Photo-induced semimetallic states realised +in electron–hole coupled insulators, Nat. Commun. 9(1), 4322 (2018), doi:10.1038/s41467- +018-06801-1. +[16] M. Fishman, S. R. White and E. M. Stoudenmire, The ITensor software library for tensor +network calculations, 2007.14822. +8 + diff --git a/N9E3T4oBgHgl3EQfZQq9/content/tmp_files/load_file.txt b/N9E3T4oBgHgl3EQfZQq9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f59360dd6ab5d9f783927d5ae1f139306e5259c --- /dev/null +++ b/N9E3T4oBgHgl3EQfZQq9/content/tmp_files/load_file.txt @@ -0,0 +1,324 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf,len=323 +page_content='CONTENTS CONTENTS Photoinduced pairing in Mott insulators Satoshi Ejima1,2⋆ and Holger Fehske1,3 1 Institut für Physik, Universität Greifswald, 17489 Greifswald, Germany 2 Institut für Softwaretechnologie, Abteilung High-Performance Computing, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 22529 Hamburg, Germany 3 Erlangen National High Performance Computing Center, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany satoshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='ejima@dlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='de January 12, 2023 International Conference on Strongly Correlated Electron Systems (SCES 2022) Amsterdam, 24-29 July 2022 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='21468/SciPostPhysProc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Abstract Utilizing time-evolution techniques in (infinite) matrix-product-state representation, we study the non-equilibrium dynamics of driven Mott insulators and demonstrate photoinduced η pairing directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Analyzing the time evolution of the correspond- ing pairing correlations, we determine the optimal laser pump parameters for which long- range η-pairing becomes dominant after pulse irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The time-dependent photoemis- sion spectra for this optimal pump parameter set show clear signatures of the photoinduced insulator-to-metal phase transition related to the formation of η pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Model 2 3 Pairing correlations 3 4 Non-equilibrium dynamics 5 5 Conclusions 6 References 7 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='04496v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='str-el] 11 Jan 2023 SCES Amsterdam 2022 金2 MODEL 1 Introduction η pairing, proposed first by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Yang in 1989 [1], gives rise to a pairing-density-wave-like off- diagonal long-range order in the Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' While it can be used to construct exact eigen- states of this model, η pairing is absent in the Hubbard model’s ground state, and therefore has attracted only specific attention, mostly from mathematical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Recently, however, it was pointed out that the η-pairing state will be enforced by pulse irradiation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The respective enhancement of pairing correlations emerged in time-dependent exact diagonalisations: Calcu- lating all eigenstates as well as pairing correlations for a small cluster and taking the selection rule of η pairs into account, Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' showed that this photoinduced state is related to the η-pairing state [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Meanwhile, as a result of on-going developments in (time-depenent) density-matrix renormal- isation group [(t-)DMRG] technique [3,4], optically driven systems in (quasi-)one-dimension can be simulated directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' In doing so, static correlation functions such as η-pair correlations can be computed by means of the infinite time-evolving block decimation (iTEBD) technique [5], taking advantage of translational invariance in the infinite matrix-product- state (iMPS) representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Building window sites with so-called infinite boundary conditions (IBC) in the uniform update scheme [6] enables us to simulate non-equilibrium dynamics of ex- cited (quasi-)one-dimensional (1D) systems by a laser electric field [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' On this basis, in this study, we reexamine the time-evolution of photoinduced η-pairing, mainly to confirm or put in question previous small cluster results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Thereby we emphasize the impor- tance of using optimal pump pulse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Furthermore, we reconsider the relation between the η-pairing correlations and the optical spectrum in the small-amplitude regime after pulse ir- radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Finally we prove the photoinduced insulator-to-metal phase transition by simulating time-dependent photoemission spectra of driven Mott insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 2 Model Let us consider the 1D half-filled Hubbard model, ˆH = −th � j,σ � ˆc† j,σˆcj+1,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' � + U � j � ˆnj,↑ − 1/2 �� ˆnj,↓ − 1/2 � , (1) where th is the nearest-neighbor transfer amplitude and U gives the on-site part of the Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' (1), ˆc† j,σ (ˆcj,σ) creates (annihilates) a spin-σ (=↑,↓) electron at Wannier lattice site j, and ˆnj,σ = ˆc† j,σˆcj,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' In the repulsive case (U > 0) the model realizes a Mott insulating ground state with a finite charge gap ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Exact eigenstates of the Hubbard model can be constructed by means of the operators ˆη+ = � j(−1)j ˆ∆† j, ˆη− = ( ˆη+)†, and ˆηz = 1 2 � j(ˆnj,↑ + ˆnj,↓ − 1), where ˆ∆† j = ˆc† j,↓ˆc† j,↑ denotes the singlet pair-creation operator [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' These so-called η operators fulfill SU(2) commutation relations [ ˆη+, ˆη−] = 2 ˆηz and [ ˆηz, ˆη±] = ± ˆη±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Apparently, the Hubbard Hamiltonian (1) commutes with ˆη2 = 1 2( ˆη+ ˆη−+ ˆη− ˆη+)+( ˆηz)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', 〈η2〉 is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Long-ranged pairing correlations 〈 ˆη+ j ˆη− ℓ 〉 develop when the ex- pectation value 〈 ˆη2〉 becomes finite, but such η-pairing states cannot be the ground state of the Hubbard model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Pulse irradiation can establish η-paired states in Mott insulators however [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' To address this issue, we apply a pump pulse with amplitude A0, frequency ωp and width σp, 2 3 PAIRING CORRELATIONS 0 5 10 15 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='5 t · th ˜P(q = π, t) 2nd(t) U/th = 8 Figure 1: Typical time-evolution process of ˜P(q = π, t) and 2nd(t) for the photoin- duced η-pairing states in the strong-coupling regime of the driven Hubbard model with U/th = 8 and pump parameters A0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='0, σp = 2t−1 h and t0 = 10t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The iTEBD data are obtained for bond dimension χ = 1200, ensuring a truncation error smaller than 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' For the iTEBD calculations, we employ a second-order Suzuki-Trotter decomposition with time step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='1t−1 h (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='01t−1 h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' centered at time t0(> 0): A(t) = A0e−(t−t0)2/(2σ2 p) cos � ωp(t − t0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' (2) The external time-dependent electric field A(t) changes the hopping amplitude by a Peierls phase [8]: thˆc† j,σˆcj+1,σ → theiA(t)ˆc† j,σˆcj+1,σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', ˆH → ˆH(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' As a result, the system being initially in the ground state, is driven out of equilibrium, |ψ(0)〉 → |ψ(t)〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 3 Pairing correlations The η-pairing state can be detected evaluating the time evolution of the pair-correlation function P(r, t) = 1 L � j 〈ψ(t)| ˆ∆† j+r ˆ∆j + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=')|ψ(t)〉 (3) and its Fourier transform ˜P(q, t) = � r eiqr P(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' As found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' [2,9] for small clusters, ˜P(π, t) is enhanced after pulse irradiation, indicating the formation of an η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' By means of iTEBD this is confirmed directly in the thermodynamic limit which is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 1 for a pump with A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='0 and σp = 2t−1 h centered at t0 = 10t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' ˜P(π, t) shows a clear response to pulse irradiation and is strengthened as the system progresses in time until saturation is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Obviously, the nonlocal contributions have a stronger impact on ˜P(π, t) than the double occupancy nd(t) = (1/L) � j〈ψ(t)|ˆnj,↑ˆnj,↓|ψ(t)〉 [note that P(r = 0, t) = 2nd(t), where nd(0) > 0 for the finite U values considered].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The enhancement process of η-pairing can be described as follows [2]: The initial state before pulse irradiation is the ground state of the Hubbard chain with |η = 0,ηz = 0〉, which is consis- tent with the numerical finding: ˜P(0, t = 0) ≃ 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Turning on the pump pulse, the 3 3 PAIRING CORRELATIONS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='2 4 6 8 10 12 A0 ωp/th 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='2 2 4 6 8 10 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='2 ω/th, ωp/th Im χJJ(ω) 0 2 4 6 8 10 � P(π, t)/A2 0 A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='04 A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='06 A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='08 A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='10 Figure 2: (a): Contour plots of ˜P(q = π, t) in the ωp-A0 plane at t = 15t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Again U/th = 8, and the pump is parametrized by σp = 2t−1 h at t0 = 10t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' (b): ˜P(π, t) at t = 15t−1 h in the small-A0 area enclosed by the dashed square in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Dividing by A2 0, data can be rescaled to Imχ(ω) (black line), where Imχ(ω) is the imaginary part of the optical spectrum χJJ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Hamiltonian does not commute with the η-operators anymore, [ ˆH(t),η+] = [ ˆH,η+]cos[A(t)] + � k F(k, t)ˆc† π−k,↓ˆc† k,↑ , (4) where F(k, t) = 4th sin[A(t)]sin k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' This alters the initial state to a state with a finite expectation value 〈 ˆη2〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Even though the commutation relation is recovered for t ≫ t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', [ ˆH(t), ˆη+] → [ ˆH, ˆη+] [since A(t) → 0], |ψ(t)〉 now includes components of |η > 0,ηz = 0〉 leading to the enhancement of ˜P(π, t), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 1 for t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Note that the enhancement of η pairing after pulse irradiation depends, however, strongly on the pump pulse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The optimal parameter set for inducing η-pairing states can be determined examining the A0 and ωp dependences of ˜P(π, t) by iTEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Figure 2(a) shows the contour plot of ˜P(π, t) after pulse irradiation (t = 15t−1 h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' We find a single maximum around A0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4 and ωp/th ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='0 (marked by the “×" symbol), instead of the stripe structure observed in the finite-system (L = 14) exact diagonalisation (ED) simulations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Another notable results of previous ED calculations [2] was that the peak structure of ˜P(π, t) as a function of ωp for small A0 is essentially the same as those of the ground-state optical spectrum, χJJ(ω > 0) = − 1 L 〈ψ0|ˆJ 1 E0 − ˆH + ħhω + iηL ˆJ|ψ0〉, (5) where |ψ0〉 is the ground state having energy E0 and Lorentzian width ηL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' In (5), the Hubbard- model charge-current operator is ˆJ = ith � j,σ(ˆc† j,σˆcj+1,σ − ˆc† j+1,σˆcj,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Figure 2(b) compares the iTEBD data, obtained for ˜P(π, t) at various small A0 and t = 15t−1 h , with the t-DMRG results for χJJ(ω) (using ηL/th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='2), in dependence on ωp respectively ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Most notably, ˜P(π, t) divided by A2 0 scales to the imaginary part of the optical spectrum Imχ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' This can be understood as follows: The hopping term including the Peierls phase can be divided 4 4 NON-EQUILIBRIUM DYNAMICS −π π −π π −π π 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='2 A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) t = 5t−1 h t = 10t−1 h t = 15t−1 h 8 −8 −4 0 4 0 ω/th k (a) t = 5t−1 h 0 k (b) t = 10t−1 h 0 k (c) t = 15t−1 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='1 (d) Figure 3: Snapshots of the photoemission spectra A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) indicating photoinduced η- pairing during the pump at times t = 5t−1 h (a), 10t−1 h (b) and 15t−1 h (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The pump is parametrized by A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='0 [see ‘×’-symbol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 2(a)], and σp = 2t−1 h at t0 = 10t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' The transient integrated density of states A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) obtained from the data of panels (a)-(c) is depicted in panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' All data are obtained by the (i)TEBD technique with IBC for the 1D half-filled Hubbard model with U/th = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Note that the time cutoff in the simulation of time-dependent correlation functions is T = 5t−1 h , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 7 extends only over the interval −T ≤ τ1,τ2 ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' As a compromise between time and frequency resolutions we have chosen a probe pulse width σpr = 2t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' into kinetic and current operators as −th � j,σ � eiA(t)ˆc† j,σˆcj+1,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' � = ˆK cos[A(t)] + ˆJ sin[A(t)], (6) where ˆK = −th � j,σ(ˆc† j,σˆcj+1,σ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' For small A0 and large t, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' (6) can be approximated by ˆJA0, yielding a significant contribution of A2 0 to the pair correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Needless to say that the finite-size effects are eliminated by simulating the pair correlations directly in thermodynamic limit by iTEBD, leading to the single-peak structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 2(b), in strong contrast to the multiple-peak structure observed in the ED calculations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 4 Non-equilibrium dynamics We now analyze the non-equilibrium photoemission spectra A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) = � σ=↑,↓ Aσ(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) for the optimal pump parameter set marked by the “×"-symbol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' To explore the system dynamics in a non-equilibrium situation, time-dependent spectral functions of the form [10] Aσ(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) = � r e−ikr � ∞ −∞ � ∞ −∞ dτ1dτ2 f (τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='ω) · Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) (7) are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Here, the non-equilibrium two-point correlator Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) = 〈φ(t)|ˆc† j+r,σ(τ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t)ˆcj,σ(τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t)|φ(t)〉 (8) 5 5 CONCLUSIONS is defined relative to t, and f (τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='ω) = eiω(τ1−τ2)g(τ1)g(τ2), g(τ) = exp[−τ2/2σ2 pr]/ � 2πσpr (9) specify the shape of the probe pulse, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', in a time-dependent photoemission spectroscopy exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' How numerically simulate two-time-dependent quantities such as Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) has been explained in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' [7] [see paragraphs below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Figure 3 displays our (i)TEBD results for the 1D half-filled Hubbard model in the strong- coupling regime (U/th = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Before pump irradiation the state is a Mott insulator with a noticable single-particle gap, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 3(a) for t = 5t−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' In the midst of the pump (t = 10t−1 h ), an extra dispersion above Fermi energy (ω > EF) appears and persists afterwards [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Evaluating the integrated density of states A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) = 1 L � k A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t), (10) we see more clearly how the spectral weight is shifted from ω < EF to ω > EF due to the photoin- duced η-pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Figure 3(d) gives A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' t) for the photoinduced η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Obviously, the spectral weight for ω > EF increases distinctly over time, indicating a photoinduced phase tran- sition from a Mott insulator to a metallic η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' This photoinduced insulator-to-metal transition should be observed in time- and angle-resolved photoemission spectroscopy, when the pure Hubbard model is realized experimentally, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=', in optical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' We note that the pho- toinduced phase transition cannot be observed by simulating the time-dependent photoemission spectra with not-optimized pump-pulse parameters, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' 5 Conclusions To summarize, combining tensor-network algorithms with infinite time-evolving block decimation techniques, we revisited the problem of photoinducing η-pairing states in the one-dimensional Hubbard model at half band filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' This allowed us to prove the enhancement of the pairing correlations directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' We also determined the optimal pump-pulse parameter set that maximizes the η-pairing tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' An η-pairing related Mott insulator to metal transition could be extracted from the time-dependent photoemission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' We wish to stress that the numerical approach presented here can be applied to simulate the non-equilibrium dynamics of any (quasi-)one-dimensional translational-invariant system in entire ranges of interacting and driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' For example, the photoinduced metallization of excitonic insulators was demonstrated quite recently in accordance with time- and angle-resolved photoemission spectroscopy experiments on Ta2NiSe5 [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Acknowledgements The iTEBD simulations were performed using the ITensor library [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content=' Funding information S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'} +page_content='E.' metadata={'source': 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Urbana-Champaign*, USA +JINTAO JIANG, University of Illinois at Urbana-Champaign*, USA +XIAOLAN KE, University of Illinois at Urbana-Champaign*, USA +YITAO MENG, University of Illinois at Urbana-Champaign*, USA +CONG XIE, University of Illinois at Urbana-Champaign*, USA +INDRANIL GUPTA, University of Illinois at Urbana-Champaign, USA +Machine Learning graphs (or models) can be challenging or impossible to train when either devices have limited memory, or models are +large. To split the model across devices, learning-based approaches are still popular. While these result in model placements that train +fast on data (i.e., low step times), learning-based model-parallelism is time-consuming, taking many hours or days to create a placement +plan of operators on devices. We present the Baechi system, the first to adopt an algorithmic approach to the placement problem for +running machine learning training graphs on small clusters of memory-constrained devices. We integrate our implementation of +Baechi into two popular open-source learning frameworks: TensorFlow and PyTorch. Our experimental results using GPUs show +that: (i) Baechi generates placement plans 654×–206K × faster than state-of-the-art learning-based approaches, and (ii) Baechi-placed +model’s step (training) time is comparable to expert placements in PyTorch, and only up to 6.2% worse than expert placements in +TensorFlow. We prove mathematically that our two algorithms are within a constant factor of the optimal. Our work shows that +compared to learning-based approaches, algorithmic approaches can face different challenges for adaptation to Machine learning +systems, but also they offer proven bounds, and significant performance benefits. +CCS Concepts: • Computer systems organization → Cloud computing. +Additional Key Words and Phrases: Machine Learning Systems, Placement Algorithms, Constrained Memory, TensorFlow, PyTorch, +Distributed Systems +This submission is an extended version of "Baechi: Fast Device Placement of Machine Learning Graphs - Beomyeol Jeon, Linda Cai, +Pallavi Srivastava, Jintao Jiang, Xiaolan Ke, Yitao Meng, Cong Xie, and Indranil Gupta. In Proceedings of the 11th ACM Sympo- +sium on Cloud Computing (Virtual Event, USA) (SoCC ’20). Association for Computing Machinery, New York, NY, USA, 416–430. +https://doi.org/10.1145/3419111.3421302". Document detailing the additional contributions has been attached as a supplementary material +*Work done while the authors were at University of Illinois at Urbana-Champaign, USA +Authors’ addresses: Beomyeol Jeon, University of Illinois at Urbana-Champaign, Urbana, USA, bj2@illinois.edu; Linda Cai, Princeton University, +USA, tcai@princeton.edu; Chirag Shetty, University of Illinois at Urbana-Champaign, USA, cshetty2@illinois.edu; Pallavi Srivastava, University of Illi- +nois at Urbana-Champaign*, USA; Jintao Jiang, University of Illinois at Urbana-Champaign*, USA; Xiaolan Ke, University of Illinois at Urbana-Champaign*, +USA; Yitao Meng, University of Illinois at Urbana-Champaign*, USA; Cong Xie, University of Illinois at Urbana-Champaign*, USA; Indranil Gupta, +University of Illinois at Urbana-Champaign, USA, indy@illinois.edu. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.08695v1 [cs.DC] 20 Jan 2023 + +2 +Jeon et al. +ACM Reference Format: +Beomyeol Jeon, Linda Cai, Chirag Shetty, Pallavi Srivastava, Jintao Jiang, Xiaolan Ke, Yitao Meng, Cong Xie, and Indranil Gupta. 2023. +Baechi: Fast Device Placement of Machine Learning Graphs. 1, 1 (January 2023), 37 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +Introduction +Distributed Machine Learning frameworks use more than one device in order to train large models and allow for +larger training sets. This has led to multiple open-source systems, including TensorFlow [1], PyTorch [56], MXNet [16], +Theano [71], Caffe [34], CNTK [62], and others [41, 64, 78]. Many of these systems use data parallelism, wherein each +device (GPU) runs the entire model, and multiple items are inputted and trained in parallel across devices. +Yet, the increasing size of Machine Learning (ML) models and scale of training datasets is quickly outpacing +available GPU memory. For instance the vanilla implementation of a 1000-layer deep residual network required 48 GB +memory [17], which is much larger than the amount of RAM available on a typical COTS (Commercial Off-the-Shelf) +device. Even after further optimizations to reduce memory cost, the ML model still required 7 GB memory, making it +impossible to run an entire model on a single device with limited memory, as well as prohibitively expensive on public +clouds like AWS [6], Google Cloud [24], and Azure [49]. +At the same time, today, ML training is gravitating towards being run among small collections of memory-constrained +devices. These include small groups of cheap devices like edge devices (for scenarios arising from Internet of Things +and Cyberphysical systems), Unmanned Aerial Vehicles (UAVs or drones), and to some extent even mobile devices. For +instance, real-time requirements [48, 81], privacy needs [11, 12], or budgetary constraints, necessitate training only +using nearby or in-house devices with limited resources. +These two trends—increasing model graph sizes and growing prevalence of puny devices being used to train the +model graph—together cause scenarios wherein a single device is insufficient and results in an Out of Memory (or +OOM) exception. For example, we found that the Google Neural Machine Translation (GNMT) [77] model OOMs on a 4 +GB GPU even with conservative parameters: batch size 128, 4 512-unit long short-term memory (LSTM) layers, 30K +vocabulary, and sequence length 50. +This problem is traditionally solved by adopting model parallelism, wherein the ML model graph is split across +multiple devices. Today, a popular way to accomplish model parallelism in industry is to use learning-based approaches +to generate the placement of operators on devices, most commonly by using Reinforcement Learning (RL) or variants. +Significant in this space are works from Google [50, 51] and the Placeto system [2]. A learning-based approach learns +iteratively (via RL) and adjusts the placement on the target cluster, with the goal of minimizing execution time for each +training step in the placed model, i.e., its step time. +While learning-based approaches achieve step times around those obtained by expert placements, they can unfor- +tunately take an inordinately long time to generate their placement plans. For instance, using one state-of-the-art +learning-based approach [2], NMT models require 94,000 steps during the learning-based placement, and even with a +conservatively low estimate of runtime per learning step of 2.63 seconds, the total placement time would come to 68.67 +hours. One might possibly apply parallelization techniques [35, 36, 75] to the learning model being used to perform +placement, in order to speed it up, but the total incurred resource costs would stay just as high—hence, parallelization is +orthogonal to our discussion. +Such long waits are cumbersome and even prohibitive at model development time, when the software developer +needs to make many quick and ad-hoc deployments [2]. In fact, studies of analytics clusters reveal that most analytics +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +3 +job runs tend to be short [18, 76]. For instance, the step time for a typical model graph (e.g., NMT or Inception-V3), to +train on a single data batch, is O(seconds) on a typical GPU. Overwhelming this time with learning-based placement +times which span hours, significantly inhibits the developer’s agility. +Additionally, a learning-based placement run works only for a target cluster and a given model graph with fixed +hyperparameters (e.g., batch size, learning rate, etc.). If the model graph were to be transitioned to a different cluster +with different GPU specs, the learning has to be repeated all over again, incurring the high overhead. Consider a +developer who is trying to find the right batch size for a target cluster. This process of exploration is iterative, and every +hyperparameter value trial needs a new run of the learning-based technique, making the overall undertaking slow. +For the model development process to be agile, nimble, and at the same time coherent with future real deployments, +what is needed is a new class of placement techniques for model parallelism, that: i) are significantly faster in placement +than learning-based approaches, and yet ii) achieve fast step times in the placed model. +This paper is the first to adopt a traditional algorithmic approach for the placement of ML models on memory- +constrained clusters. Subsequent to our initial work [33], a few other authors have published algorithmic or dynamic +programming-based ideas for model placement, however these are either : i) standalone and not integrated into open- +source systems [67], ii) or they are aimed at only placing specific models like transformers [47], [63], or iii) they are at +best comparable in performance to ours [25]. Orthogonal to model parallelism is pipeline parallelism [27],[31],[22],[79]— +our paper does not explore the latter, in order to keep our discussion focused on the benefits of algorithmic approaches +over learning approaches. +The contributions of this paper are: +• We adapt classical literature from parallel job scheduling to propose two memory-constrained algorithms, called +m-SCT (memory-constrained Small Communication Times) and m-ETF (memory-constrained Earliest Task First). We +also present m-TOPO (memory-constrained TOPO-logical order), a strawman. We focus on the static version of the +problem. +• We prove that under certain assumptions, both m-ETF and m-SCT steps time is within a constant factor of the +optimal. +• We present the Baechi system (Korean for placement, pronounced “Bay-Chee”). Baechi incorporates m-SCT/m-ETF +into both TensorFlow as well as PyTorch. Our exposition focuses on the multiple design decisions that were needed +in Baechi to derive performance out of the algorithmic underpinnings. +• We present tailored integration of Baechi with both TensorFlow and PyTorch to address the different programming +abstractions and architecture in these two frameworks. +• We present experimental results from a real deployment on a small cluster of GPUs, using both TensorFlow and +PyTorch which show that Baechi generates placement plans in time 654×–206K × faster than today’s learning-based +approaches, and yet the placed model’s step time (training time) is either faster than or, at worst, only up to 6.2% +higher, compared to expert-based placements. +2 +New Algorithms for Memory-Constrained Placement +This section presents the problem formulation and our three placement techniques. For each technique, we first discuss +the classical approach (not memory-aware), and then our adapted memory-constrained algorithm. Where applicable, +we prove optimality. +Manuscript submitted to ACM + +4 +Jeon et al. +Table 1. Terms and Notations. Used in the Paper. +𝐺 +Machine Learning graph to be placed +(Classical: Dependency graph of tasks to be placed) +𝑚 +Number of operators (or tasks) in 𝐺 +𝑛 +Number of devices in a cluster +𝑀 +Memory available per device +𝑑𝑖 +Size of memory required by operator (task) 𝑖 +𝑘𝑖 +Computation time of operator (task) 𝑇𝑖 +𝑐𝑖𝑗 +Communication time of the output of operator 𝑇𝑖 +𝜌 +Ratio between maximum operator-to-operator (task-to-task) communication time and minimum +per-operator (per-task) computation time +SCT assumption +Small communication time assumption: Ratio between maximum operator-to-operator (task-to- +task) communication time and minimum per-operator (per-task) computation time is ≤ 1 +makespan +Training time for one data mini-batch, i.e., runtime for executing a ML graph on one input +mini-batch +Our three approaches are: 1) a placer based on topological sorting (TOPO) 2) a placer based on Earliest Task First +(ETF), and 3) a placer based on Small Communication Time (SCT). +2.1 +Problem Formulation +Given 𝑛 devices (GPUs), each with memory size 𝑀, and a Machine Learning (ML) graph 𝐺 that is a DAG (Directed +Acyclic Graph) of operators, the device placement problem is to place nodes of 𝐺 (operators) on the devices so that the +makespan is minimized. Makespan, equivalent to step time, is traditionally defined as the total execution time to train +on one input mini-batch (i.e., unit of training data). Table 1 summarizes key terms used throughout the paper. When +discussing classical algorithms, we use the classical terms “tasks” instead of operators. +If one assumes devices have infinite (sufficient) memory, scheduling with communication delay is a well-studied +theoretical problem. The problem is NP-hard even when under the simplest of assumptions [29], such as infinite number +of devices and unit times for computation and communication (UET-UCT). +Out of the three best-performing solutions to the infinite memory problem, we choose the two that map best to ML +graphs: 1) Earliest Task First or ETF [32, 74], and 2) Small Communication Time or SCT [26]. SCT is provably close to +optimal when the ratio of maximum communication time between any two tasks to minimum computation time for +any task is ≤ 1. +We excluded a third solution, UET-UCT [53], since it assumes unit computation and communication times, but ML +graphs have heterogeneous operators. +2.2 +m-TOPO: Topological Sort Placer +Background: Topological Sort (Not Memory-Aware). Topological sort [37] is a linear ordering of vertices in a DAG, +such that for each directed edge 𝑢 → 𝑣, 𝑢 appears before 𝑣 in the linear ordering. +New Memory-Constrained Version (m-TOPO). Our modified version, called m-TOPO, works as follows. It calculates +the maximum load-balanced memory that will be used per device, by dividing total required memory by number of +devices, and then accounting for outlier operators. Concretely, this per-device cap is𝐶𝑎𝑝 = (� +𝑖 ∈[𝑚] 𝑑𝑖/𝑛+max𝑖 ∈[𝑚] 𝑑𝑖). +Then m-TOPO works iteratively, and assigns operators to devices in increasing order of device ID. For the current +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +5 +device, m-TOPO places operators until the device memory usage reaches 𝐶𝑎𝑝. At that point, m-TOPO moves on to the +next device ID, and so on. At runtime, m-TOPO executes the operators at a device in the topologically sorted order +2.3 +m-ETF: Earliest Task First Placer +Background: ETF (Not Memory-Aware). ETF [32] maintains two lists: a sorted task list 𝑇 containing unscheduled +tasks, and a device list 𝑃. In 𝑇, tasks are sorted by earliest schedulable time. The earliest schedulable time of task 𝑖 is the +latest finish time of 𝑖’s parents in the DAG, plus time for their data to reach 𝑖. In 𝑃, each device is associated with its +earliest available time, i.e., last finish time of its assigned tasks (so far). +ETF iteratively: i) places the head of the task queue 𝑇 at that device from 𝑃 which has the earliest available time, ii) +then updates the earliest available time of that device to be the completion time of the placed task, and iii) updates +earliest schedulable time of that task’s dependencies in queue 𝑇 (if applicable). +The earliest schedulable time of task 𝑗 on device 𝑝 is the later of two times: (i) device 𝑝’s earliest available time +(𝑓 𝑟𝑒𝑒(𝑝)), and (ii) all predecessor tasks of 𝑗 have completed and have communicated their data to 𝑗. More formally, +let: a) Γ−(𝑗) be the set of 𝑗’s predecessors; b) for 𝑖: start time is 𝑠𝑖, computation time is 𝑘𝑖; c) 𝑥𝑖𝑝 = 0 when task 𝑖 is on +device 𝑝, otherwise 𝑥𝑖𝑝 = 1; d) commmunication time from task 𝑖 to 𝑗 is 𝑐𝑖𝑗. Then, the earliest schedulable time of task +𝑗 across all devices is: +min +𝑝 ∈𝑃 +� +max �𝑓 𝑟𝑒𝑒(𝑝), max +𝑖 ∈Γ−(𝑗)(𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗𝑥𝑖𝑝)�� +. +(1) +Under the SCT assumption (Table 1), ETF’s makespan was shown [32] to have an approximation ratio of (2 + 𝜌 − 1 +𝑚 ) +within optimal, where 𝜌 is the ratio of the maximum communication time to minimum computation time, and 𝑚 is the +number of devices. This approximation ratio approaches 3 when 𝜌 approaches 1 and 𝑚 ≫ 1. +New Memory-Constrained Version (m-ETF). +Our new modified algorithm, called m-ETF, maintains a queue 𝑄 of operator-device pairs (𝑖, 𝑝) for all unscheduled +operators and all devices. Elements (𝑖, 𝑝) in 𝑄 are sorted in increasing order of the earliest schedulable time for operator +𝑖 on device 𝑝. This earliest schedulable time takes into account dependent parents of 𝑖 as well as the earliest time that +device 𝑝 is available, given operators already scheduled at 𝑝. The reader will notice that m-ETF’s modified queue can +also be used for ETF–the key reason to use (𝑖, 𝑝) pairs is for m-ETF to do fast searches, since the earliest available +device(s) may have insufficient memory. +m-ETF iteratively looks at the head of the queue. If the head element (𝑖, 𝑝) is not schedulable because device 𝑝’s +leftover memory is insufficient, then the head is removed. If the head is schedulable, then operator 𝑖 is assigned to start +on device 𝑝 at that earliest time, and we: i) update 𝑝’s earliest available time to be the completion time of 𝑖, and ii) +update 𝑖’s child operators’ earliest schedulable times in queue 𝑄 (if applicable). +2.4 +m-SCT: Small Communication Time Placer +Background: SCT (Not Memory-Aware). The classical SCT algorithm [26] first develops a solution assuming an +infinite number of available devices, and then specializes for a finite number of devices. We elaborate details, as they +are relevant to our memory-constrained version. +Classical SCT: Infinite Number of Devices. SCT uses integer linear programming (ILP). The key idea is to find the +favorite child of a given task 𝑖, and prioritize its scheduling on the same device as task 𝑖. For a task 𝑖, denote its favorite +child as 𝑓 (𝑖). +Manuscript submitted to ACM + +6 +Jeon et al. +The original ILP specification from [26] solves for variables 𝑥𝑖𝑗 ∈ {0, 1}, where 𝑥𝑖𝑗 = 0 if and only if 𝑗 is 𝑖’s favorite +child. +For completeness, we provide this full ILP specification below [26] (Section 3.2 in that paper). Below, the machine +learning graph is 𝐺 = (𝑉, 𝐸); and 𝑖, 𝑗 refer to operators. + + +min𝑤∞ +Minimize makespan. +∀𝑖 → 𝑗 ∈ 𝐸(𝐺), 𝑥𝑖𝑗 ∈ {0, 1} +𝑥𝑖𝑗 = 0 when 𝑗 is 𝑖’s favorite child. +∀𝑖 ∈ 𝑉 (𝐺), 𝑠𝑖 ≥ 0 +All tasks start after time=0. +∀𝑖 ∈ 𝑉 (𝐺), 𝑠𝑖 + 𝑘𝑖 ≤ 𝑤∞ +All tasks should complete before makespan. +∀𝑖 → 𝑗 ∈ 𝐸(𝐺), 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗𝑥𝑖𝑗 ≤ 𝑠𝑗 +Given edge 𝑖 → 𝑗, then 𝑗must start after 𝑖 completes. +If on different devices, communication cost should be added. +∀𝑖 ∈ 𝑉 (𝐺), +∑︁ +𝑗 ∈Γ+(𝑖) +𝑥𝑖𝑗 ≥ |Γ+(𝑖)| − 1 +Every task has at most one favorite child. +∀𝑖 ∈ 𝑉 (𝐺), +∑︁ +𝑗 ∈Γ−(𝑖) +𝑥𝑗𝑖 ≥ |Γ−(𝑖)| − 1 +Every task is the favorite child of at most one predecessor. +(2) +We modify the above as follows. We allow 𝑥𝑖𝑗 to take any real value between 0 and 1, thus making the ILP a relaxed +LP. This can be solved in polynomial time using the interior point method [39]. Then the SCT algorithm simply rounds +the LP solution 𝑥𝑖𝑗 to be 1 if 𝑥𝑖𝑗 ≥ 0.1, setting it to 0 otherwise. 𝑥𝑖𝑗 can be used to determine the favorite child of each +task: 𝑗 is 𝑖’s favorite child if and only if after rounding, 𝑥𝑖𝑗 = 0. +This infinite device algorithm’s makespan was shown [26] to achieve an approximation ratio 2+2𝜌 +2+𝜌 +within optimal, +where 𝜌 is the ratio of the maximum communication time to the minimum computation time. +We note that the ILP has a meaningful LP relaxation if and only if: (i) infinite number of devices are available, and (ii) +the SCT assumption is satisfied, i.e., the ratio of the maximum inter-task communication time to the minimum task +computation time is ≤ 1. Nevertheless, even if this assumption were not true for an ML graph and devices, we show +later that SCT can still be attractive. +Classical SCT: Extension to Finite Number of Devices. +For a finite number of devices, SCT schedules tasks similar to ETF [32], but: a) prefers placing the favorite child +of a task 𝑖 on the same devices as 𝑖 (each task has at most one favorite child, and at most one favorite parent), and b) +prioritizes urgent tasks, i.e., a task that can begin right away on any device. +It was proved that SCT’s makespan has an approximation ratio of ( 4+3𝜌 +2+𝜌 − +2+2𝜌 +𝑚(2+𝜌) ) within optimal [26], which is +strictly better than ETF’s (Section 2.3). For instance, when 𝜌 approaches 1 and 𝑚 ≫ 1, then SCT is within 7 +3 of optimal +while ETF is 3 times worse than optimal. +New Memory-Constrained Version (m-SCT). Our proposed memory-constrained algorithm, called m-SCT, works +as follows. First, m-SCT determines scheduling priority and selects devices in the same way as the finite case SCT +algorithm just described. Second, when a device 𝑝 runs out of available memory, m-SCT excludes 𝑝 from future operator +placements. +In spite of the seemingly small difference, Figure 1 shows that m-SCT can succeed where SCT fails. SCT achieves a +makespan of 8 time units with infinite memory but OOMs for finite memory. With finite memory, m-SCT succeeds and +increases makespan to only 9 time units. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +7 +Fig. 1. Classical SCT vs. m-SCT. When per-device memory is limited to 4 memory units, SCT OOMs but m-SCT succeeds. m-SCT’s +training time (makespan) is only slightly higher (9) than SCT with infinite memory (8). Dashed arrows show data transfers. +2.5 +Optimality of m-ETF and m-SCT +Classical ETF and SCT were originally proposed to schedule a DAG of tasks on 𝑛 processors. The original work [26, 32] +derived upper bounds on the makespan achieved by them. Processor memory was not considered in that original +formulation, and infinite memory was assumed. +On the contrary, in ML model training, each device has a memory constraint. Impact of memory on scheduling is +further pronounced due to the persistent memory that each task requires. Consequently, the schedules obtained by +m-SCT/m-ETF can differ significantly from the SCT/ETF schedules (i.e., without memory constraint)—Fig. 1 shows an +example. It is not clear how much worse m-SCT/m-ETF makespan is, compared to the optimal. Thus we derive upper +bound on makespan of both m-ETF and m-SCT by extending the proofs in [32] and [26] to the memory constrained +case. We show that m-SCT/m-ETF makespans are within a constant factor of the optimal makespan. +Because the proofs for optimality of m-ETF and m-SCT are involved, we show them in Appendix A and Appendix B +respectively. We summarize our results and approach here: +Result 1: The completion time of m-ETF under realistic communication cost and limited memory, is within a known +factor of the optimal schedule possible under zero communication cost but with infinite memory. +Result 2: The completion time of m-SCT under realistic communication cost and limited memory, is within a known +factor of the optimal schedule under infinite memory. +Intuitively, our derived upper bounds are proportional to the ratio 𝑛 +𝑟 ,where 𝑛 is the total number of devices available, +and 𝑟 is the number of devices out of 𝑛 that still have spare memory after all the tasks have been placed in a way +that “fills up” memory device by device. Note that 𝑛 +𝑟 > 1, otherwise the problem is unsolvable. Intuitively, a smaller +Manuscript submitted to ACM + +op1 +op2 +0p3 +op4 +1(1) +3(2) +3(2) +1(1) +op5 +op6 +3(1) +1(1) +op1 +op2 +op3 +op4 +op5 +op6 +0p2 +op6 +op1 +op5 +op3 +op48 +Jeon et al. +𝑛 +𝑟 (i.e., larger 𝑟) indicates looser memory constraint and thus better makespan. As 𝑛 +𝑟 approaches 1, m-SCT’s solution +(respectively m-ETF’s) starts to approach that of SCT (respectively ETF). +3 +Baechi Design +This section describes how we implement Baechi in a way that works modularly with TensorFlow [1] as well as +PyTorch [56], two popular open-source learning platforms originally developed by Alphabet and Meta respectively. +At a high level—for both target systems, Baechi first creates a computation graph of the input model, where each +node is annotated with its memory requirements and time to complete. This graph is then fed to the chosen algorithm +(Section 2’s m-SCT, m-ETF, or m-TOPO) to generate the placement. Finally, training is automatically executed with the +given placement and without requiring the developer to modify the code for the model. +However, because of two key differences in abstractions and architectures between TensorFlow and PyTorch, Baechi’s +design for each is slightly different. First, the “nodes” in the computation graph are operators in TensorFlow while in +PyTorch they are modules. The former are fine-grained mathematical operations on tensors, while the latter are coarser +structures similar to classes in object-oriented languages. Second, in TensorFlow a model is a static graph of operators, +while in PyTorch, the computation graph is constructed only during the forward run. Because of foreknowledge of +the graph, TensorFlow can automatically insert rendezvous operators [68] for cross-device communication. However, +PyTorch does not automatically insert these essential cross-device communication primitives. It requires PyTorch +developers to write explicit code that moves tensors across devices during execution. A side benefit of our work is the +automatic generation of these communication primitives. +In the remainder of this paper, we refer to the integration of Baechi into TensorFlow as Baechi-TF, and Baechi’s +integration into PyTorch as Baechi-PY. +Next, we describe our techniques and optimizations for Baechi-TF in Section 3.1, and then additional changes and +differences required for Baechi-PY design in Section 3.2. +3.1 +Design of Baechi-TF +To work with TensorFlow, Baechi needs to address four challenges: +1) Satisfying TensorFlow’s colocation constraints, 2) Minimizing Data Transfer via Co-Placement, 3) Optimizations to +reduce the number of operators to be placed, and 4) Accommodating Sequential and Parallel Communications. Baechi +solves these using a mix of both new ideas (Sections 3.1.1,3.1.3,3.1.4) and ideas similar to past work (Sections 3.1.2,3.1.3). +Working Example. We use Figure 2 as a working example throughout this section. It is a simplified TensorFlow graph +for linear regression training with stochastic gradient descent (SGD). +3.1.1 +TensorFlow Colocation Constraints The first challenge arises from the fact that TensorFlow (TF) +requires certain operators to be colocated. For instance, TensorFlow offers a variable operator, tf.Variable, which is +used to store persistent state such as an ML model parameter. The assignment and read operators of a variable are +implemented as separate operators in TensorFlow, but need to be placed on the same device as the variable operator. +TensorFlow represents this placement requirement as a colocation group involving all these operators. E.g., in Figure 2 +there are two colocation groups: one containing Weight and ApplyGrad, and another containing Step and UpdateStep. +Baechi’s initial placement (using the algorithms of Section 2) ignores colocation requirements. Our first attempt +was to post-adjust placement, i.e., to “adjust” the device placement, which was generated ignoring colocation, by +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +9 +Fig. 2. Working Example. ML Graph for Linear Regression. +Fig. 3. Co-Placement. Subgraph of tf.tensordot Generat- +ing Data Transfers by m-ETF. +“moving” operators from one device to another, in order to satisfy TF’s colocation constraints. We explored multiple +post-adjustment approaches including: i) preferring the device on which the compute-dominant operator in the +group is placed, ii) preferring the device on which the memory-dominant operator in the group is placed, and iii) +preferring the device on which a majority of operators in the group are placed. We found all these three approaches +produced inconsistent performance gains, some giving step times up to 406% worse than the expert. We concluded that +post-adjusting was not a feasible design pathway. +Baechi’s novel contribution is to co-adjust placement, using colocation constraint-based grouping while creating the +schedule. (In comparison, e.g., ColocRL [51] groups before placement.) Concretely, whenever Baechi places the first +operator from a given colocation group, all other operators in that group are immediately placed on that same device. +Baechi tracks the available memory on each device given its assigned operators. If the device cannot hold the entire +colocation group, then Baechi moves to the algorithm’s next device choice. We found this approach the most effective +in practice, and it is thus the default setting in Baechi. +3.1.2 +Co-Placement Optimization Different from TensorFlow’s colocation constraints (Section 3.1.1), Baechi +further prefers to do co-placement of certain operators. This is aimed at minimizing data transfer overheads. Common +instances include: (i) groups of communicating operators whose computation times are much shorter than their +communication times, and (ii) matched forward and backward (gradient-calculating) operators. +Figure 3 shows an example for case (i). This subgraph generated by tf.tensordot API is a frequent pattern +occurring inside TensorFlow graphs. The subgraph permutes the dimensions of opin output according to the perm’s +output (Transpose) and then changes the tensor shape by Shape’s output (Reshape). +When m-ETF places this subgraph on a cluster of 3 devices, it places opin, perm, and Shape on different devices. +Computation costs for perm and Shape are very short (because they process predefined values), whereas subsequent +communication times are much larger. Thus, m-ETF’s initial placement results in a high execution time. +Baechi’s co-placement heuristic works as follows. If the output of an operator is only used by its next operator, we +place both operators on the same device. This is akin to similar heuristics used in ColocRL [51]. In Figure 3, +Manuscript submitted to ACM + +Input +Output +Step +Update +MatMul +Loss +Grad +Step +Apply +Weight +GradoPin +Transpose +Reshape +oPout +perm +Shape10 +Jeon et al. +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 4. Operator Fusion Without Creating Cycles. (a) shows a fused ML Graph Example. When 𝑜𝑝𝑠𝑟𝑐 and 𝑜𝑝𝑑𝑠𝑡 are fused, some +scenarios create a cycle (b), while others do not (c, d, e, f). Baechi fuses operators in a subset of “safe” cases, particularly (e, f). +Baechi’s co-placement optimization places all of the operators on one device, avoiding any data transfers among the +operators. +For case (ii), to calculate gradients in the ML model, TensorFlow generates a backward operator for each forward +operator. Baechi co-places each backward operator on the same device as its respectively-matched forward operator. +Upon placing the first operator in a colocation group, Baechi uses both the co-placement heuristic and the colocation +constraints (Section 3.1.1) to determine which other operators to also place on the same device. Co-placement not only +minimizes communication overheads but also speeds up the placement time by reducing the overhead of calculating +schedulable times on devices. +3.1.3 +Operator Count Minimization Placement time can be decreased by reducing the number of opera- +tors/groups to be placed. We do this via two additional methods: +i) Operator Fusion: Fusing operators that are directly connected and in the same co-placement group; and +ii) Forward-Operator-Based Placement: Placing operators by only considering the forward operators. +Operator Fusion. Baechi fuses operators using either the colocation constraints (Section 3.1.1) or co-placement +optimizations (Section 3.1.2). This is new and different from TensorFlow’s fusion of operations. One challenge that +appears here is that this may introduce cycles in the graph, violating the DAG required by our algorithms. +Manuscript submitted to ACM + +Input +Output +Step +MatMul +Loss +Grad +Update +Cycle +Step +Apply +Weight +.i +GradoP1 +0P2 +oPsrc +opdstop1 +0P2 +oPsrc +oPdstop1 +0P2 +opsrc +oPdstop +oPsrc +oPdstop +oPsrc +opdstBaechi: Fast Device Placement of Machine Learning Graphs +11 +(a) +(b) +Fig. 5. Operator Fusion. Avoiding Data Transfer Example. (a) Before Fusion. (b) After Fusion. +Figure 4a shows an example resulting from Figure 2—a cycle is created when Step and UpdateStep are fused into a +new meta-operator, and Weight and ApplyGrad are fused. +Consider two nodes–source and destination–with an edge from source to destination. Merging source and destination +creates a cycle if and only if there is at least one additional path from source to destination, other than the direct edge. +Note that there cannot be a reverse destination to source path as this means the original graph would have had a cycle. +In Figure 4b, fusing opsrc and opdst creates a cycle. Unfortunately, we found that pre-checking existence of such +additional paths before fusing two operators is unscalable, because the model graph is massive. +Instead, Baechi realizes that a necessary condition for an additional path to exist is that the source has an out-degree +at least 2 and the destination has an in-degree at least 2 (otherwise there wouldn’t be additional paths). Thus Baechi +uses a conservative approach wherein it fuses two operators only if the negation is true, i.e., either the source has an +out-degree of at most 1, or the destination has an in-degree of at most 1 (Figures 4e, 4f). This fusion rule misses a few +fusions (Figures 4c, 4d) but it catches common patterns we observed, like Figure 4e. +Forward-Operator-Based Placement. When memory is sufficient (i.e., one device could run the entire model), Baechi +considers only forward operators for placement and thereafter co-places each corresponding backward (gradient) +operators on the same respective device as their forward counterparts. This is a commonly-used technique [2, 51]. This +significantly cuts placement time. When device memory is insufficient, Baechi runs the placement algorithms using +both forward and backward operators, forcing corresponding pairs to be co-placed using the heuristic of Section 3.1.2. +Example: Benefits of Fusion. Figure 5a shows the placement of a subgraph of Figure 2 on two devices. Baechi +first places Grad on device-1. Baechi places the next operator, Step on the idle device-2, and colocates (due to TF +constraints) UpdateStep on device-2. This creates communication between the devices. Assuming operators’ compute +costs are 1, and communication cost between Grad and UpdateStep is 5, this results in an execution time of 7 time +units. +On the other hand, Figure 5b shows that Baechi merges Step and UpdateStep with operator fusion. Since this +meta-operator’s schedulable time on device-1 is earlier than on device-2 due to communication overhead, Baechi +places it on device-1. Fusion lowers total execution time to 3 time units. +Manuscript submitted to ACM + +Step +Update +Grad +StepStep +Grad +Update +Step12 +Jeon et al. +Loops in the Original Model Graph. Different from the cycles discussed above, some network graphs consist of +loops, e.g., RNNs. We use the unrolled ML graph [4] to turn the graph into a DAG, and then apply Baechi’s techniques. +3.1.4 +Sequential vs. Parallel Communication Our algorithms from Sections 2.3 and 2.4 assume that each +operator can send data simultaneously to its children. Baechi also proposes a new way to deal with environments +involving constrained networks (including our deployment in Section 5), where data transfer is sequential. For networks +that limit each device to do at most one transfer at a time (out or in), Baechi assumes communication queues at devices. +Concretely, when a data transfer between two devices is requested, Baechi assumes the request is put into the respective +devices’ communication queues and processed sequentially at both ends. During placement, Baechi calculates the wait +time at the communication queues and adds it to the earliest schedulable time computed for the operator. Specifically +the queue wait time is added to equation (1) in Section 2.3. Otherwise, normal m-SCT/m-ETF apply, as described earlier. +3.2 +Design of Baechi-PY +We remind the reader that unlike TensorFlow’s fine-grained operators and known communication graph, PyTorch: (i) +has coarser modules, and (ii) requires the programmer to explicitly program cross-device communication. +Concretely—first, PyTorch models are built by composing different modules. The model is not natively available as a +graph unlike TensorFlow. To feed the model to Baechi’s algorithms, Section 3.2.1 describes how we construct a graph by +using PyTorch’s Autograd [56] which tracks the flow of tensors among the modules of the model. Second, the primitive +.to() API provided by PyTorch for developers to program communication is inefficient and manual. To address this, +Section 3.2.2 presents our communication protocol to handle cross-device transfers efficiently and automatically. +3.2.1 +Baechi-PyTorch Graph Developers build PyTorch models by composing modules, i.e, classes inherited +from PyTorch’s nn.Module. Each module contains its tensor parameters (e.g., weights of a linear layer) and a forward() +method, which defines how the module modifies the input. By default Baechi treats modules as the nodes of the graph +in Baechi-PY. Placing a module on a device means moving all its parameters to the device before beginning the training. +During the training, our communication protocol (next Section 3.2.2) ensures that the input to that module is also +moved to the same device. Subsequently the forward() operation of the module will be invoked at runtime, and it will +be executed on the assigned device. A model may also include operations not defined as PyTorch modules. For instance, +arithmetic operations like scaling (e.g: x=x/2). But these operations usually do not have any associated parameters that +must be assigned to a device by Baechi-PY. Hence we exclude them from the graph during the placement planning. By +default, this associated operation will be executed on the device of the input tensor, i.e., x’s device in this case1. +Baechi-PY constructs the model’s graph in two steps. First we obtain all modules that constitute the model and +co-place modules occurring in common design patterns. Second we obtain the edges of the graph by tracking the flow +of tensors using PyTorch’s Autograd [56]. This approach is similar to PipeDream [27]. +Co-placement: Treating modules as nodes, we observed it is common for models to contain specific subgraphs (of +modules) that occur as common design patterns throughout the model graph. Baechi-PY groups such subgraphs into a +single node—this is called co-placement (unlike Baechi-TF’s co-location constraints in Section 3.1, this co-placement is +a performance optimization in PyTorch). For instance in Inception models, the subgraph (Conv2d)→ (Batch Norm) +→ (inplace ReLU) occurs commonly, and Baechi-PY groups each occurrence as one node in the computation graph. +1Arithmetic operations may still take some time to complete and memory to store their outputs, but we observed that their impact on the overall step +time and memory budget is small, thus we ignore them in generating placements. If desired, such operations may be defined as nn.Modules and be +included in the placement graph. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +13 +This co-placement allows Baechi-PY to avoid communication of tensors along the two edges of this subgraph. An +additional benefit of co-placement is that it significantly reduces the size of the computation graph, thus making +Baechi’s algorithms (Section 2) run faster. For instance, co-placement reduces the number of nodes in Inception-V3 +PyTorch by 60%, from 325 to 133. By default, Baechi-PY uses the most atomic modules, i.e., not further divisible, e.g., 2D +Convolution module (Conv2d). If the developer wishes, they can programmatically specify which modules Baechi-PY +should be co-placed and treated as individual nodes. +Building the graph: Next, Baechi obtains edges between the nodes. To do this, we run a training step of the model +with dummy data. (We run 20 such dummy training steps, also helping us profile memory usage and computation +times of all the nodes.) +During the forward run, we annotate each intermediate output tensor with the node that +generated it. Meanwhile, PyTorch’s Autograd automatically fills in the gradient function (grad_fn) for each tensor +created. Further, each such grad_fn has a list of grad_fn of tensors used as inputs in creating this tensor. Autograd +stores this list to perform back-propagation later. We use this information that traces the tensor gradient functions, +along with our tensor to node annotation, to construct a dependency graph among modules of the model. +It is possible that some gradient functions may include operations not related to any module, e.g., Autograd-specific +operations such as SelectBackward or gradients of arithmetic operations. But since these operations do not need a +device placement, they are removed to obtain the dependency graph only between nodes of the model (they are added +back in after the model is placed) . +3.2.2 +Communication Protocol PyTorch provides native support for synchronous communication, which can +be inefficient. If asynchronous data transfer is used, the developer is required to carefully insert synchronizations to +ensure correctness. +We design a general communication protocol for cross-device communication that is efficient and automated, thus +relieving the developer from specifying manual configurations. We do so by leveraging CUDA streams, an abstraction +that allows overlapping multiple sequences of operations in a GPU [54]. +To move a tensor T to 𝐺𝑃𝑈0, PyTorch provides an API T.to(0). To ensure correctness, .to() conservatively blocks +both sending and receiving devices until all the operations submitted to both devices so far are completed. This can be +avoided by leveraging the abstraction of CUDA streams [54], in order to overlap communication with computation. +A CUDA stream in a GPU is a FIFO queue of operations that will be sequentially executed on the GPU. By default, +all operations submitted to a GPU are placed on a single stream. To perform two operations in parallel, they must be +placed on two separate streams on the GPU. Accordingly, on each GPU, Baechi-PY defines one compute stream and +multiple communication streams. The compute stream queues the computations corresponding to the modules placed +on the GPU, while the communication streams concurrently move the relevant tensors across the devices. +However, CUDA streams need to be programmed carefully to specify synchronization points that obey dependencies +in the model graph. We use CUDA Events provided by the runtime API [57] to synchronize the independent streams. +None of the existing ways of using CUDA streams in PyTorch fits our needs. Concretely—first, in PyTorch-Distributed +[46] and PipeDream [27], training steps proceed in stages, each working on a different batch of data. All tensors generated +in a given stage are transferred to the next device at the end of the stage. This pattern, where all communication +happens synchronously only after all computations are complete requires few, if any, synchronizations. In contrast, +Nimble [43], deals with multiple parallel streams working on the same batch of data, like in Model Parallelism. Compute +and communication events may asynchronously occur at any time and it requires synchronizations to preserve data +dependencies. However, Nimble is a single-GPU system. +Manuscript submitted to ACM + +14 +Jeon et al. +In Baechi-PY, we use a greedy-wait strategy. Concretely, first we greedily push out the output of a node, as soon as it +is computed, to the devices of its children nodes. Second, before starting the compute operation, a child node must wait +for all its incoming input stream or if the input has already been transferred it must pull the copy of the inputs on its +devices. We implement our greedy-wait communication protocol as a wrapper around the forward() function of each +node. +Algorithm 1: Communication protocol built around each node’s forward(input) +1 for each parent of the node in graph do +2 +(node’s compute_stream) wait for (rx_stream from parent’s device); +3 end +4 On node’s compute_stream: +5 +input_local = local copies of input on node’s device; +6 +output = forward_operation(input_local); +7 for each child of the node in graph do +8 +(tx_stream to child’s device) wait for (node_compute_stream); +9 end +10 for each child of the node in graph do +11 +Using (node’s tx-rx_stream pair to child’s device ): +12 +send output to child; +13 end +14 return output +Algorithm 1 shows the complete communication protocol, and we describe it in detail. Before we start the training, +for a given node and its child node on a different device, we create two streams - a tx-stream on the node’s device and a +rx-stream on the child node’s device. Only one such stream pair is sufficient for a child device, even if multiple children +nodes are on that device. So if a node on 𝐺𝑃𝑈0 has two children, one on 𝐺𝑃𝑈1 and another on 𝐺𝑃𝑈2, then for that node +we create one tx-rx stream pair each to 𝐺𝑃𝑈1 and 𝐺𝑃𝑈2. We do this for every node and for every device any of its child +nodes reside in. +Each device’s compute stream queues the computations of nodes assigned to that device. When a node reaches the +head of the compute stream of its device, the compute stream is made to wait for all the rx-streams to that device from +all the parent nodes (line 2). The rx-streams carry a copy of the output tensors of the parent nodes to the device of the +current node. Once all rx-streams have completed the transfer, these tensors are passed as inputs to the actual forward +computation of the node (lines 4-6). All the tx-streams egressing from this node are made to wait for the compute +stream to finish the computation (lines 7-9). The tx-stream carries the outputs of the node to the child node’s device and +the corresponding rx-stream on the child node’s device receives it. As soon as the output is ready, it is asynchronously +sent to all the child devices through the tx-streams and received asynchronously at the child devices through their +respective rx-streams (lines 10-12). The compute stream can move to the next node assigned to it while the tx-streams +are transferring out the output. Similarly, the compute streams on child devices are not interrupted by incoming tensors +on their rx-streams. Note that the output is sent to a child device only once even if multiple children reside on that child +device. The tx-rx streams serve as synchronization points in addition to overlapping communication and computation. +Our m-ETF and m-SCT algorithms from Sections 2.3 and 2.4 assume that each node can send data simultaneously to +its children. Our protocol mimics this communication with multiple outgoing tx-streams transferring the tensors to +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +15 +Input Model +In TF as tf.Graph +In PT as nn.Module +Profiler +Graph Optimizer (TF only) +Co-Placement Grouper +Operator Fuser +Execution Simulator +Global Scheduler +(m-SCT, m-ETF, m-TOPO) +Placed +Graph +TF/PT +Runtime +Device +Device +Device +Assigner +(PT only) +Node +to Device +Placement +TF = TensorFlow, PT = PyTorch +Graph +Generator +(PT only) +Fig. 6. Baechi System Architecture +child devices in parallel. While such overlapping tx-streams from a device may slightly increase the communication +times in each of the streams, we observed that the associated effect on step time is small. Also as long as we facilitate and +synchronize the forward run correctly, PyTorch’s Autograd [56] ensures that the back-propagation correctly executes in +the reverse order of the forward sequence. It automatically manages input-output dependencies and synchronizations +in the reversed order. +4 +Implementation +In order to integrate modularly with TensorFlow (TF) [1] v1.12 and PyTorch v1.9 [56], Baechi adopts the architecture +shown in Figure 6. Baechi executes the following steps: 1) its Graph Generator and Profiler constructs the graph +annotated with each node’s time and memory requirements, 2) Baechi’s Graph Optimizer for TensorFlow uses the design +of Section 3.1 to account for TF colocation constraints, and applies co-placement and operator fusion, 3) Baechi’s +Execution Simulator (ES) (Section 4.2) executes our algorithms (m-TOPO, m-ETF, or m-SCT) and generates the placement +(in TensorFlow a ready-to-use placed graph is output), and 4) the Assigner for PyTorch (Section 4.3) modifies the model +to allow execution according to the generated placement. The placed graph can then be used in the training script as +the drop-in replacement for the single-GPU model, in both TensorFlow and PyTorch. Next we describe each of the +components in detail. +4.1 +Graph Generator, Profiler and Optimizer +In Baechi-TF, the model is already given as a static graph. The Profiler then measures and annotates each node with +its time and memory requirements. Baechi parses this annotated graph and generates an equivalent intermediate +NetworkX [61] graph. The NetworkX format allows Baechi to both store operator execution metadata (computation and +communication times, memory needed, etc.), and to easily manipulate the graph (e.g., fuse operators). Then, in case of +Baechi-TF, we additionally apply the co-placement and operator fusion optimizations (Sections 3.1.2, 3.1.3) to the graph. +Manuscript submitted to ACM + +16 +Jeon et al. +Memory +Inference +Training +Permanent +(a) +(a) + (b) + (c) +Temporary +(b) + (e) +(e) + (d) +Table 2. Memory Consumption in PyTorch +In Baechi-PY, the Graph Generator constructs the graph corresponding to the input model as per Section 3.2.1. +For communication time, we use a linear model proportional to data size. Concretely, we implemented a microbench- +mark tool to measure communication times for various data sizes, and generated a communication cost function through +linear regression. +4.1.1 +Profiler: The Profiler measures computation times and memory requirements of each node in the graph. In +Baechi-TF, we use the standard TensorFlow profiling tool to obtain computation time and memory allocation for each +operator. TensorFlow profiler returns allocation information for temporary, permanent, and output tensor memory. +The temporary memory is allocated at the beginning of an operation and deallocated when the operation finishes. +The permanent memory is allocated and used over the entire execution, e.g., to store persistent states such as weights. +For Baechi-PY we build a simple profiler, akin to that in [27], for measuring time using hooks [58], and for measuring +memory 2. +Baechi-PY Profiler Memory Estimation +In PyTorch, GPU Memory required to hold all the tensors is reserved once during the first training step and reused in +subsequent steps. To model the memory usage pattern, we categorize each node’s memory as consisting of 5 components: +(a) Parameters memory: Memory occupied by parameters of the node; +(b) Output Memory: forward output of the node; +(c) Parameter gradient: gradients of parameters of the node; +(d) Upstream gradient: gradient of output of the node; +(e) Memory temporarily used in computing the output/gradient. +Table 2 summarizes how these five metrics are used in training and inference, and whether they are used as temporary +memory or permanent memory. We describe each term. During the training phase, each node requires memory to +store: (a) its parameters, (b) the node’s forward output tensors, and (c) parameters’ gradient information. In PyTorch, +memory to store parameter gradient information is acquired once in the beginning of training and permanently held +until the end of training. Similarly, output tensors ((b)) are treated as permanent memory since during each forward run, +outputs of all nodes must be stored. They are required later during back-propagation. In contrast during the inference +phase (forward only runs), output of a node is temporary since it is immediately released after being consumed by +the subsequent node. During back-propagation in training, memory is also required to hold the gradient of output +of the node ((d)). This is a temporary requirement since this memory is released after the output gradient has been +used to compute the node’s parameter gradients and its input gradient. Nodes may also require temporary memory +while performing these computations ((e)). For example, in computing the parameter gradients, a temporary matrix is +used to store all the gradients and then the node’s parameter gradients are updated at once. For in-place nodes like the +2While PyTorch has an internal profiler using it would have required us to handle dependencies and operation granularity carefully. Our simple approach +avoids these. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +17 +in-place ReLU, (b) is set to 0 since no new output tensor is generated ((a) and (c) are also 0 incase of ReLU since it has +no associated parameters). +4.2 +Execution Simulator +Baechi’s Execution Simulator (ES) executes the algorithms on the profiled graph and generates the placement. The +ES takes as input the NetworkX operator graph, the number of GPUs and the memory capacity of each of them. The +output is the graph in which all operators are assigned to devices. We describe the common parts of the ES across both +Baechi-TF and Baechi-PY, explicitly pointing out differences as necessary. +Our initial attempt was in fact to try re-purposing TensorFlow ’s (existing) simulator, but using that required us to +assume operators were already placed, zero communication cost, and no caching—these were inapplicable to Baechi. +Motivated by this, we designed Baechi’s new ES uniquely for memory-constrained placement. +The ES consists of: a) a global scheduler, and b) simulated devices (with specs identical to deployment). The global +scheduler maintains a single queue with operators that are ready to run. The scheduler extracts operators from its +queue and applies our scheduling algorithms (m-TOPO, m-ETF, m-SCT) to place them on devices. +In ES, each device has two FIFO queues, one for operators and one for data transfer. This allows data transfer to +overlap with operator execution. When a device receives a tensor from another device, it caches the tensor to avoid +duplicate data transfer. +Dynamic Memory Allocation. +Calculating a device’s memory usage as the sum total of all its assigned operators (assigned over the entire duration) +clearly overestimates memory. For example in TensorFlow, Inception-V3 with batch size 32 can execute using 4 GB +even though its operators’ memory needs add up to 22 GB. +In generating the placements, the ES calculates memory in a way that parallels how the frameworks manage memory. +Concretely Baechi’s ES tracks an estimate of memory usage during its placement. When an operator executes on a +device, the device allocates temporary memory, and separate memory for its output tensors. The temporary memory is +deallocated when the operator finishes. There are minor differences in the ES for TensorFlow and PyTorch. TensorFlow +uses separate operators for forward and backward computation. The output memory of an operator is deallocated +after all its successors finish. In PyTorch, the forward and backward computation runs in a single module. The output +memory of a module is held until its backward computation is completed. The output memory is treated as a part of +the permanent memory as explained in the previous Section 4.1. If a device’s memory becomes full, the device can be +removed–this never happens in practice as usually a device has at least a few bytes left. +Note that in Baechi-TF, memory is reserved for a colocation group at device 𝑝 when the first operator is placed on 𝑝 +(Section 3.1.1). The reserved memory is deallocated when all the operators in the group finish. +Linear Programming Solver. To solve the SCT LP problem, we use the interior point method [13]. This is preferable +over other solvers such as simplex [9] as it guarantees polynomial execution time [72]. Concretely, we use the primal +dual interior-point solver via Mosek optimization [7], which has a run time complexity of 𝑂(𝑛3.5𝐿), where 𝐿 is the +maximum number of bits in the LP input, and 𝑛 is the number of variables. +4.3 +Assigner in Baechi-PY +Once the mapping of graph nodes to devices is decided, the Assigner contains the mechanism to initialize the assignment. +The Assigner for TensorFlow merely changes the device attribute of the nodes. The Assigner for PyTorch requires +Manuscript submitted to ACM + +18 +Jeon et al. +multiple steps—it needs to: i) move nodes’ parameters to the assigned devices, ii) send output tensors from a node to +devices of child nodes at the device boundaries, and iii) avoid naive use of .to() for cross-device communication as +this leads to inflated step times owing to unnecessary blocking of the devices (see Section 3.2.2). +Baechi-PY’s Assigner enables this process by automatically adding wrappers around the forward() methods of the +nodes. The wrapper transparently handles the communication and caching of the tensors using the communication +protocol in Section 3.2.2. This way, the developer does not need to make any changes to the input model code written +for a single-GPU execution. The output of the Assigner is a model assignment that can be used as a drop-in in any +existing training script. +4.4 +Miscellaneous Issues +We discuss a few key miscellaneous aspects. +LP Modifications. The ILP solutions (Section 2.4) resulted in more than one favorite child (or parent) being selected +for certain nodes. In Baechi we lowered the rounding threshold from 0.5 to below 0.2. This eliminated all violations, +and avoided nodes from having multiple favorite children. (We use threshold = 0.1 in practice.) +Ignoring Bootstrap Steps in Profiling. 1) In a training run of a model graph, step times are initially high due to +TensorFlow bootstrapping. We estimate step times in steady state, after a few iterations have passed. 2) Some TensorFlow +operators are implemented with multiple GPU kernels. When profiling these operators, we include multiple kernel +executions, in order to avoid underestimation. This is similar to TensorFlow’s cost model [1]. +Reordering Layers in PyTorch. Dynamic line-by-line execution in PyTorch means that the modules’ forward() +functions will be called in the order in which they appear in the code rather than in the topological order followed +by Baechi’ ES (Sec. 4.2). We reorder the actual execution of modules’ computations on GPUs by launching a thread +when a module’s forward() is called. The thread waits until its topological parent (according to the ES) has submitted +the computation task to the GPU and only then submits its task. However, such reordering does not give a noticeable +advantage over just executing the code order since the two orders do not differ significantly in most cases. +5 +Evaluation +Our evaluation answers the following six questions: +1. How fast is Baechi’s placement time, i.e., how quickly do our algorithms find placements? (Section 5.2) +2. How fast are the step times of the placement generated by Baechi, i.e., training time per step of the placed model? +(Section 5.3) +3. How do the step times for Baechi compare to single GPU and expert placements? (Section 5.3) +4. How do the Baechi’s step times change when there is insufficient memory per GPU? (Section 5.4) +5. How much is the benefit due to Baechi’s optimizations from Sections 3.1.2, 3.1.3 and 3.2.2? (Section 5.5) +6. Are algorithmic approaches preferable over RL approaches for model parallelism? (all subsections). +5.1 +Experimental Settings +We use two popular ML benchmarks for each framework: A) for TensorFlow, we use Inception-V3 and Google Neural +Machine Translation System (GNMT), and B) for PyTorch, we use Inception-V3 and a Transformer model. The former +choice is because: i) Inception-V3 and GNMT are respectively considered the best representatives of vision and Natural +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +19 +Language Processing (NLP) models, and ii) past work [2, 50, 51] used Inception-V3 and NMT (GNMT is a more complex +version), thus allowing us to compare. For PyTorch we replace GNMT with Transformer as the former is implemented +using the LSTM module [59], making the (latter) Transformer a more complex and generalized version. We describe the +three benchmark configurations in detail below: +Inception-V3 Benchmark Configuration. Inception-V3 [66] is a convolutional neural network architecture that is +widely used for image classification. This model is composed of multiple blocks called Inception modules. The Inception +modules consist of branches of convolutional and pooling operators. To train the model, we use RMSProp [28] and +batch sizes of both 32 and 64. +GNMT Benchmark Configuration. Google Neural Machine Translation System (GNMT) [77] is a language model +for automated translation. GNMT consists of: encoder and decoder modules, each a stack of recurrent neural networks +(RNNs); and the attention module to process long sequences effectively. We use 4 long short-term memory (LSTM) +layers of the encoder and the decoder layers with residual connections, and the Bahdanau attention mechanism [8]. We +use the LSTM hidden size of 512, the vocabulary size of 30,000, the unrolled RNNs with the sequence length of 40 and +50, and the batch size of 128 and 256. Baechi-TF applies the co-placement optimization to LSTM cell operators and also +to attention operators. +Compared to Inception-V3, GNMT has fewer barriers (sync points) inside its model graph, indicating that GNMT has +a higher potential to benefit from Baechi-TF’s parallel placements. +Transformer Benchmark Configuration. Transformers are a versatile family of models used in vision as well as +language. Like Neural Machine Translation (NMT), Transformers have an encoder-attention-decoder architecture. +But while NMT processes one word at a time, Transformers use multi-head attention modules that process the entire +sequence at once. In PyTorch, we implement an attention operation in the traditional way [23]—as one large matrix +multiplication and hence as a single module. For concreteness, we use the base Transformer model from [73] (without +weight sharing) with a vocabulary size of 30,000, sequence length of 50 and batch sizes of 64, and 128. +Machine Setup. All experiments are run on our local server that has 4 NVIDIA GTX 2080 GPUs, with 8 GB per- +GPU memory (the machine also has an Intel i9-7960X CPU, but this is not used to execute operators). GPUs are +connected to CPUs via PCIe 3.0 x16 (we do not use NVLink [55]). All data transfers go through the host memory (no +P2P communication among GPUs). This results in a slow IO bus, and we believe this high ratio of communication +overhead to computation overhead is representative of realistic scenarios like the kinds outlined in Section 1. We place +all GPU-supported operators only on GPUs. +Approach to Comparison. To quantify the benefits of using an algorithmic approach to model parallelism over a +Reinforcement Learning (RL) approach for model parallelism, we compare Baechi to the best RL-based model parallelism +techniques: [2, 50, 51]. Directly running these other systems was complicated by lack of uniform availability of working +code—Placeto’s code [2] missed key optimizations; ColocRL [51] is proprietary; only HierarchicalRL’s code [50] was +available, but it was slow and generated inefficient placements. E.g., For GNMT, HierarchicalRL took 12 hours+ to +run placement (batch size 128, length 50) and the resultant step time was much higher than expert’s, contrary to +HierarchicalRL paper’s claims. Essentially, direct comparison would be unfair to these other papers without knowing +the exact hyperparameters they used to achieve their “best” performance. In light of this, our comparison gives the +benefit of doubt to, and uses the best performance from, these learning-based placement papers. All the above papers +compared step times to experts, and we do too. We do not compare to other algorithmic techniques for model parallelism +Manuscript submitted to ACM + +20 +Jeon et al. +Model +HierarchicalRL [50] +Placeto [2] +Baechi (m-SCT) +Inception-V3 +11 hrs 50 mins +1 hr 49 mins +1-10 seconds +NMT (GNMT) +1 day 21 hrs 14 mins +2 days 20 hrs 40 mins +1.2-48 seconds +Transformer +N/A +N/A +1-3 seconds +Table 3. Placement Time. Time to Generate a Placement for our target machine with 4 GPUs. +(listed in Section 1) because of either: their standalone nature [67] (making a TensorFlow/PyTorch comparison unfair), or +their limitation to Transformers [47, 63], or because they are already shown to be comparable to our performance [25]. +This also keeps our evaluation focused on comparing algorithmic approaches to RL approaches for model parallelism. +5.2 +Placement Time +Table 3 shows both: 1) measured placement times of Baechi, and 2) calculated placement times for two learning-based +techniques, namely: HierarchicalRL [50] and Placeto [2]. The numbers for HierarchicalRL and Placeto are normalized +quantities, both derived from numbers reported in Addanki et al. [2]. For these two systems, we multiply the fastest +step time among its reported placements, by the number of placement samples3. For instance, HierarchicalRL’s [50] +Inception-V3 placement training time is derived as a product of the reported final step time (1.19 s) and the number of +samples (35,800), giving 42,602 s, or 11 hrs 50 mins. +Hence, the numbers for these learning-based placers are their best-case performance. In comparison, we use the +worst-case placement times from Baechi, specifically from m-SCT which took the longest to generate a placement. Note +that all times in Table 3 exclude time to profile the graph, as profiling is a common baseline encountered by all the three +approaches shown. We find the profiling time to be low: about 10–12 s total for Inception-V3 and GNMT in Baechi-TF, +and about 11–14 s for Inception-V3 and Transformer in Baechi-PY. For instance, in Baechi-TF, this breaks down as 2-4 s +for warmup execution, 1–3 s for graph execution for profiles, and less an 1 s for parsing profile results. +Table 3 shows that Baechi places ML models orders of magnitude faster than the learning-based approaches. For +Inception-V3, Baechi reduces placement time, from 1.8–11.8 hours (using existing techniques [2, 50]), to under 10 s +in both Baechi-TF and Baechi-PY. Thus Baechi is 654×–42.6K× faster at placing Inception-V3. For GNMT, Baechi-TF +reduces placement time from several days to under 48 s. Thus Baechi is 3392×–206K× faster at placing GNMT. For +Transformer, Baechi-PY places it under 3 s. Because HierarchicalRL and Placeto [2, 50] did not include Transformers in +their evaluation, the corresponding entries are marked as not available in Table 3. +Overall, Baechi is 654×–206K× faster at placement compared to today’s learning-based approaches [2, 50]. +5.3 +Placement with Sufficient Memory +We next evaluate the effectiveness of the generated placement by measuring the step time of the placed model, i.e., +its time to execute 1 training step on an input data batch. Step time is a key metric as completion time on a training +set is directly proportional to step time. We first explore the scenario when each GPU has sufficient memory to run +the entire model. We compare against both: i) step time on a single GPU, which might be fast because it avoids the +overheads of communication, and ii) an expert-based placement scheme for placement on multiple GPUs. +3Even if one were to parallelize the learning-based placers, their resource usage would be similar to the normalized time metric we show. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +21 +Speedup over +Single GPU +Expert (4 GPUs) +Model +Batch +Size +Single +GPU +Expert +m-TOPO +m-ETF +m-SCT +m-ETF +m-SCT +m-ETF +m-SCT +Inception-V3 +32 +0.269 +0.269 +0.286 +0.269 +0.269 +0.00% (1 GPU Expert) +64 +0.491 +0.491 +0.521 +0.491 +0.491 +0.00% (1 GPU Expert) +GNMT +(length: 40) +128 +0.251 +0.214 +0.265 +0.224 +0.212 +12.1% +18.4% +-4.5% +0.9% +256 +0.474 +0.376 +0.481 +0.354 +0.369 +33.9% +28.5% +6.2% +1.9% +GNMT +(length: 50) +128 +0.319 +0.259 +0.348 +0.264 +0.267 +20.9% +19.5% +-1.9% +-3.0% +TensorFlow +256 +0.618 +0.484 +0.609 +0.502 +0.516 +23.1% +19.8% +-3.6% +-6.2% +Inception-V3 +32 +0.240 +0.240 +0.274 +0.241 +0.241 +0.00% (1 GPU Expert) +64 +0.461 +0.461 +0.537 +0.465 +0.462 +0.00% (1 GPU Expert) +Transformer +(length: 50) +64 +0.249 +0.257 +0.262 +0.242 +0.244 +2.9% +2.0% +6.2% +5.3% +PyTorch +128 +0.465 +0.462 +0.466 +0.451 +0.453 +3.0% +2.6% +2.4% +2.0% +Table 4. Baechi with Sufficient Memory. Average Step Times (Training) in seconds of Placed Model Graphs, and Speedup over Single +GPU and Expert Placements. 4 GPUs (unless otherwise mentioned). +The expert is a manual process and we do it as follows. For GNMT in TensorFlow, we use the technique of Wu et al. +[77]. Each LSTM layer in the encoder and decoder modules are placed on different GPUs. The embedding layer is placed +on the same GPU as the first LSTM layer. The output projection layer is placed on the same GPU as the last decoder +LSTM layer. For Inception-V3 in both TensorFlow and PyTorch, the expert is the single GPU placement, similar to +HierarchicalRL [50]. +For the Transformer model in PyTorch we use the common practice of putting the encoder on one device and the +decoder on another device [21]. +m-ETF, m-SCT –VS.– Single GPU, Expert. Table 4 shows the step times for the three algorithms in Baechi–namely +m-TOPO, m-ETF, and m-SCT—as well as the single GPU and expert. We show numbers for 2 batch sizes in each model, +and 2 sequence lengths in GNMT. We observe that for Inception-V3: 1) in TensorFlow, m-ETF and m-SCT find the same +device placements as the expert, i.e., place all operators in a single GPU. 2) In PyTorch m-ETF and m-SCT placements +use three and two GPUs respectively, but have the same step time as 1-GPU expert. 3) Compared to the expert, m-TOPO +step time’s is higher by 6.1–6.3% in Baechi-TF and by 14–17% in Baechi-PY for Inception-V3. This occurs because +m-TOPO splits the neural network between the Inception blocks, and hence the next inception block(s) are unable to +run until the previous block(s) finish. +In TensorFlow GNMT, first, compared to single GPU placement, m-ETF’s placements have step times that are +12.1–33.9% faster. The step time speedups for m-SCT over single GPU are between 18.4–28.5%. These observations +show that Baechi’s m-ETF and m-SCT are able to extract benefits of parallelism in spite of communication overheads. +Second, in GNMT, compared to the expert, m-ETF is between 4.5% slower and 6.2% faster in step times. Compared to +the expert, m-SCT is between 6.2% slower and 1.9% faster. +In PyTorch Transformer, m-SCT and m-ETF placements are 2.0-6.0% faster than single-GPU and expert placements. +They place only the decoder’s embedding and first multi-head attention layer on a separate device. Since this computation +is independent of the encoder, m-SCT and m-ETF exploit the parallelism in the model. The rest of the decoder requires +output of the encoder and is hence placed on the same device as the encoder (in contrast to the expert) to minimize +communication. +Manuscript submitted to ACM + +22 +Jeon et al. +Table 5. Baechi with Insufficient Memory. Average Step Times (Training) in seconds of Placed Model Graphs (Parentheses show +Slowdown compared to Sufficient Memory for the same algorithm). +Model +Batch +Size +Memory +Fraction +Single +GPU +Expert +m-TOPO +m-ETF +m-SCT +Inception-V3 +32 +0.3 +OOM +OOM +0.690 +(58.6%) +0.312 +(13.8%) +0.292 +(7.9%) +TensorFlow +GNMT +32 +0.3 +OOM +0.221 +(3.2%) +0.272 +(2.6%) +0.230 +(2.6%) +0.212 +(0.0%) +Inception-V3 +32 +0.3 +OOM +OOM +0.275 +(0.0%) +0.250 +(3.7%) +0.254 +(5.4%) +Inception-V3 +64 +0.4 +OOM +OOM +0.537 +(0.0%) +0.527 +(13.3%) +0.535 +(16.1%) +PyTorch +Transformer +64 +0.3 +OOM +0.257 +0.262 +(0.0%) +0.240 +(0.0%) +0.241 +(0.0%) +These observations show that Baechi’s m-ETF and m-SCT are able to generate placements with step times in the +same ballpark as the expert, while taking significantly less time to create a placement than the manual expert which +takes minutes to hours. +m-TOPO. Table 4 also shows that, Baechi-TF’s m-TOPO is significantly slower than m-ETF and m-SCT. m-TOPO’s step +times are 5.8%–26.4% slower than m-ETF and 5.8%–23.3% slower than m-SCT. After analysing m-TOPO we found that it +places most of the encoder’s LSTM layers at the first two GPUs, and most of the decoder LSTM layers at the other two +GPUs. However, this parallelization is offset negatively by the high data transfers between the kernel weight and the +LSTM cell operators for LSTM layers. Similarly, with Transformer in Baechi-PY, m-TOPO’s step time is 3.3%–8.2% +slower than m-ETF and m-SCT. Essentially m-TOPO fails to exploit the parallelism between the encoder and the decoder. +m-SCT vs. m-ETF. Theoretical analysis in [26] shows SCT beating ETF and one would expect the same with m-SCT +and m-ETF. In practice, the reverse is true—Table 4 shows that m-ETF’s step times are faster than m-SCT’s for 5 out of 6 +settings in Baechi-TF (it is slower only under sequence length 40, batch size 128), and faster or equal in 3 out of all 4 +settings in Baechi-PY (it is slower only under Inception-V3, batch size 64). +This behavior of m-SCT is because of two reasons. First, SCT’s optimality proof relies on the assumption that the +minimum operator computation time is larger than or equal to the maximum communication time. This does not +hold in our experimental machine—a 4 B GPU-GPU transfer takes 50–200 ms while, in TensorFlow, many operators +execute within 1 ms, and 67% of Inception-V3’s operators take under 50 ms. The m-SCT LP model (Section 2.4) assumes +parallel data transfers from an operator to all its children. Our experimental machine only allows sequential transfers +(Section 3.1.4)4. Overall, m-SCT and m-ETF are comparable in practice, with m-ETF having a slight edge in both +placement time and step time. +4Faster data transfers between GPUs, e.g., via NVLink [55], have the potential to make m-SCT more competitive than m-ETF, but this is outside our scope. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +23 +(a) + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 +GPU0 +GPU1 +GPU2 +GPU3 +Normalized Peak Memory + (i.e 1.0 = 30% of Max GPU memory, 2.4GB) +Inception V3 (100% memory) +Inception V3 (30% memory) +GNMT-40 (100% memory) +GNMT-40 (30% memory) +Memory load distribution with Baechi-Tensorflow +(b) + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 +GPU0 +GPU1 +GPU2 +GPU3 +Inception V3 (100% memory) +Inception V3 (30% memory) +Transformer (100% memory) +Transformer (30%memory) +Memory load distribution with Baechi-PyTorch +Fig. 7. Baechi Load Balance of Memory Usage using m-SCT. Dashed line is memory limit for each GPU (normalized). Note that 1.0 +on y axis corresponds to 30% of the max GPU memory (i.e. 2.4 GB in a 8 GB GPU) +5.4 +Placement with Insufficient Memory +Next, we limit per-GPU memory to a fraction of maximum available memory on the GPUs. Table 5 shows results for: 1) +Baechi TensorFlow with memory limited to 30%, i.e., from 8 GB down to 2.4 GB, for: Inception-V3 with batch size of 32, +and GNMT with batch size of 128 and sequence length 40; and 2) Baechi PyTorch: memory limited to 30% (Inception-V3 +with batch size of 32, Transformer with batch size 64) and 40% memory limit (Inception-V3 with batch size 64). +A few notes follow on configuration changes in the experiments with Baechi-TF. For GNMT, co-placement (Section +3.1.2) remains enabled and we use the same configuration as Section 5.3. For Inception-V3 we disable co-placement as +otherwise it generated a massive operator group, causing an Out of Memory error (OOM). Disabling co-placement +increases the number of operators to be placed from 2,620 to 7,077, and placement time from 1 s to 10.3 s. No configuration +changes were required for Baechi-PY experiments. +Effect on Step Time. Table 5 shows that the single GPU placer always suffers an OOM (Out of Memory) error. The +expert placer OOMs for Inception-V3 (in both TensorFlow and PyTorch), but succeeds for TensorFlow GNMT and +PyTorch Transformer. In comparison, all three variants of Baechi (m-TOPO, m-ETF, m-SCT) succeed in placing under +insufficient memory under all five settings. +For Inception-V3 in both Baechi-TF and Baechi-PY, only Baechi succeeds in placement. Compared to the sufficient +memory cases (Table 4), m-ETF and m-SCT provide step times that are only 13.8% and 7.9% worse in Baechi-TF and, +13.3% and 16.1% worse in Baechi-PY respectively. m-TOPO in TensorFlow degrades by 58.6% because of its disabled +co-placement, which ballooned communication along the graph’s critical path. In PyTorch there is no change in m-TOPO +since the algorithm does not depend on the maximum limit as long as it is more than m-TOPO’s per device cap. +For GNMT and Transformer, the overheads of all three Baechi algorithms and the expert are small (shown as % +numbers within parentheses), meaning that with insufficient memory Baechi is nearly as fast as when memory is +sufficient. +Load Distribution. Figure 7 shows the peak memory usage, normalized to the memory limit for each GPU (insufficient +memory case) for Baechi-TF and Baechi-PY. In both Baechi-TF and Baechi-PY, for Inception-V3, +Manuscript submitted to ACM + +24 +Jeon et al. +(a) + 0.95 + 1 + 1.05 + 1.1 + 1.15 + 1.2 + 1.25 + 1.3 + 1.35 + 1.4 +GNMT-40, m-SCT +(128, 100%) +GNMT-40, m-ETF +(128, 100%) +GNMT-40, m-SCT +(256, 100%) +GNMT-40, m-ETF +(256, 100%) +Noramalized Step Time +(i.e 1.0 = Same as un-perturbed step time) +GNMT-40 = GNMT with sequence length 40 +Max, Min step-time +Average step-time +Step-times +Sensitivity test on Baechi-Tensorflow +(b) + 0.96 + 0.98 + 1 + 1.02 + 1.04 + 1.06 + 1.08 + 1.1 +IV3, m-ETF +(32, 30%) +IV3, m-SCT +(32, 30%) +Tr, m-ETF +(64, 30%) +Tr, m-SCT +(64, 30%) +IV3, m-ETF +(64, 100%) +IV3, m-SCT +(64, 100%) +IV3 = Inception V3, Tr = Transformer +Max, Min step-time +Average step-time +Step-times +Sensitivity test on Baechi-PyTorch +Fig. 8. Baechi Sensitivity to Profiling Errors. All computation and communication times are perturbed randomly by up to 20% and +the step time for placement generated by Baechi is measured. Values in X-axis parentheses are (Batch Size, Memory Fraction Available) +with a 30% memory cap, a single GPU does not suffice, and that m-SCT relies on a mix of multiple GPUs. In particular, +2 of the 4 GPUs appear to be used more. This is because Inception-V3 has more barriers (sync points) than GNMT in +TensorFlow, limiting Inception-V3’s ability to parallelize effectively. +For TensorFlow GNMT and PyTorch Transformer, Baechi’s m-SCT is able to load-balance more evenly (than Inception- +V3) across the GPUs, even when memory is sufficient. In fact, for both these cases we found that m-SCT generates an +identical placement in both cases with sufficient and with insufficient memory. This fact is also true for the expert, m- +TOPO, and m-ETF. However specifically in case of GNMT with Baechi-TF, their step times are 2.6–3.2% higher than the +sufficient memory cases (Table 4).This slowdown is because of TensorFlow runtime memory optimizations. Concretely, +when the memory usage approaches its limit, the TensorFlow runtime resorts to certain memory optimizations to +decrease peak memory usage. For the expert placement, peak memory usage for one GPU device decreases from 2 +GB (83% of the memory limit) to 1.45 GB and thus the number of memory operations increases 6% under insufficient +memory. These memory optimizations do not kick in for m-SCT, making it faster than the expert. +Profile Sensitivity. To measure Baechi’s sensitivity to profiling errors, we perform runs where in each run all +computation and communication profiles are randomly and independently perturbed by up to ±20%. This should +account for errors in our time measurements as well as small speed differences between the device used to profile +the model and devices where the model actually runs. Figure 8 shows the perturbation in step-times of the resulting +placements, w.r.t. to step-time without any perturbation of the profiles. Compared to the unperturbed step times, the +step times with perturbed profiles remain within a fraction of 0.99× to 1.3× in Baechi-TF, and between 0.97× and 1.08× +times in Baechi-PY Thus we conclude that m-SCT and m-ETF are resilient to reasonable levels of errors in profiled +values. +5.5 +Benefit of Baechi Optimizations +5.5.1 +Benefit of Baechi-TF Optimizations. Table 6 shows the benefit from the combined optimizations of +Section 3.1.2 and 3.1.3 in Baechi-TF. We use Inception-V3 with batch size 32 and GNMT with batch size of 128. We use +the m-SCT variant of Baechi. The experimental setup has 4 GPUs with sufficient memory. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +25 +Table 6. Benefits of Baechi-TF Optimizations (Section 3.1.1). Number of Operators to be Placed, Placement Times in seconds, and +Average Step Times in seconds. m-SCT. +Model +Un-Optimized +Optimized +Num. +Ops +Placement +(seconds) +Step +(seconds) +Num. +Ops +Placement +(seconds) +Step +(seconds) +Inception-V3 +6884 +68.0 +0.302 +17 +0.9 +0.269 +GNMT +(length: 40) +18050 +275.1 +0.580 +542 +1.2 +0.212 +GNMT +(length: 50) +22340 +406.1 +0.793 +706 +2.4 +0.267 +Table 7. Benefits of communication protocol in Baechi-PY (Section 3.2.2). Step times in seconds without and with the protocol +Model +Algorithm +Without Protocol +With Protocol +% Change +Inception V3 (32, 30% memory) +m-ETF +0.252 +0.250 +0.0% +m-SCT +0.268 +0.254 +5.5% +Inception V3 (64, 40% memory) +m-ETF +0.551 +0.528 +4.3% +m-SCT +0.550 +0.535 +2.8% +Transformer (64, 100% memory) +m-ETF +0.246 +0.242 +0.0% +m-SCT +0.246 +0.244 +0.0% +Overall, we observe that Baechi-TF’s combined optimizations achieve 75.6×–229.3× speedup in placement times, +and 1.1×–3.0× speedup in step times. We discuss a few interesting aspects. Operator fusion (Section 3.1.3) reduces +both number of operators to be placed and thus also placement time. Forward-operator-based placement (Section +3.1.3) significantly speeds up placement. Concretely the latter optimization reduces the number of operators to be +placed 2.7× for Inception-V3 and 6.5×–7.0× for GNMT. This accelerates the placement times 13.7× for Inception-V3 and +20.2×–31.4× for GNMT. +Co-placement (Section 3.1.2) is efficient because it clusters operators. This reduces step times. While co-placement +does not change the operator count to be placed, it decreases placement time by reducing the overhead of calculating +schedulable times. +5.5.2 +Benefit of Baechi-PY Communication Protocol. To evaluate the communication protocol in Baechi- +PY (Section 3.2.2), we create a baseline plain wrapper. In it, each node transfers the inputs from devices of its parent +(if different from module’s device) by simply using blocking calls to .to() instead of using streams. Table 7 shows +the comparison of step times. Baechi-PY’s communication protocol gives up to 5.5% benefit with Inception-V3, and +very little benefit under Transformer. This is because, in PyTorch, both these models have a strong linear spine, which +creates fewer opportunities for parallelism. +6 +Discussion and Limitations +Algorithmic Approaches vs Learning-based Approaches. +When we first implemented m-ETF and m-SCT, the placed models had very high step times because communication- +intensive operators violated the SCT assumption (Table 1). We whittled away at this with a persistent effort at systems +design and optimizations (outlined in Section 3.1), which played a major role in bringing the step times down. Although +Manuscript submitted to ACM + +26 +Jeon et al. +our exploration was efficient and we cycled new techniques and optimizations on a weekly basis, it took 1 person-year +of effort to converge to what now appears in this paper for Baechi-TF, and an additional 1 person-year for Baechi-PY. +This is indicative of the difficulties associated with implementing scheduling algorithms on today’s open-source ML +systems (and in a sense shows why existing learning-based approaches are so attractive!). Nevertheless, our results +show that the benefits of algorithmic design were worth the exploratory pain. +Our experience also indicates reasons why developers (and companies!) often choose to “jump” so quickly towards +using learning-based (including RL-based) solutions for scheduling problems: fast time to design (optimization of +parameters and hyperparameters can be often be done as a rote task, rather than a creative task), and hence fast time +to production. However, this comes at the expense of latter pain points in generalizing learning-based approaches to +different architectures and models (in comparison, Baechi runs as-is, given an arbitrary model and a machine profile), +as well as the high times to generate a placement using learning-based approaches, which becomes a bottleneck in +the exploratory design phase when the developer is iteratively building and revising their model. We conclude that +learning-based techniques (for any problem) should not be built in isolation from, or in lieu of, algorithmic-based +approaches—but rather hand-in-hand with them. +Limitations of Baechi-TF. The peak memory usage of TensorFlow is highly dependent on the execution order of +operators [3]. So Baechi would benefit most if the framework (TensorFlow or PyTorch) faithfully executed operators in +the same order as specified by Baechi’s ES. For Baechi-PY we enforce this via the “Reordering Problem” (Section 4.4). +For TensorFlow while we do not enforce this ordering, and we observed in several runs of Baechi-TF that TensorFlow +deviated from this order, yet memory caps were not violated for m-SCT and m-ETF in Baechi-TF runs. It is possible +that if memory caps were tightened further (than 30%, compared to our experiments), engineering may be required for +Baechi-TF to force TensorFlow to follow the ES execution order. +Limitations of Baechi-PY. +(i) Correctness issues with in-place operations: Inplace operations may lead to race conditions and incorrectness. +Concretely, select modules in PyTorch can be made to modify the input tensors in-place, e.g., ReLU with in-place flag +set. Baechi-PY’s communication protocol (in Section 3.2.2) uses an independent tx stream to move out the tensors +from a device. If the subsequent module in the compute stream is in-place, it may modify the tensor while it is being +transferred out. This may lead to an incorrect tensor. While we did not encounter such a cases in evaluation, a simple +fix is to turn off the in-place feature for the module in question. This may however increase the memory consumption. +(ii) Weight sharing in Transformers: Currently the Assigner (Section 4.3) does not support cases where weights +are shared across multiple modules in the model (e.g., Transformers with embedder weight sharing [73]). +(iii) Model code modifications: Some operations like concatenate and add, which take multiple inputs, must be +defined as PyTorch modules. Only then can the Assigner ensure that tensors being concatenated or added are on the +same device. In most cases, this is a few lines of code change. For instance, in Inception-V3, concatenate is used 7 times +and add is used only once. +7 +Related work +Data Parallelism (DP). Data parallelism (a.k.a. DP) refers to training the same model replicas with multiple partitioned +data in parallel. This is motivated by increasing sizes of datasets. MALT [45] is a fault-tolerant, network-cost effective +solution for data parallel ML. Another common data parallelism framework is NESL [10], a first-order functional +language that enables developers to put irregular-parallel program in parallel devices. OptiML [65] is a domain-specific +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +27 +language (DSL). Most major ML frameworks offer support for data parallelism [1, 16, 56]. While DP typically replicates +the model on each device, ZeRO [60] eliminates this redundancy and reduces memory consumption in DP. DP is +orthogonal to model parallelism, and therefore DP techniques can be integrated into Baechi. +Model Parallelism. Compared to data parallelism, relatively fewer solutions exist for model parallelism. DistBelief [19] +and STRADS [40] require the user to manually specify device placement, while the systems in [42, 44] do not generalize +to arbitrary ML models. +As discussed in Section 1, reinforcement-learning based approaches have been popular lately to perform placement +for model parallelism, including work from Google [50, 51] and the Placeto system [2]. ColocRL [51] trains a sequence-to- +sequence model by RL to generate placements of manually grouped subsets of TensorFlow operators. HierarchicalRL [50] +substitutes the human intervention for grouping operators with an ML model and jointly trains the ML models for +operator grouping and device placements. Placeto [2] proposes an approach that transfers learned device placement +models to new ML models in order to minimize training times for the new model placements. +The original version of our paper [33] both inspired follow-up work [25], and also had parallel work [67], on +algorithmic approaches to model parallelism. However [67] is standalone, meaning that it is not integrated with +TensorFlow or PyTorch, making a fair comparison with Baechi difficult. Pesto [25] presents direct comparisons with +Baechi (Baechi-TF)—the most important metric of placement times are similar for Pesto and Baechi, with small +improvements for step time. Thus for practical purposes we consider Pesto to be comparable in performance to Baechi. +Works like PipeDream [27], GPipe [31], DAPPLE [22], PipeMare [79] introduce and optimize various aspects of +Pipeline Parallelism. In Pipeline Parallelism, the model is usually vertically split into contiguous stages. Amazon +Sagemaker recently introduced automating Model and Pipeline Parallelism on their platform recently, though their +code is proprietary[5, 38]. Techniques for pipeline parallelism can be integrated orthogonally into Baechi. +Model Parallelism for large language models. Recent progress on very large language models like GPT [14, 15] have +given rise to works that focus specifically on Model Parallelism for such models like Megatron-LM [63] and TeraPipe +[47]. However, these systems focus narrowly on Transformer models. Alpa [82] combines intra and inter-operator +parallelism. The inter-operator parallelism is limited to vertical splits and will not, for instance, place multiple parallel +branches of a model on different devices, thus making it different from model parallelism. +Classical Parallel Scheduling. Classical parallel scheduling, e.g., ETF [32] and SCT [26], has been widely used in task +scheduling on multiple computers. ETF and SCT are used as baselines by many subsequent works [20, 30, 52, 74, 80]. +None of these address memory constraints and a finite number of devices. For instance, Eyraud-Dubois et al. [20] +investigate the execution of tree-shaped task graphs using multiple processors, but without always obeying memory +restrictions. +TensorFlow Graph Optimizations. Existing techniques [69, 70] work only after the graph has been placed—e.g., to +improve operations’ performance—and thus are inapplicable. E.g., Running Grappler (TensorFlow’s graph optimizer) +generates an optimized graph protobuf, but it is unusable as it lacks certain metadata. Baechi’s targeted problem is +harder as we have to both optimize the graph and do placement. +8 +Conclusions +We have proposed algorithmic solutions to model parallelism, useful in scenarios where devices are memory-constrained +or neural networks are massive. Among our three algorithms (m-ETF, m-TOPO, m-SCT), the m-SCT algorithm is +Manuscript submitted to ACM + +28 +Jeon et al. +provably within a constant factor of the optimal achievable training time. We have implemented these algorithms into +our new Baechi system, as two systems Baechi-TF and Baechi-PY which are respectively usable in a modular manner +with TensorFlow and PyTorch. +Experimental results showed that, across TensorFlow and PyTorch, our approaches reduce placement time by a +factor of between 654×–206000× compared to today’s state-of-the-art placement approaches which are learning-based, +while increasing step time (makespan) by only up to 6.2% compared to expert placers. When memory is constrained +further, while single GPU and expert placers suffer OOM errors, Baechi’s algorithms, especially m-SCT and m-ETF, +were able to place successfully. Compared to sufficient memory the step times suffered an increase of only up to 13.8% +in TensorFlow and 16.1% in PyTorch. Further, Baechi-TF’s optimizations help reduce placement time by 75.6×–229.3×, +and step time by 1.1×–3.0×. We also conclude that m-SCT and m-ETF perform comparably, with m-ETF having a slight +edge for slower networks. +The original version of our paper [33] inspired follow-up work [25] along with parallel work [67], on algorithmic +approaches to model parallelism. Together, this new generation of algorithms for model parallelism offers the promise +of speed, generalizability, predictability, and analyzability. These will be invaluable as learning models, both training +and inference, move closer to edge devices and human-facing devices. +Code. Baechi’s code is openly available at the following link: +http://dprg.cs.uiuc.edu/downloads.php +Acknowledgments +This work was supported in part by the following grants: NSF IIS 1909577, and NSF CNS 1908888; as well as by generous +gifts from Capital One, Schlumberger, and Microsoft. 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We extend it to the case where the devices have finite memory. For any given 𝑝 devices, let 𝜔𝑝 +m-etf be the m-ETF +makespan and 𝜔𝑝 +opt be the optimal makespan achieved using devices with infinte memory and zero device-to-device +communication costs. +Theorem 1. The makespan of m-ETF, 𝜔𝑛 +m-etf is at most (2 + 𝜌)𝜔𝑅 +opt, where 𝑅 is an integer < 𝑛 (computed in equation 11). +Proof. Let 𝐾 = +𝑛 · 𝑀 +�𝑚 +𝑖=1 𝑑𝑖 +, where for any operator (task) 𝑖 in 𝐺, 𝑑𝑖 is the size of memory required by 𝑖. Intuitively, 𝐾 is +the ratio of the total memory available from all devices to the total memory required by the model. Thus 𝐾 > 1, and for +practical purposes we can assume that 𝐾 is sufficiently larger than 1. At each step, m-ETF greedily matches a ready task +to an available device. Specifically, a device is said to be available if there is neither a task currently running nor has +been scheduled to run on that device. A task is said to be ready if all of its predecessors have completed. Let 𝐼 and 𝐴 be +the set of available devices and ready tasks at a given step respectively. When a task completes, 𝐼 is updated to include +the recently free device and all of the task’s children that are ready are added to 𝐴. +At each such step, a device 𝑑 is said to be memory-sufficient (MS) if the remaining free memory on 𝑑 is greater than +the memory requirement of each task in 𝐴. If 𝑑 has insufficient memory for even a single task in 𝐴, we say 𝑑 is not MS +thereafter. It is removed from 𝐼 and is not considered for any further placement. +The time (0,𝜔m-etf) can be partitioned into two distinct sets. Set A containing the time-periods when all the MS +devices (in that time-period) are busy and set B when at least one MS is idle. Suppose B is the disjoint union of intervals +(𝑏𝑙𝑖,𝑏𝑟𝑖) i.e, +B = (𝑏𝑙1,𝑏𝑟1) ∪ (𝑏𝑙2,𝑏𝑟2) ∪ · · · ∪ (𝑏𝑙𝑞,𝑏𝑟𝑞) +where 𝑏𝑙1 < 𝑏𝑟1 < 𝑏𝑙2 < 𝑏𝑟2 · · · < 𝑏𝑙𝑞 < 𝑏𝑟𝑞. +Lemma (Theorem 3.2 in [32]). We can find a chain of tasks, +𝑋 : 𝑇𝑙 → 𝑇𝑙−1 → · · · → 𝑇1 +such that +𝑞 +∑︁ +𝑖=1 +(𝑏𝑟1 − 𝑏𝑙1) ≤ +𝑙∑︁ +𝑗=1 +𝑡(𝑇𝑗) + +𝑙−1 +∑︁ +𝑗=1 +𝑐𝑗 (𝑗+1) +That is, the total time period of B will be covered by computation and communication times along the chain 𝑋. We +will denote �𝑙−1 +𝑗=1 𝑐𝑗 (𝑗+1) by 𝐶𝑋 . For proof, we refer the reader to Theorem 3.2 in [32]. +Let 𝑟 be the number of devices that remain MS until the end of m-ETF (i.e at time 𝜔m-etf). With 𝐾 > 1, we will have +𝑟 >= 1. Let ˆB be the set of all time-periods when atleast one of these 𝑟 devices is idle. Note that ˆB ⊆ B, thus the chain +𝑋 from A will cover ˆB as well. Thus we have: +∑︁ +𝑟 +𝑡(𝜙𝑖) ≤ 𝑟 × +∑︁ +𝑇𝑗 ∈𝑋 +𝑡(𝑇𝑗) + 𝑟 × 𝐶𝑋 +(3) +where 𝜙𝑖 is the set of times when the device 𝑑𝑖 is idle in (0,𝜔m-etf) and 𝑡(𝑇) is computation time of task 𝑇. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +33 +Let 𝜔𝑝 +opt be the optimal makespan on 𝑝 devices with no memory limits and no communication costs. Since the +makespan on any number of devices is at least as large as a chain in the graph, we have +𝜔𝑟 +opt ≥ +∑︁ +𝑋 +𝑡(𝑇𝑗) . . . (𝑖), +𝜔𝑛 +opt ≥ +∑︁ +𝑋 +𝑡(𝑇𝑗) . . . (𝑖𝑖) +(4) +Also, the net computation time of 𝐺 can be bounded as: +∑︁ +𝑇𝑗 ∈𝐺 +𝑡(𝑇𝑗) ≤ 𝑟 × 𝜔𝑟 +opt . . . (𝑖), +∑︁ +𝑇𝑗 ∈𝐺 +𝑡(𝑇𝑗) ≤ 𝑛 × 𝜔𝑛 +opt . . . (𝑖𝑖) +(5) +Now we bound the m-ETF makespan. Consider the 𝑟 devices, their idle time and the jobs running on them: +𝜔𝑛 +m-etf = 1 +𝑟 +�∑︁ +𝑟 +𝑡(𝑇𝑗) + +∑︁ +𝑟 +𝑡(𝜙𝑖) +� +(6) +≤ 1 +𝑟 +�∑︁ +𝐺 +𝑡(𝑇𝑗) + +∑︁ +𝑟 +𝑡(𝜙𝑖) +� +(7) +Using 3, 4(i) and 5(i), +𝜔𝑛 +m-etf ≤ 1 +𝑟 +� +2𝑟 × 𝜔𝑟 +opt + 𝑟 × 𝐶𝑋 +� += 2𝜔𝑟 +opt + 𝐶𝑋 +Similarly, using 3, 4(ii) and 5(ii), +𝜔𝑛 +m-etf ≤ 1 +𝑟 +� +(𝑛 + 𝑟) × 𝜔𝑛 +opt + 𝑟 × 𝐶𝑋 +� += +�𝑛 + 𝑟 +𝑟 +� +𝜔𝑛 +opt + 𝐶𝑋 +Note that 𝑟 will vary depending on the exact topological order considered for the m-ETF. So we define 𝑅 as the +minimum 𝑟 across all possible topological ordering of the graph. Thus we have, +𝜔𝑛 +m-etf ≤ min +� +2𝜔𝑅 +opt, +�𝑛 + 𝑅 +𝑅 +� +𝜔𝑛 +opt +� ++ 𝐶𝑋 +(8) +Further, with 𝜌 as the ratio between maximum communication ttime and minimum computation time, we have +𝐶𝑋 ≤ 𝜌𝜔𝑛 +opt (also 𝐶𝑋 ≤ 𝜌𝜔𝑅 +opt ) +Thus we have, +𝜔𝑛 +m-etf ≤ +� +1 + 𝑛 +𝑅 + 𝜌 +� +𝜔𝑛 +opt +(9) +Alternatively, using the bound with 𝜔𝑅 +opt, +𝜔𝑛 +m-etf ≤ (2 + 𝜌)𝜔𝑅 +opt +(10) +Here we note that bound in Equation 10 is of the same form as the original bound on ETF given in [32], where 𝑅 +replaces 𝑛. Thus the makespan of m-ETF, like ETF, is within a constant factor of the optimal +Finally, 𝑅 can be computed as follows. Let the largest memory requirement of any task in 𝐺 be 𝐽 × 𝑀. Since the +devices become non-MS when they can not place any of the available task in A , a memory of only (1 − 𝐽)𝑀 is use-able +at each device in the worst case. Thus by greedily filling in the tasks onto devices, we get: +Manuscript submitted to ACM + +34 +Jeon et al. +𝑅 ≥ 𝑛 − +� �𝑚 +𝑖=1 𝑑𝑖 +(1 − 𝐽)𝑀 +� += 𝑛 − +𝑛 +(1 − 𝐽)𝐾 +Rounding it up, we have, +𝑅 = +� +𝑛 +� +1 − +1 +(1 − 𝐽)𝐾 +�� +(11) +□ +B +Optimality Analysis of m-SCT +We now formally prove that m-SCT’s approximation ratio to optimal is an additive constant away from SCT’s approxi- +mation ratio. Since SCT itself was known to be within a constant factor of optimal [26], our result means that m-SCT +is also within a constant factor of optimal. Recall that we assume 𝜌 - the ration of maximum communication time to +minimum computation time (defined in Table: 1) - is less than 1. +We will use similar notation to our analysis for m-ETF. Let 𝐾 = +𝑛 · 𝑀 +�𝑚 +𝑖=1 𝑑𝑖 +, where for any operator (task) 𝑖 in 𝐺, 𝑑𝑖 is +the size of memory required by 𝑖. We will define 𝐽 as the ratio between largest memory requirement from a single task +and 𝑀. Formally, 𝐽 = max𝑖 ∈[𝑚] +𝑑𝑖 +𝑀 . +Let 𝑠𝑖 be the start time of task 𝑖 in m-SCT, and 𝑠∞ +𝑖 +be the start time of task i in the infinite device SCT algorithm. Let +𝑢𝑗 be the time where a task 𝑗 becomes urgent, which is exactly the earliest time when task 𝑗 can start on any device. +Formally, 𝑢𝑗 = 𝑚𝑎𝑥𝑖→𝑗 ∈𝐸(𝐺)𝑠𝑖 + 𝑝𝑖 + 𝑐𝑖𝑗. +Similar to m-ETF, we will say a device 𝑑 is memory sufficient (abbreviated as MS) at time 𝑇 if and only if remaining +free memory on 𝑑 is greater than the memory requirement of each task in 𝐴. Finally, we will use 𝑟 to denote the number +of devices that are MS throughout the scheduling process. +We will now analyse the approximation ratio of m-SCT with three steps. First we will show that not many tasks are +impacted by devices going out of memory. Next we will show that any MS device must be idle only for a limited time. +Our proof for this step follows a similar outline to the proof of Theorem 3 in [26], but our proof is significantly shorter +due to a condensed case analysis. Finally, we will bound the makespace of m-SCT by summing up the idle and busy +time on MS devices. +Lemma 2. There are at most 𝑛 − 𝑟 task pairs (𝑖, 𝑗) such that 𝑗 is 𝑖’s favourite child, however when 𝑗 is scheduled, the +device 𝑑 where 𝑖 is scheduled on does not have sufficient memory for task 𝑗. +Proof. Since 𝑖 is scheduled on device 𝑑, 𝑑 must be memory sufficient when 𝑖 is scheduled, but is no longer memory +sufficient sometime after 𝑖 is scheduled (since 𝑑 does not have sufficient memory for task 𝑗). Since there are in total 𝑛 +devices and 𝑟 devices are always memory sufficient throughout m-SCT, there must only be 𝑛 − 𝑟 events where a device +transition from being memory sufficient to not memory sufficient. +□ +Lemma 3 (Variant of Lemma 6 in [26]). Given two time units 𝑠′ ≤ 𝑠 such that 𝑠 − 𝑠′ ≤ 𝑐𝑚𝑎𝑥, let 𝑖 be a task such that +𝑠𝑖 ≤ 𝑠′ ≤ 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑚𝑎𝑥, then any busy or awake device at 𝑠′ is free for at most max(𝑠′ − 𝑠𝑖,𝑐𝑚𝑎𝑥) time during [𝑠𝑖,𝑠]. +Proof. +(1) If a device 𝑑 is busy at 𝑠′. Let task 𝑎 be the task that is running at time 𝑠′. There are two possibilities: +• If task 𝑎 started before 𝑠𝑖, then device 𝑑 is busy for at least 𝑠′ −𝑠𝑖 time, thus free for at most 𝑠 −𝑠′ ≤ 𝑐𝑚𝑎𝑥 time. +• On the other hand, if task 𝑎 started at some time 𝑠∗ where 𝑠𝑖 ≤ 𝑠∗ ≤ 𝑠′, then either task 𝑎 is still being executed +at 𝑠, or task 𝑎 has completed at 𝑠. In the first case device 𝑑 is free for at most 𝑠∗ −𝑠𝑖 ≤ 𝑠′ −𝑠𝑖 time. In the second +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +35 +case device 𝑑 is busy for at least 𝑘𝑎 ≥ 𝑐𝑚𝑎𝑥 ≥ 𝑠 − 𝑠′ time, and thus is free for at most 𝑠 − 𝑠𝑖 − (𝑠 − 𝑠′) = 𝑠′ − 𝑠𝑖 +time. +We conclude that device 𝑑 must be busy for at least min{𝑠′ − 𝑠𝑖,𝑠 − 𝑠′} time during [𝑠𝑖,𝑠]. +(2) If a device is awake at 𝑠′, let 𝑎 be the last task on 𝑑 before 𝑠′ and let 𝑠∗ be when 𝑎 finishes. Then we know by the +nature of our algorithm that some task 𝑏 will start on 𝑑 no later than 𝑠∗ + 𝑐𝑚𝑎𝑥. Since 𝑠 − 𝑠′ ≤ 𝑐𝑚𝑎𝑥, we know +that 𝑏 is not yet finished at time 𝑠. Therefore either task 𝑎 starts after 𝑠𝑖 (which means the device is busy for at +least 𝑘𝑎 ≥ 𝑐𝑚𝑎𝑥 time and free for at most 𝑠 − 𝑠𝑖 − 𝑐𝑚𝑎𝑥 ≤ 𝑠′ − 𝑠𝑖 time), or the device is vacant for at most 𝑐𝑚𝑎𝑥 +time. +□ +Lemma 4. Assume that task 𝑗’s favourite parent𝑖∗’s device is MS during time period [𝑠𝑖,𝑠𝑗]. Then there exists a predecessor +𝑖 of 𝑗 such that the total amount of idle time during [𝑠𝑖,𝑠𝑗] on any device 𝑑 that is MS throughout the period is at most +𝑠∞ +𝑗 − 𝑠∞ +𝑖 . +Proof. Note that since task 𝑗’s favourite predecessor 𝑖∗’s device is MS during [𝑠𝑖,𝑠𝑗], it is possible to schedule 𝑗 on +the same device as 𝑖∗. This fact will be used in the case analysis. +Let 𝑖 be a predecessor of 𝑗 such that 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗 is maximized (namely, 𝑢𝑗 = 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗). Notice also that after 𝑗 +becomes urgent at 𝑢𝑗 and before 𝑗 is scheduled, all memory sufficient devices must be busy (otherwise 𝑗 would have +been scheduled on a device). Hence for any 𝑇 < 𝑠𝑗, the total vacant time for an MS device during [𝑇,𝑠𝑗] is equal to its +total vacant time during [𝑇,𝑢𝑗]. Now we will discuss three different scenarios and prove that in each scenario, an MS +device is vacant for at most 𝑠∞ +𝑗 − 𝑠∞ +𝑖 +time during [𝑠𝑖,𝑠𝑗]. +(1) When 𝑗 is not the favorite child of 𝑖, we know that in the infinite device SCT algorithm, 𝑖 and 𝑗 are scheduled on +different devices. Hence 𝑠∞ +𝑗 +≥ 𝑠∞ +𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗. On the other hand, in m-SCT, after 𝑗 becomes urgent (𝑗 becomes +urgent at time 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗) and before 𝑗 is scheduled, any MS device must be busy. Therefore the amount of +vacant time on each MS device during [𝑠𝑖,𝑠𝑗] must be at most 𝑘𝑖 + 𝑐𝑖𝑗. +(2) When 𝑗 is the favorite child of 𝑖, but 𝑖’s device is not awake when task 𝑖 ends, we know that the time 𝑗 can be ready +on𝑖’s device is the same as 𝑗’s urgent time, which means there is another task𝑤 such that𝑠𝑤+𝑘𝑤+𝑐𝑤𝑗 = 𝑠𝑖+𝑘𝑖+𝑐𝑖𝑗, +but 𝑗 is not 𝑤’s favorite child. We can now use exactly the same argument as in the first case to prove that vacant +time on any MS device during [𝑠𝑖,𝑠𝑗] must be at most 𝑘𝑖 + 𝑐𝑖𝑗. +(3) When 𝑗 is the favorite child of 𝑖, and 𝑖’s device is awake when task 𝑖 ends. Denote the time 𝑗 becomes ready on +𝑖’s device as 𝑟𝑒𝑎𝑑𝑦(𝑗), there must exist some 𝑗’s predecessor 𝑦 ≠ 𝑖 such that 𝑠𝑦 + 𝑘𝑦 + 𝑐𝑦𝑗 = 𝑟𝑒𝑎𝑑𝑦(𝑗). Since 𝑖’s +device is awake when task 𝑖 ends, task 𝑗 will be scheduled on 𝑖’s device if it is still idle by 𝑟𝑒𝑎𝑑𝑦(𝑗). Hence a task +𝑤 (which is either 𝑗 or an urgent task) has to be scheduled on the device of 𝑖 at or before 𝑟𝑒𝑎𝑑𝑦(𝑗). We will now +consider the predecessor successor pair (𝑦, 𝑗), and prove that during [𝑠𝑦,𝑢𝑗] the vacant time on any MS machine +is at most 𝑠∞ +𝑗 − 𝑠∞ +𝑦 . +• If 𝑤 = 𝑗, note that 𝑗 is not 𝑦’s favorite child. Hence in the infinite device SCT, 𝑗 and 𝑦 are not on the same +device. We hence conclude that 𝑠𝑗 − 𝑠𝑦 = 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 = 𝑘𝑦 + 𝑐𝑦𝑗 ≤ 𝑠∞ +𝑗 − 𝑠∞ +𝑦 . +• If 𝑤 ≠ 𝑗, then 𝑤 must be urgent (the only tasks that are allowed to be scheduled on an awake machine is the +favorite child and urgent tasks). Hence at the start time of 𝑤, it must be the case that all MS devices are either +busy or awake (because if there is a free MS device, k would have been scheduled on it). By Lemma 3, any +busy or awake device at the start time of 𝑤 can only be vacant for at most 𝑚𝑎𝑥(𝑠𝑤 − 𝑠𝑦,𝑐𝑚𝑎𝑥) time during +[𝑠𝑦,𝑢𝑗]. Now we will upper bound 𝑚𝑎𝑥(𝑠𝑤 −𝑠𝑦,𝑐𝑚𝑎𝑥) using the facts 1) 𝑤 happens before 𝑟𝑒𝑎𝑑𝑦(𝑗), but after +Manuscript submitted to ACM + +36 +Jeon et al. +task 𝑖 is completed (namely, after 𝑠𝑖 + 𝑘𝑖) and 2) 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 ≥ 𝑘𝑦 ≥ 𝑐𝑚𝑎𝑥. +𝑚𝑎𝑥(𝑠𝑤 − 𝑠𝑦,𝑐𝑚𝑎𝑥) ≤ 𝑚𝑎𝑥(𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦,𝑐𝑚𝑎𝑥) = 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦. +Since in the infinite device SCT, 𝑗 and 𝑦 are not on the same device, we now conclude that the total vacant +time on an MS device must be at most 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 = 𝑘𝑦 + 𝑐𝑦𝑗 ≤ 𝑠∞ +𝑗 − 𝑠∞ +𝑦 . +□ +Lemma 5. Assume that task 𝑗’s favourite parent 𝑖∗’s device is not MS during time period [𝑠𝑖,𝑠𝑗]. Then there exists a +predecessor 𝑖 of 𝑗 such that the total amount of idle time during [𝑠𝑖,𝑠𝑗] on any device 𝑑 that is MS throughout the period is +at most 𝑠∞ +𝑗 − 𝑠∞ +𝑖 + 𝑐𝑖𝑗. +Proof. As argued in Lemma 4, any MS device must be busy after 𝑢𝑗. Let 𝑖 be a predecessor of 𝑗 such that 𝑠𝑖 +𝑘𝑖 +𝑐𝑖𝑗 +is maximized (namely, 𝑢𝑗 = 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗). Then 𝑢𝑗 − 𝑠𝑖 = 𝑘𝑖 + 𝑐𝑖𝑗 ≤ 𝑠∞ +𝑗 − 𝑠∞ +𝑖 + 𝑐𝑖𝑗 (because even with infinite number of +devices, task 𝑖 must be fully executed before task 𝑗 starts). +□ +Let 𝑅 be the minimum 𝑟 across all possible graph configurations with memory availability parameter 𝐾. We will use +𝜔𝑝 +m-sct and 𝜔𝑝 +sct to denote the makespan of m-SCT (with memory limit) and SCT (without memory limit) with 𝑝 devices +respectively. Analogously, we will use 𝜔𝑝 +m-opt and 𝜔𝑝 +opt be the optimal makespan on 𝑝 devices with memory limit and +with no memory limit respectively. We will use 𝛼 to denote the approximation of the infinite device SCT (against the +optimal makespan with infinite devices). +Theorem 6. The makespan of m-SCT is at most ( 𝑝 +𝑅 + 𝛼) · 𝜔𝑝 +opt + (𝑛−𝑅) +𝑅 +· 𝑐𝑚𝑎𝑥 for any 𝑝. +Proof. Let 𝐷𝑀𝑆 be the set of all devices that are MS throughout the m-SCT algorithm. We know that |𝐷𝑀𝑆 | = 𝑟. It +is clear that the total amount of computation time spent on devices in 𝐷𝑀𝑆 is at most the sum of computational time +for all tasks, which is at most 𝜔𝑟 +opt · 𝑟. +Now we will count the amount of time a device 𝑑 ∈ 𝐷𝑀𝑆 is idle. WLOG, let 𝑇1 be the task that finishes last in m-SCT +and let 𝑇𝑙 → 𝑇𝑙−1 → · · · → 𝑇1 be the chain of task in 𝐺 ending at 𝑇1 such that 𝑇𝑙 is a source. Before the start time 𝑠𝑙 of +𝑇𝑙, all MS devices must be busy, because 𝑇𝑙 is urgent from time 0 and would have been scheduled on a device as soon +as it becomes idle. Let 𝑛𝑑 be the number of task pairs (𝑖, 𝑗) such that 𝑖 is 𝑗’s favourite parent but 𝑖’s parent is not MS +when task 𝑗 starts. By Lemma 2, 4 and 5 we know that during [𝑠𝑙,𝜔𝑝 +m-sct] = [𝑠𝑙,𝑠1 + 𝑘1] (𝑠1, 𝑘1 are the start time and +computation time of 𝑇1 respectively) , the amount of time 𝑑 is idle is at most +�� +� +𝑙−1 +∑︁ +𝑗=1 +� +𝑠∞ +𝑗 − 𝑠∞ +𝑗+1 +��� +� ++ 𝑛𝑑 · 𝑐𝑚𝑎𝑥 ≤ 𝜔∞ +sct + 𝑛𝑑 · 𝑐𝑚𝑎𝑥. +Summing these all up for all devices in 𝐷𝑀𝑆 we get that the total idle time across all devices in 𝐷𝑀𝑆 is at most +𝑟 · 𝜔∞ +sct + (𝑛 − 𝑟) · 𝑐𝑚𝑎𝑥. +Manuscript submitted to ACM + +Baechi: Fast Device Placement of Machine Learning Graphs +37 +We now conclude that +𝑟 · 𝜔𝑝 +m-sct ≤ 𝑟 · 𝜔𝑟 +opt + 𝑟 · 𝜔∞ +sct + (𝑛 − 𝑟) · 𝑐𝑚𝑎𝑥 +⇒ 𝜔𝑝 +m-sct ≤ 𝜔𝑟 +opt + 𝜔∞ +sct + 𝑛 − 𝑟 +𝑟 +· 𝑐𝑚𝑎𝑥 +≤ 𝜔𝑟 +opt + 𝛼 · 𝜔∞ +opt + 𝑛 − 𝑟 +𝑟 +· 𝑐𝑚𝑎𝑥 +≤ 𝜔𝑅 +opt + 𝛼 · 𝜔∞ +opt + 𝑛 − 𝑅 +𝑅 +· 𝑐𝑚𝑎𝑥 +(because 𝑅 ≤ 𝑟). +Lastly, observe that (without memory limit), the optimal makespan with 𝑅 devices is at most 𝑝 +𝑅 times the optimal +makespan with 𝑝 devices. Also, 𝜔∞ +opt ≤ 𝜔𝑅 +opt. Hence 𝜔𝑛 +m-sct ≤ ( 𝑝 +𝑅 + 𝛼) · 𝜔𝑝 +opt + (𝑛−𝑅) +𝑅 +· 𝑐𝑚𝑎𝑥. One could minimize the +RHS over all 𝑝 to get the best upper bound. +□ +Using the same analysis as for m-ETF, we know that +𝑅 = +� +𝑛 +� +1 − +1 +(1 − 𝐽)𝐾 +�� +. +Manuscript submitted to ACM + diff --git a/OtFAT4oBgHgl3EQfyx73/content/tmp_files/load_file.txt b/OtFAT4oBgHgl3EQfyx73/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af08b67bbe2e5b01dd8c0dfd0646874d2d7d3ad9 --- /dev/null +++ b/OtFAT4oBgHgl3EQfyx73/content/tmp_files/load_file.txt @@ -0,0 +1,2004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf,len=2003 +page_content='Baechi: Fast Device Placement of Machine Learning Graphs BEOMYEOL JEON,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA LINDA CAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA CHIRAG SHETTY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA PALLAVI SRIVASTAVA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA JINTAO JIANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA XIAOLAN KE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA YITAO MENG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA CONG XIE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA INDRANIL GUPTA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' USA Machine Learning graphs (or models) can be challenging or impossible to train when either devices have limited memory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' or models are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To split the model across devices, learning-based approaches are still popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While these result in model placements that train fast on data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', low step times), learning-based model-parallelism is time-consuming, taking many hours or days to create a placement plan of operators on devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We present the Baechi system, the first to adopt an algorithmic approach to the placement problem for running machine learning training graphs on small clusters of memory-constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We integrate our implementation of Baechi into two popular open-source learning frameworks: TensorFlow and PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our experimental results using GPUs show that: (i) Baechi generates placement plans 654×–206K × faster than state-of-the-art learning-based approaches, and (ii) Baechi-placed model’s step (training) time is comparable to expert placements in PyTorch, and only up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% worse than expert placements in TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We prove mathematically that our two algorithms are within a constant factor of the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our work shows that compared to learning-based approaches, algorithmic approaches can face different challenges for adaptation to Machine learning systems, but also they offer proven bounds, and significant performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' CCS Concepts: • Computer systems organization → Cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Additional Key Words and Phrases: Machine Learning Systems, Placement Algorithms, Constrained Memory, TensorFlow, PyTorch, Distributed Systems This submission is an extended version of "Baechi: Fast Device Placement of Machine Learning Graphs - Beomyeol Jeon, Linda Cai, Pallavi Srivastava, Jintao Jiang, Xiaolan Ke, Yitao Meng, Cong Xie, and Indranil Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Proceedings of the 11th ACM Sympo- sium on Cloud Computing (Virtual Event, USA) (SoCC ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 416–430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1145/3419111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3421302".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Document detailing the additional contributions has been attached as a supplementary material Work done while the authors were at University of Illinois at Urbana-Champaign, USA Authors’ addresses: Beomyeol Jeon, University of Illinois at Urbana-Champaign, Urbana, USA, bj2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Linda Cai, Princeton University, USA, tcai@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Chirag Shetty, University of Illinois at Urbana-Champaign, USA, cshetty2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Pallavi Srivastava, University of Illi- nois at Urbana-Champaign*, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Jintao Jiang, University of Illinois at Urbana-Champaign*, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Xiaolan Ke, University of Illinois at Urbana-Champaign*, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Yitao Meng, University of Illinois at Urbana-Champaign*, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Cong Xie, University of Illinois at Urbana-Champaign*, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Indranil Gupta, University of Illinois at Urbana-Champaign, USA, indy@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='08695v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='DC] 20 Jan 2023 2 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ACM Reference Format: Beomyeol Jeon, Linda Cai, Chirag Shetty, Pallavi Srivastava, Jintao Jiang, Xiaolan Ke, Yitao Meng, Cong Xie, and Indranil Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi: Fast Device Placement of Machine Learning Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1, 1 (January 2023), 37 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='nnnnnnn 1 Introduction Distributed Machine Learning frameworks use more than one device in order to train large models and allow for larger training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This has led to multiple open-source systems, including TensorFlow [1], PyTorch [56], MXNet [16], Theano [71], Caffe [34], CNTK [62], and others [41, 64, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Many of these systems use data parallelism, wherein each device (GPU) runs the entire model, and multiple items are inputted and trained in parallel across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Yet, the increasing size of Machine Learning (ML) models and scale of training datasets is quickly outpacing available GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance the vanilla implementation of a 1000-layer deep residual network required 48 GB memory [17], which is much larger than the amount of RAM available on a typical COTS (Commercial Off-the-Shelf) device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Even after further optimizations to reduce memory cost, the ML model still required 7 GB memory, making it impossible to run an entire model on a single device with limited memory, as well as prohibitively expensive on public clouds like AWS [6], Google Cloud [24], and Azure [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At the same time, today, ML training is gravitating towards being run among small collections of memory-constrained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' These include small groups of cheap devices like edge devices (for scenarios arising from Internet of Things and Cyberphysical systems), Unmanned Aerial Vehicles (UAVs or drones), and to some extent even mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, real-time requirements [48, 81], privacy needs [11, 12], or budgetary constraints, necessitate training only using nearby or in-house devices with limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' These two trends—increasing model graph sizes and growing prevalence of puny devices being used to train the model graph—together cause scenarios wherein a single device is insufficient and results in an Out of Memory (or OOM) exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For example, we found that the Google Neural Machine Translation (GNMT) [77] model OOMs on a 4 GB GPU even with conservative parameters: batch size 128, 4 512-unit long short-term memory (LSTM) layers, 30K vocabulary, and sequence length 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This problem is traditionally solved by adopting model parallelism, wherein the ML model graph is split across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Today, a popular way to accomplish model parallelism in industry is to use learning-based approaches to generate the placement of operators on devices, most commonly by using Reinforcement Learning (RL) or variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Significant in this space are works from Google [50, 51] and the Placeto system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A learning-based approach learns iteratively (via RL) and adjusts the placement on the target cluster, with the goal of minimizing execution time for each training step in the placed model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', its step time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While learning-based approaches achieve step times around those obtained by expert placements, they can unfor- tunately take an inordinately long time to generate their placement plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, using one state-of-the-art learning-based approach [2], NMT models require 94,000 steps during the learning-based placement, and even with a conservatively low estimate of runtime per learning step of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='63 seconds, the total placement time would come to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='67 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' One might possibly apply parallelization techniques [35, 36, 75] to the learning model being used to perform placement, in order to speed it up, but the total incurred resource costs would stay just as high—hence, parallelization is orthogonal to our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Such long waits are cumbersome and even prohibitive at model development time, when the software developer needs to make many quick and ad-hoc deployments [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In fact, studies of analytics clusters reveal that most analytics Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 3 job runs tend to be short [18, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, the step time for a typical model graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', NMT or Inception-V3), to train on a single data batch, is O(seconds) on a typical GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Overwhelming this time with learning-based placement times which span hours, significantly inhibits the developer’s agility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Additionally, a learning-based placement run works only for a target cluster and a given model graph with fixed hyperparameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', batch size, learning rate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the model graph were to be transitioned to a different cluster with different GPU specs, the learning has to be repeated all over again, incurring the high overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Consider a developer who is trying to find the right batch size for a target cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This process of exploration is iterative, and every hyperparameter value trial needs a new run of the learning-based technique, making the overall undertaking slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For the model development process to be agile, nimble, and at the same time coherent with future real deployments, what is needed is a new class of placement techniques for model parallelism, that: i) are significantly faster in placement than learning-based approaches, and yet ii) achieve fast step times in the placed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This paper is the first to adopt a traditional algorithmic approach for the placement of ML models on memory- constrained clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Subsequent to our initial work [33], a few other authors have published algorithmic or dynamic programming-based ideas for model placement, however these are either : i) standalone and not integrated into open- source systems [67], ii) or they are aimed at only placing specific models like transformers [47], [63], or iii) they are at best comparable in performance to ours [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Orthogonal to model parallelism is pipeline parallelism [27],[31],[22],[79]— our paper does not explore the latter, in order to keep our discussion focused on the benefits of algorithmic approaches over learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The contributions of this paper are: We adapt classical literature from parallel job scheduling to propose two memory-constrained algorithms, called m-SCT (memory-constrained Small Communication Times) and m-ETF (memory-constrained Earliest Task First).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We also present m-TOPO (memory-constrained TOPO-logical order), a strawman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We focus on the static version of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We prove that under certain assumptions, both m-ETF and m-SCT steps time is within a constant factor of the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We present the Baechi system (Korean for placement, pronounced “Bay-Chee”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi incorporates m-SCT/m-ETF into both TensorFlow as well as PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our exposition focuses on the multiple design decisions that were needed in Baechi to derive performance out of the algorithmic underpinnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We present tailored integration of Baechi with both TensorFlow and PyTorch to address the different programming abstractions and architecture in these two frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We present experimental results from a real deployment on a small cluster of GPUs, using both TensorFlow and PyTorch which show that Baechi generates placement plans in time 654×–206K × faster than today’s learning-based approaches, and yet the placed model’s step time (training time) is either faster than or, at worst, only up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% higher, compared to expert-based placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2 New Algorithms for Memory-Constrained Placement This section presents the problem formulation and our three placement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For each technique, we first discuss the classical approach (not memory-aware), and then our adapted memory-constrained algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Where applicable, we prove optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM 4 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Terms and Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Used in the Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝐺 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Machine Learning graph to be placed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='(Classical: Dependency graph of tasks to be placed) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Number of operators (or tasks) in 𝐺 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Number of devices in a cluster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑀 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Memory available per device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑑𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Size of memory required by operator (task) 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑘𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Computation time of operator (task) 𝑇𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝑐𝑖𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Communication time of the output of operator 𝑇𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='𝜌 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Ratio between maximum operator-to-operator (task-to-task) communication time and minimum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='per-operator (per-task) computation time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='SCT assumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Small communication time assumption: Ratio between maximum operator-to-operator (task-to- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='task) communication time and minimum per-operator (per-task) computation time is ≤ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='makespan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Training time for one data mini-batch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', runtime for executing a ML graph on one input mini-batch Our three approaches are: 1) a placer based on topological sorting (TOPO) 2) a placer based on Earliest Task First (ETF), and 3) a placer based on Small Communication Time (SCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Problem Formulation Given 𝑛 devices (GPUs), each with memory size 𝑀, and a Machine Learning (ML) graph 𝐺 that is a DAG (Directed Acyclic Graph) of operators, the device placement problem is to place nodes of 𝐺 (operators) on the devices so that the makespan is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Makespan, equivalent to step time, is traditionally defined as the total execution time to train on one input mini-batch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', unit of training data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 1 summarizes key terms used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When discussing classical algorithms, we use the classical terms “tasks” instead of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If one assumes devices have infinite (sufficient) memory, scheduling with communication delay is a well-studied theoretical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The problem is NP-hard even when under the simplest of assumptions [29], such as infinite number of devices and unit times for computation and communication (UET-UCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Out of the three best-performing solutions to the infinite memory problem, we choose the two that map best to ML graphs: 1) Earliest Task First or ETF [32, 74], and 2) Small Communication Time or SCT [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' SCT is provably close to optimal when the ratio of maximum communication time between any two tasks to minimum computation time for any task is ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We excluded a third solution, UET-UCT [53], since it assumes unit computation and communication times, but ML graphs have heterogeneous operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 m-TOPO: Topological Sort Placer Background: Topological Sort (Not Memory-Aware).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Topological sort [37] is a linear ordering of vertices in a DAG, such that for each directed edge 𝑢 → 𝑣, 𝑢 appears before 𝑣 in the linear ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' New Memory-Constrained Version (m-TOPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our modified version, called m-TOPO, works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It calculates the maximum load-balanced memory that will be used per device, by dividing total required memory by number of devices, and then accounting for outlier operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, this per-device cap is𝐶𝑎𝑝 = (� 𝑖 ∈[𝑚] 𝑑𝑖/𝑛+max𝑖 ∈[𝑚] 𝑑𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then m-TOPO works iteratively, and assigns operators to devices in increasing order of device ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For the current Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 5 device, m-TOPO places operators until the device memory usage reaches 𝐶𝑎𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At that point, m-TOPO moves on to the next device ID, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At runtime, m-TOPO executes the operators at a device in the topologically sorted order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 m-ETF: Earliest Task First Placer Background: ETF (Not Memory-Aware).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ETF [32] maintains two lists: a sorted task list 𝑇 containing unscheduled tasks, and a device list 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In 𝑇, tasks are sorted by earliest schedulable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The earliest schedulable time of task 𝑖 is the latest finish time of 𝑖’s parents in the DAG, plus time for their data to reach 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In 𝑃, each device is associated with its earliest available time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', last finish time of its assigned tasks (so far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ETF iteratively: i) places the head of the task queue 𝑇 at that device from 𝑃 which has the earliest available time, ii) then updates the earliest available time of that device to be the completion time of the placed task, and iii) updates earliest schedulable time of that task’s dependencies in queue 𝑇 (if applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The earliest schedulable time of task 𝑗 on device 𝑝 is the later of two times: (i) device 𝑝’s earliest available time (𝑓 𝑟𝑒𝑒(𝑝)), and (ii) all predecessor tasks of 𝑗 have completed and have communicated their data to 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' More formally, let: a) Γ−(𝑗) be the set of 𝑗’s predecessors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' b) for 𝑖: start time is 𝑠𝑖, computation time is 𝑘𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' c) 𝑥𝑖𝑝 = 0 when task 𝑖 is on device 𝑝, otherwise 𝑥𝑖𝑝 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' d) commmunication time from task 𝑖 to 𝑗 is 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then, the earliest schedulable time of task 𝑗 across all devices is: min 𝑝 ∈𝑃 � max �𝑓 𝑟𝑒𝑒(𝑝), max 𝑖 ∈Γ−(𝑗)(𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗𝑥𝑖𝑝)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (1) Under the SCT assumption (Table 1), ETF’s makespan was shown [32] to have an approximation ratio of (2 + 𝜌 − 1 𝑚 ) within optimal, where 𝜌 is the ratio of the maximum communication time to minimum computation time, and 𝑚 is the number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This approximation ratio approaches 3 when 𝜌 approaches 1 and 𝑚 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' New Memory-Constrained Version (m-ETF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our new modified algorithm, called m-ETF, maintains a queue 𝑄 of operator-device pairs (𝑖, 𝑝) for all unscheduled operators and all devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Elements (𝑖, 𝑝) in 𝑄 are sorted in increasing order of the earliest schedulable time for operator 𝑖 on device 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This earliest schedulable time takes into account dependent parents of 𝑖 as well as the earliest time that device 𝑝 is available, given operators already scheduled at 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The reader will notice that m-ETF’s modified queue can also be used for ETF–the key reason to use (𝑖, 𝑝) pairs is for m-ETF to do fast searches, since the earliest available device(s) may have insufficient memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-ETF iteratively looks at the head of the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the head element (𝑖, 𝑝) is not schedulable because device 𝑝’s leftover memory is insufficient, then the head is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the head is schedulable, then operator 𝑖 is assigned to start on device 𝑝 at that earliest time, and we: i) update 𝑝’s earliest available time to be the completion time of 𝑖, and ii) update 𝑖’s child operators’ earliest schedulable times in queue 𝑄 (if applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 m-SCT: Small Communication Time Placer Background: SCT (Not Memory-Aware).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The classical SCT algorithm [26] first develops a solution assuming an infinite number of available devices, and then specializes for a finite number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We elaborate details, as they are relevant to our memory-constrained version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Classical SCT: Infinite Number of Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' SCT uses integer linear programming (ILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The key idea is to find the favorite child of a given task 𝑖, and prioritize its scheduling on the same device as task 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For a task 𝑖, denote its favorite child as 𝑓 (𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM 6 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The original ILP specification from [26] solves for variables 𝑥𝑖𝑗 ∈ {0, 1}, where 𝑥𝑖𝑗 = 0 if and only if 𝑗 is 𝑖’s favorite child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For completeness, we provide this full ILP specification below [26] (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 in that paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Below, the machine learning graph is 𝐺 = (𝑉, 𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' and 𝑖, 𝑗 refer to operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 min𝑤∞ Minimize makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 → 𝑗 ∈ 𝐸(𝐺), 𝑥𝑖𝑗 ∈ {0, 1} 𝑥𝑖𝑗 = 0 when 𝑗 is 𝑖’s favorite child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 ∈ 𝑉 (𝐺), 𝑠𝑖 ≥ 0 All tasks start after time=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 ∈ 𝑉 (𝐺), 𝑠𝑖 + 𝑘𝑖 ≤ 𝑤∞ All tasks should complete before makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 → 𝑗 ∈ 𝐸(𝐺), 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗𝑥𝑖𝑗 ≤ 𝑠𝑗 Given edge 𝑖 → 𝑗, then 𝑗must start after 𝑖 completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If on different devices, communication cost should be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 ∈ 𝑉 (𝐺), ∑︁ 𝑗 ∈Γ+(𝑖) 𝑥𝑖𝑗 ≥ |Γ+(𝑖)| − 1 Every task has at most one favorite child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ∀𝑖 ∈ 𝑉 (𝐺), ∑︁ 𝑗 ∈Γ−(𝑖) 𝑥𝑗𝑖 ≥ |Γ−(𝑖)| − 1 Every task is the favorite child of at most one predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (2) We modify the above as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We allow 𝑥𝑖𝑗 to take any real value between 0 and 1, thus making the ILP a relaxed LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This can be solved in polynomial time using the interior point method [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then the SCT algorithm simply rounds the LP solution 𝑥𝑖𝑗 to be 1 if 𝑥𝑖𝑗 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1, setting it to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 𝑥𝑖𝑗 can be used to determine the favorite child of each task: 𝑗 is 𝑖’s favorite child if and only if after rounding, 𝑥𝑖𝑗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This infinite device algorithm’s makespan was shown [26] to achieve an approximation ratio 2+2𝜌 2+𝜌 within optimal, where 𝜌 is the ratio of the maximum communication time to the minimum computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We note that the ILP has a meaningful LP relaxation if and only if: (i) infinite number of devices are available, and (ii) the SCT assumption is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', the ratio of the maximum inter-task communication time to the minimum task computation time is ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Nevertheless, even if this assumption were not true for an ML graph and devices, we show later that SCT can still be attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Classical SCT: Extension to Finite Number of Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For a finite number of devices, SCT schedules tasks similar to ETF [32], but: a) prefers placing the favorite child of a task 𝑖 on the same devices as 𝑖 (each task has at most one favorite child, and at most one favorite parent), and b) prioritizes urgent tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', a task that can begin right away on any device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It was proved that SCT’s makespan has an approximation ratio of ( 4+3𝜌 2+𝜌 − 2+2𝜌 𝑚(2+𝜌) ) within optimal [26], which is strictly better than ETF’s (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, when 𝜌 approaches 1 and 𝑚 ≫ 1, then SCT is within 7 3 of optimal while ETF is 3 times worse than optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' New Memory-Constrained Version (m-SCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our proposed memory-constrained algorithm, called m-SCT, works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' First, m-SCT determines scheduling priority and selects devices in the same way as the finite case SCT algorithm just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second, when a device 𝑝 runs out of available memory, m-SCT excludes 𝑝 from future operator placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In spite of the seemingly small difference, Figure 1 shows that m-SCT can succeed where SCT fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' SCT achieves a makespan of 8 time units with infinite memory but OOMs for finite memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' With finite memory, m-SCT succeeds and increases makespan to only 9 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Classical SCT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When per-device memory is limited to 4 memory units, SCT OOMs but m-SCT succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-SCT’s training time (makespan) is only slightly higher (9) than SCT with infinite memory (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Dashed arrows show data transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5 Optimality of m-ETF and m-SCT Classical ETF and SCT were originally proposed to schedule a DAG of tasks on 𝑛 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The original work [26, 32] derived upper bounds on the makespan achieved by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Processor memory was not considered in that original formulation, and infinite memory was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' On the contrary, in ML model training, each device has a memory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Impact of memory on scheduling is further pronounced due to the persistent memory that each task requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Consequently, the schedules obtained by m-SCT/m-ETF can differ significantly from the SCT/ETF schedules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', without memory constraint)—Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is not clear how much worse m-SCT/m-ETF makespan is, compared to the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus we derive upper bound on makespan of both m-ETF and m-SCT by extending the proofs in [32] and [26] to the memory constrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We show that m-SCT/m-ETF makespans are within a constant factor of the optimal makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Because the proofs for optimality of m-ETF and m-SCT are involved, we show them in Appendix A and Appendix B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We summarize our results and approach here: Result 1: The completion time of m-ETF under realistic communication cost and limited memory, is within a known factor of the optimal schedule possible under zero communication cost but with infinite memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Result 2: The completion time of m-SCT under realistic communication cost and limited memory, is within a known factor of the optimal schedule under infinite memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Intuitively, our derived upper bounds are proportional to the ratio 𝑛 𝑟 ,where 𝑛 is the total number of devices available, and 𝑟 is the number of devices out of 𝑛 that still have spare memory after all the tasks have been placed in a way that “fills up” memory device by device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that 𝑛 𝑟 > 1, otherwise the problem is unsolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Intuitively, a smaller Manuscript submitted to ACM op1 op2 0p3 op4 1(1) 3(2) 3(2) 1(1) op5 op6 3(1) 1(1) op1 op2 op3 op4 op5 op6 0p2 op6 op1 op5 op3 op48 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 𝑛 𝑟 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', larger 𝑟) indicates looser memory constraint and thus better makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' As 𝑛 𝑟 approaches 1, m-SCT’s solution (respectively m-ETF’s) starts to approach that of SCT (respectively ETF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3 Baechi Design This section describes how we implement Baechi in a way that works modularly with TensorFlow [1] as well as PyTorch [56], two popular open-source learning platforms originally developed by Alphabet and Meta respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At a high level—for both target systems, Baechi first creates a computation graph of the input model, where each node is annotated with its memory requirements and time to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This graph is then fed to the chosen algorithm (Section 2’s m-SCT, m-ETF, or m-TOPO) to generate the placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Finally, training is automatically executed with the given placement and without requiring the developer to modify the code for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, because of two key differences in abstractions and architectures between TensorFlow and PyTorch, Baechi’s design for each is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' First, the “nodes” in the computation graph are operators in TensorFlow while in PyTorch they are modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The former are fine-grained mathematical operations on tensors, while the latter are coarser structures similar to classes in object-oriented languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second, in TensorFlow a model is a static graph of operators, while in PyTorch, the computation graph is constructed only during the forward run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Because of foreknowledge of the graph, TensorFlow can automatically insert rendezvous operators [68] for cross-device communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, PyTorch does not automatically insert these essential cross-device communication primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It requires PyTorch developers to write explicit code that moves tensors across devices during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A side benefit of our work is the automatic generation of these communication primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In the remainder of this paper, we refer to the integration of Baechi into TensorFlow as Baechi-TF, and Baechi’s integration into PyTorch as Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Next, we describe our techniques and optimizations for Baechi-TF in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1, and then additional changes and differences required for Baechi-PY design in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Design of Baechi-TF To work with TensorFlow, Baechi needs to address four challenges: 1) Satisfying TensorFlow’s colocation constraints, 2) Minimizing Data Transfer via Co-Placement, 3) Optimizations to reduce the number of operators to be placed, and 4) Accommodating Sequential and Parallel Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi solves these using a mix of both new ideas (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4) and ideas similar to past work (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Working Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use Figure 2 as a working example throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is a simplified TensorFlow graph for linear regression training with stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 TensorFlow Colocation Constraints The first challenge arises from the fact that TensorFlow (TF) requires certain operators to be colocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, TensorFlow offers a variable operator, tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Variable, which is used to store persistent state such as an ML model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The assignment and read operators of a variable are implemented as separate operators in TensorFlow, but need to be placed on the same device as the variable operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' TensorFlow represents this placement requirement as a colocation group involving all these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', in Figure 2 there are two colocation groups: one containing Weight and ApplyGrad, and another containing Step and UpdateStep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s initial placement (using the algorithms of Section 2) ignores colocation requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our first attempt was to post-adjust placement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', to “adjust” the device placement, which was generated ignoring colocation, by Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Working Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ML Graph for Linear Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Co-Placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Subgraph of tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='tensordot Generat- ing Data Transfers by m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' “moving” operators from one device to another, in order to satisfy TF’s colocation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We explored multiple post-adjustment approaches including: i) preferring the device on which the compute-dominant operator in the group is placed, ii) preferring the device on which the memory-dominant operator in the group is placed, and iii) preferring the device on which a majority of operators in the group are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We found all these three approaches produced inconsistent performance gains, some giving step times up to 406% worse than the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We concluded that post-adjusting was not a feasible design pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s novel contribution is to co-adjust placement, using colocation constraint-based grouping while creating the schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (In comparison, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', ColocRL [51] groups before placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=') Concretely, whenever Baechi places the first operator from a given colocation group, all other operators in that group are immediately placed on that same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi tracks the available memory on each device given its assigned operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the device cannot hold the entire colocation group, then Baechi moves to the algorithm’s next device choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We found this approach the most effective in practice, and it is thus the default setting in Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Co-Placement Optimization Different from TensorFlow’s colocation constraints (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1), Baechi further prefers to do co-placement of certain operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is aimed at minimizing data transfer overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Common instances include: (i) groups of communicating operators whose computation times are much shorter than their communication times, and (ii) matched forward and backward (gradient-calculating) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Figure 3 shows an example for case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This subgraph generated by tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='tensordot API is a frequent pattern occurring inside TensorFlow graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The subgraph permutes the dimensions of opin output according to the perm’s output (Transpose) and then changes the tensor shape by Shape’s output (Reshape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When m-ETF places this subgraph on a cluster of 3 devices, it places opin, perm, and Shape on different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Computation costs for perm and Shape are very short (because they process predefined values), whereas subsequent communication times are much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus, m-ETF’s initial placement results in a high execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s co-placement heuristic works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the output of an operator is only used by its next operator, we place both operators on the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is akin to similar heuristics used in ColocRL [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Figure 3, Manuscript submitted to ACM Input Output Step Update MatMul Loss Grad Step Apply Weight GradoPin Transpose Reshape oPout perm Shape10 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Operator Fusion Without Creating Cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (a) shows a fused ML Graph Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When 𝑜𝑝𝑠𝑟𝑐 and 𝑜𝑝𝑑𝑠𝑡 are fused, some scenarios create a cycle (b), while others do not (c, d, e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi fuses operators in a subset of “safe” cases, particularly (e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s co-placement optimization places all of the operators on one device, avoiding any data transfers among the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For case (ii), to calculate gradients in the ML model, TensorFlow generates a backward operator for each forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi co-places each backward operator on the same device as its respectively-matched forward operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Upon placing the first operator in a colocation group, Baechi uses both the co-placement heuristic and the colocation constraints (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1) to determine which other operators to also place on the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Co-placement not only minimizes communication overheads but also speeds up the placement time by reducing the overhead of calculating schedulable times on devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 Operator Count Minimization Placement time can be decreased by reducing the number of opera- tors/groups to be placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We do this via two additional methods: i) Operator Fusion: Fusing operators that are directly connected and in the same co-placement group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' and ii) Forward-Operator-Based Placement: Placing operators by only considering the forward operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Operator Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi fuses operators using either the colocation constraints (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1) or co-placement optimizations (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is new and different from TensorFlow’s fusion of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' One challenge that appears here is that this may introduce cycles in the graph, violating the DAG required by our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Input Output Step MatMul Loss Grad Update Cycle Step Apply Weight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='i GradoP1 0P2 oPsrc opdstop1 0P2 oPsrc oPdstop1 0P2 opsrc oPdstop oPsrc oPdstop oPsrc opdstBaechi: Fast Device Placement of Machine Learning Graphs 11 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Operator Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Avoiding Data Transfer Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (a) Before Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (b) After Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Figure 4a shows an example resulting from Figure 2—a cycle is created when Step and UpdateStep are fused into a new meta-operator, and Weight and ApplyGrad are fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Consider two nodes–source and destination–with an edge from source to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Merging source and destination creates a cycle if and only if there is at least one additional path from source to destination, other than the direct edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that there cannot be a reverse destination to source path as this means the original graph would have had a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Figure 4b, fusing opsrc and opdst creates a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Unfortunately, we found that pre-checking existence of such additional paths before fusing two operators is unscalable, because the model graph is massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Instead, Baechi realizes that a necessary condition for an additional path to exist is that the source has an out-degree at least 2 and the destination has an in-degree at least 2 (otherwise there wouldn’t be additional paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus Baechi uses a conservative approach wherein it fuses two operators only if the negation is true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', either the source has an out-degree of at most 1, or the destination has an in-degree of at most 1 (Figures 4e, 4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This fusion rule misses a few fusions (Figures 4c, 4d) but it catches common patterns we observed, like Figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Forward-Operator-Based Placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When memory is sufficient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', one device could run the entire model), Baechi considers only forward operators for placement and thereafter co-places each corresponding backward (gradient) operators on the same respective device as their forward counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is a commonly-used technique [2, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This significantly cuts placement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When device memory is insufficient, Baechi runs the placement algorithms using both forward and backward operators, forcing corresponding pairs to be co-placed using the heuristic of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Example: Benefits of Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Figure 5a shows the placement of a subgraph of Figure 2 on two devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi first places Grad on device-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi places the next operator, Step on the idle device-2, and colocates (due to TF constraints) UpdateStep on device-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This creates communication between the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Assuming operators’ compute costs are 1, and communication cost between Grad and UpdateStep is 5, this results in an execution time of 7 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' On the other hand, Figure 5b shows that Baechi merges Step and UpdateStep with operator fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since this meta-operator’s schedulable time on device-1 is earlier than on device-2 due to communication overhead, Baechi places it on device-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Fusion lowers total execution time to 3 time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Step Update Grad StepStep Grad Update Step12 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Loops in the Original Model Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Different from the cycles discussed above, some network graphs consist of loops, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use the unrolled ML graph [4] to turn the graph into a DAG, and then apply Baechi’s techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 Sequential vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Parallel Communication Our algorithms from Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 assume that each operator can send data simultaneously to its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi also proposes a new way to deal with environments involving constrained networks (including our deployment in Section 5), where data transfer is sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For networks that limit each device to do at most one transfer at a time (out or in), Baechi assumes communication queues at devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, when a data transfer between two devices is requested, Baechi assumes the request is put into the respective devices’ communication queues and processed sequentially at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' During placement, Baechi calculates the wait time at the communication queues and adds it to the earliest schedulable time computed for the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Specifically the queue wait time is added to equation (1) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Otherwise, normal m-SCT/m-ETF apply, as described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Design of Baechi-PY We remind the reader that unlike TensorFlow’s fine-grained operators and known communication graph, PyTorch: (i) has coarser modules, and (ii) requires the programmer to explicitly program cross-device communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely—first, PyTorch models are built by composing different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The model is not natively available as a graph unlike TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To feed the model to Baechi’s algorithms, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 describes how we construct a graph by using PyTorch’s Autograd [56] which tracks the flow of tensors among the modules of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second, the primitive .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='to() API provided by PyTorch for developers to program communication is inefficient and manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To address this, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 presents our communication protocol to handle cross-device transfers efficiently and automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Baechi-PyTorch Graph Developers build PyTorch models by composing modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e, classes inherited from PyTorch’s nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Each module contains its tensor parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', weights of a linear layer) and a forward() method, which defines how the module modifies the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By default Baechi treats modules as the nodes of the graph in Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Placing a module on a device means moving all its parameters to the device before beginning the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' During the training, our communication protocol (next Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) ensures that the input to that module is also moved to the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Subsequently the forward() operation of the module will be invoked at runtime, and it will be executed on the assigned device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A model may also include operations not defined as PyTorch modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, arithmetic operations like scaling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g: x=x/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' But these operations usually do not have any associated parameters that must be assigned to a device by Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence we exclude them from the graph during the placement planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By default, this associated operation will be executed on the device of the input tensor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', x’s device in this case1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY constructs the model’s graph in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' First we obtain all modules that constitute the model and co-place modules occurring in common design patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second we obtain the edges of the graph by tracking the flow of tensors using PyTorch’s Autograd [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This approach is similar to PipeDream [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Co-placement: Treating modules as nodes, we observed it is common for models to contain specific subgraphs (of modules) that occur as common design patterns throughout the model graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY groups such subgraphs into a single node—this is called co-placement (unlike Baechi-TF’s co-location constraints in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1, this co-placement is a performance optimization in PyTorch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance in Inception models, the subgraph (Conv2d)→ (Batch Norm) → (inplace ReLU) occurs commonly, and Baechi-PY groups each occurrence as one node in the computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1Arithmetic operations may still take some time to complete and memory to store their outputs, but we observed that their impact on the overall step time and memory budget is small, thus we ignore them in generating placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If desired, such operations may be defined as nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Modules and be included in the placement graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 13 This co-placement allows Baechi-PY to avoid communication of tensors along the two edges of this subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' An additional benefit of co-placement is that it significantly reduces the size of the computation graph, thus making Baechi’s algorithms (Section 2) run faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, co-placement reduces the number of nodes in Inception-V3 PyTorch by 60%, from 325 to 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By default, Baechi-PY uses the most atomic modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', not further divisible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', 2D Convolution module (Conv2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the developer wishes, they can programmatically specify which modules Baechi-PY should be co-placed and treated as individual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Building the graph: Next, Baechi obtains edges between the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To do this, we run a training step of the model with dummy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (We run 20 such dummy training steps, also helping us profile memory usage and computation times of all the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=') During the forward run, we annotate each intermediate output tensor with the node that generated it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Meanwhile, PyTorch’s Autograd automatically fills in the gradient function (grad_fn) for each tensor created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Further, each such grad_fn has a list of grad_fn of tensors used as inputs in creating this tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Autograd stores this list to perform back-propagation later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use this information that traces the tensor gradient functions, along with our tensor to node annotation, to construct a dependency graph among modules of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is possible that some gradient functions may include operations not related to any module, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', Autograd-specific operations such as SelectBackward or gradients of arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' But since these operations do not need a device placement, they are removed to obtain the dependency graph only between nodes of the model (they are added back in after the model is placed) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Communication Protocol PyTorch provides native support for synchronous communication, which can be inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If asynchronous data transfer is used, the developer is required to carefully insert synchronizations to ensure correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We design a general communication protocol for cross-device communication that is efficient and automated, thus relieving the developer from specifying manual configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We do so by leveraging CUDA streams, an abstraction that allows overlapping multiple sequences of operations in a GPU [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To move a tensor T to 𝐺𝑃𝑈0, PyTorch provides an API T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='to(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To ensure correctness, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='to() conservatively blocks both sending and receiving devices until all the operations submitted to both devices so far are completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This can be avoided by leveraging the abstraction of CUDA streams [54], in order to overlap communication with computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A CUDA stream in a GPU is a FIFO queue of operations that will be sequentially executed on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By default, all operations submitted to a GPU are placed on a single stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To perform two operations in parallel, they must be placed on two separate streams on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Accordingly, on each GPU, Baechi-PY defines one compute stream and multiple communication streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The compute stream queues the computations corresponding to the modules placed on the GPU, while the communication streams concurrently move the relevant tensors across the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, CUDA streams need to be programmed carefully to specify synchronization points that obey dependencies in the model graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use CUDA Events provided by the runtime API [57] to synchronize the independent streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' None of the existing ways of using CUDA streams in PyTorch fits our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely—first, in PyTorch-Distributed [46] and PipeDream [27], training steps proceed in stages, each working on a different batch of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All tensors generated in a given stage are transferred to the next device at the end of the stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This pattern, where all communication happens synchronously only after all computations are complete requires few, if any, synchronizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In contrast, Nimble [43], deals with multiple parallel streams working on the same batch of data, like in Model Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compute and communication events may asynchronously occur at any time and it requires synchronizations to preserve data dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, Nimble is a single-GPU system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM 14 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Baechi-PY, we use a greedy-wait strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, first we greedily push out the output of a node, as soon as it is computed, to the devices of its children nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second, before starting the compute operation, a child node must wait for all its incoming input stream or if the input has already been transferred it must pull the copy of the inputs on its devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We implement our greedy-wait communication protocol as a wrapper around the forward() function of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Algorithm 1: Communication protocol built around each node’s forward(input) 1 for each parent of the node in graph do 2 (node’s compute_stream) wait for (rx_stream from parent’s device);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3 end 4 On node’s compute_stream: 5 input_local = local copies of input on node’s device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 6 output = forward_operation(input_local);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 7 for each child of the node in graph do 8 (tx_stream to child’s device) wait for (node_compute_stream);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 9 end 10 for each child of the node in graph do 11 Using (node’s tx-rx_stream pair to child’s device ): 12 send output to child;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 13 end 14 return output Algorithm 1 shows the complete communication protocol, and we describe it in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Before we start the training, for a given node and its child node on a different device, we create two streams - a tx-stream on the node’s device and a rx-stream on the child node’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Only one such stream pair is sufficient for a child device, even if multiple children nodes are on that device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' So if a node on 𝐺𝑃𝑈0 has two children, one on 𝐺𝑃𝑈1 and another on 𝐺𝑃𝑈2, then for that node we create one tx-rx stream pair each to 𝐺𝑃𝑈1 and 𝐺𝑃𝑈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We do this for every node and for every device any of its child nodes reside in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Each device’s compute stream queues the computations of nodes assigned to that device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When a node reaches the head of the compute stream of its device, the compute stream is made to wait for all the rx-streams to that device from all the parent nodes (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The rx-streams carry a copy of the output tensors of the parent nodes to the device of the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Once all rx-streams have completed the transfer, these tensors are passed as inputs to the actual forward computation of the node (lines 4-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All the tx-streams egressing from this node are made to wait for the compute stream to finish the computation (lines 7-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The tx-stream carries the outputs of the node to the child node’s device and the corresponding rx-stream on the child node’s device receives it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' As soon as the output is ready, it is asynchronously sent to all the child devices through the tx-streams and received asynchronously at the child devices through their respective rx-streams (lines 10-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The compute stream can move to the next node assigned to it while the tx-streams are transferring out the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Similarly, the compute streams on child devices are not interrupted by incoming tensors on their rx-streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that the output is sent to a child device only once even if multiple children reside on that child device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The tx-rx streams serve as synchronization points in addition to overlapping communication and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our m-ETF and m-SCT algorithms from Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 assume that each node can send data simultaneously to its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our protocol mimics this communication with multiple outgoing tx-streams transferring the tensors to Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 15 Input Model In TF as tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Graph In PT as nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='Module Profiler Graph Optimizer (TF only) Co-Placement Grouper Operator Fuser Execution Simulator Global Scheduler (m-SCT, m-ETF, m-TOPO) Placed Graph TF/PT Runtime Device Device Device Assigner (PT only) Node to Device Placement TF = TensorFlow, PT = PyTorch Graph Generator (PT only) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi System Architecture child devices in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While such overlapping tx-streams from a device may slightly increase the communication times in each of the streams, we observed that the associated effect on step time is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Also as long as we facilitate and synchronize the forward run correctly, PyTorch’s Autograd [56] ensures that the back-propagation correctly executes in the reverse order of the forward sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It automatically manages input-output dependencies and synchronizations in the reversed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4 Implementation In order to integrate modularly with TensorFlow (TF) [1] v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='12 and PyTorch v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9 [56], Baechi adopts the architecture shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi executes the following steps: 1) its Graph Generator and Profiler constructs the graph annotated with each node’s time and memory requirements, 2) Baechi’s Graph Optimizer for TensorFlow uses the design of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 to account for TF colocation constraints, and applies co-placement and operator fusion, 3) Baechi’s Execution Simulator (ES) (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) executes our algorithms (m-TOPO, m-ETF, or m-SCT) and generates the placement (in TensorFlow a ready-to-use placed graph is output), and 4) the Assigner for PyTorch (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) modifies the model to allow execution according to the generated placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The placed graph can then be used in the training script as the drop-in replacement for the single-GPU model, in both TensorFlow and PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Next we describe each of the components in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Graph Generator, Profiler and Optimizer In Baechi-TF, the model is already given as a static graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The Profiler then measures and annotates each node with its time and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi parses this annotated graph and generates an equivalent intermediate NetworkX [61] graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The NetworkX format allows Baechi to both store operator execution metadata (computation and communication times, memory needed, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ), and to easily manipulate the graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', fuse operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then, in case of Baechi-TF, we additionally apply the co-placement and operator fusion optimizations (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM 16 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Memory Inference Training Permanent (a) (a) + (b) + (c) Temporary (b) + (e) (e) + (d) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Memory Consumption in PyTorch In Baechi-PY, the Graph Generator constructs the graph corresponding to the input model as per Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For communication time, we use a linear model proportional to data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, we implemented a microbench- mark tool to measure communication times for various data sizes, and generated a communication cost function through linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Profiler: The Profiler measures computation times and memory requirements of each node in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Baechi-TF, we use the standard TensorFlow profiling tool to obtain computation time and memory allocation for each operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' TensorFlow profiler returns allocation information for temporary, permanent, and output tensor memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The temporary memory is allocated at the beginning of an operation and deallocated when the operation finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The permanent memory is allocated and used over the entire execution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', to store persistent states such as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Baechi-PY we build a simple profiler, akin to that in [27], for measuring time using hooks [58], and for measuring memory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY Profiler Memory Estimation In PyTorch, GPU Memory required to hold all the tensors is reserved once during the first training step and reused in subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To model the memory usage pattern, we categorize each node’s memory as consisting of 5 components: (a) Parameters memory: Memory occupied by parameters of the node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (b) Output Memory: forward output of the node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (c) Parameter gradient: gradients of parameters of the node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (d) Upstream gradient: gradient of output of the node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (e) Memory temporarily used in computing the output/gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 2 summarizes how these five metrics are used in training and inference, and whether they are used as temporary memory or permanent memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We describe each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' During the training phase, each node requires memory to store: (a) its parameters, (b) the node’s forward output tensors, and (c) parameters’ gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In PyTorch, memory to store parameter gradient information is acquired once in the beginning of training and permanently held until the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Similarly, output tensors ((b)) are treated as permanent memory since during each forward run, outputs of all nodes must be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' They are required later during back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In contrast during the inference phase (forward only runs), output of a node is temporary since it is immediately released after being consumed by the subsequent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' During back-propagation in training, memory is also required to hold the gradient of output of the node ((d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is a temporary requirement since this memory is released after the output gradient has been used to compute the node’s parameter gradients and its input gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Nodes may also require temporary memory while performing these computations ((e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For example, in computing the parameter gradients, a temporary matrix is used to store all the gradients and then the node’s parameter gradients are updated at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For in-place nodes like the 2While PyTorch has an internal profiler using it would have required us to handle dependencies and operation granularity carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our simple approach avoids these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 17 in-place ReLU, (b) is set to 0 since no new output tensor is generated ((a) and (c) are also 0 incase of ReLU since it has no associated parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Execution Simulator Baechi’s Execution Simulator (ES) executes the algorithms on the profiled graph and generates the placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The ES takes as input the NetworkX operator graph, the number of GPUs and the memory capacity of each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output is the graph in which all operators are assigned to devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We describe the common parts of the ES across both Baechi-TF and Baechi-PY, explicitly pointing out differences as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our initial attempt was in fact to try re-purposing TensorFlow ’s (existing) simulator, but using that required us to assume operators were already placed, zero communication cost, and no caching—these were inapplicable to Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Motivated by this, we designed Baechi’s new ES uniquely for memory-constrained placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The ES consists of: a) a global scheduler, and b) simulated devices (with specs identical to deployment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The global scheduler maintains a single queue with operators that are ready to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The scheduler extracts operators from its queue and applies our scheduling algorithms (m-TOPO, m-ETF, m-SCT) to place them on devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In ES, each device has two FIFO queues, one for operators and one for data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This allows data transfer to overlap with operator execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When a device receives a tensor from another device, it caches the tensor to avoid duplicate data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Dynamic Memory Allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Calculating a device’s memory usage as the sum total of all its assigned operators (assigned over the entire duration) clearly overestimates memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For example in TensorFlow, Inception-V3 with batch size 32 can execute using 4 GB even though its operators’ memory needs add up to 22 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In generating the placements, the ES calculates memory in a way that parallels how the frameworks manage memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely Baechi’s ES tracks an estimate of memory usage during its placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When an operator executes on a device, the device allocates temporary memory, and separate memory for its output tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The temporary memory is deallocated when the operator finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' There are minor differences in the ES for TensorFlow and PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' TensorFlow uses separate operators for forward and backward computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output memory of an operator is deallocated after all its successors finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In PyTorch, the forward and backward computation runs in a single module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output memory of a module is held until its backward computation is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output memory is treated as a part of the permanent memory as explained in the previous Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If a device’s memory becomes full, the device can be removed–this never happens in practice as usually a device has at least a few bytes left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that in Baechi-TF, memory is reserved for a colocation group at device 𝑝 when the first operator is placed on 𝑝 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The reserved memory is deallocated when all the operators in the group finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Linear Programming Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To solve the SCT LP problem, we use the interior point method [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is preferable over other solvers such as simplex [9] as it guarantees polynomial execution time [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, we use the primal dual interior-point solver via Mosek optimization [7], which has a run time complexity of 𝑂(𝑛3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5𝐿), where 𝐿 is the maximum number of bits in the LP input, and 𝑛 is the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 Assigner in Baechi-PY Once the mapping of graph nodes to devices is decided, the Assigner contains the mechanism to initialize the assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The Assigner for TensorFlow merely changes the device attribute of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The Assigner for PyTorch requires Manuscript submitted to ACM 18 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' multiple steps—it needs to: i) move nodes’ parameters to the assigned devices, ii) send output tensors from a node to devices of child nodes at the device boundaries, and iii) avoid naive use of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='to() for cross-device communication as this leads to inflated step times owing to unnecessary blocking of the devices (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY’s Assigner enables this process by automatically adding wrappers around the forward() methods of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The wrapper transparently handles the communication and caching of the tensors using the communication protocol in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This way, the developer does not need to make any changes to the input model code written for a single-GPU execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output of the Assigner is a model assignment that can be used as a drop-in in any existing training script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 Miscellaneous Issues We discuss a few key miscellaneous aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' LP Modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The ILP solutions (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4) resulted in more than one favorite child (or parent) being selected for certain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Baechi we lowered the rounding threshold from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5 to below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This eliminated all violations, and avoided nodes from having multiple favorite children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (We use threshold = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=') Ignoring Bootstrap Steps in Profiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1) In a training run of a model graph, step times are initially high due to TensorFlow bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We estimate step times in steady state, after a few iterations have passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2) Some TensorFlow operators are implemented with multiple GPU kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When profiling these operators, we include multiple kernel executions, in order to avoid underestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is similar to TensorFlow’s cost model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Reordering Layers in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Dynamic line-by-line execution in PyTorch means that the modules’ forward() functions will be called in the order in which they appear in the code rather than in the topological order followed by Baechi’ ES (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We reorder the actual execution of modules’ computations on GPUs by launching a thread when a module’s forward() is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The thread waits until its topological parent (according to the ES) has submitted the computation task to the GPU and only then submits its task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, such reordering does not give a noticeable advantage over just executing the code order since the two orders do not differ significantly in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5 Evaluation Our evaluation answers the following six questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' How fast is Baechi’s placement time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', how quickly do our algorithms find placements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' How fast are the step times of the placement generated by Baechi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', training time per step of the placed model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' How do the step times for Baechi compare to single GPU and expert placements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' How do the Baechi’s step times change when there is insufficient memory per GPU?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' How much is the benefit due to Baechi’s optimizations from Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Are algorithmic approaches preferable over RL approaches for model parallelism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (all subsections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Experimental Settings We use two popular ML benchmarks for each framework: A) for TensorFlow, we use Inception-V3 and Google Neural Machine Translation System (GNMT), and B) for PyTorch, we use Inception-V3 and a Transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The former choice is because: i) Inception-V3 and GNMT are respectively considered the best representatives of vision and Natural Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 19 Language Processing (NLP) models, and ii) past work [2, 50, 51] used Inception-V3 and NMT (GNMT is a more complex version), thus allowing us to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For PyTorch we replace GNMT with Transformer as the former is implemented using the LSTM module [59], making the (latter) Transformer a more complex and generalized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We describe the three benchmark configurations in detail below: Inception-V3 Benchmark Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Inception-V3 [66] is a convolutional neural network architecture that is widely used for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This model is composed of multiple blocks called Inception modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The Inception modules consist of branches of convolutional and pooling operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To train the model, we use RMSProp [28] and batch sizes of both 32 and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' GNMT Benchmark Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Google Neural Machine Translation System (GNMT) [77] is a language model for automated translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' GNMT consists of: encoder and decoder modules, each a stack of recurrent neural networks (RNNs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' and the attention module to process long sequences effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use 4 long short-term memory (LSTM) layers of the encoder and the decoder layers with residual connections, and the Bahdanau attention mechanism [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use the LSTM hidden size of 512, the vocabulary size of 30,000, the unrolled RNNs with the sequence length of 40 and 50, and the batch size of 128 and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-TF applies the co-placement optimization to LSTM cell operators and also to attention operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to Inception-V3, GNMT has fewer barriers (sync points) inside its model graph, indicating that GNMT has a higher potential to benefit from Baechi-TF’s parallel placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Transformer Benchmark Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Transformers are a versatile family of models used in vision as well as language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Like Neural Machine Translation (NMT), Transformers have an encoder-attention-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' But while NMT processes one word at a time, Transformers use multi-head attention modules that process the entire sequence at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In PyTorch, we implement an attention operation in the traditional way [23]—as one large matrix multiplication and hence as a single module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For concreteness, we use the base Transformer model from [73] (without weight sharing) with a vocabulary size of 30,000, sequence length of 50 and batch sizes of 64, and 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Machine Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All experiments are run on our local server that has 4 NVIDIA GTX 2080 GPUs, with 8 GB per- GPU memory (the machine also has an Intel i9-7960X CPU, but this is not used to execute operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' GPUs are connected to CPUs via PCIe 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0 x16 (we do not use NVLink [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All data transfers go through the host memory (no P2P communication among GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This results in a slow IO bus, and we believe this high ratio of communication overhead to computation overhead is representative of realistic scenarios like the kinds outlined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We place all GPU-supported operators only on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Approach to Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To quantify the benefits of using an algorithmic approach to model parallelism over a Reinforcement Learning (RL) approach for model parallelism, we compare Baechi to the best RL-based model parallelism techniques: [2, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Directly running these other systems was complicated by lack of uniform availability of working code—Placeto’s code [2] missed key optimizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ColocRL [51] is proprietary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' only HierarchicalRL’s code [50] was available, but it was slow and generated inefficient placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', For GNMT, HierarchicalRL took 12 hours+ to run placement (batch size 128, length 50) and the resultant step time was much higher than expert’s, contrary to HierarchicalRL paper’s claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Essentially, direct comparison would be unfair to these other papers without knowing the exact hyperparameters they used to achieve their “best” performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In light of this, our comparison gives the benefit of doubt to, and uses the best performance from, these learning-based placement papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All the above papers compared step times to experts, and we do too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We do not compare to other algorithmic techniques for model parallelism Manuscript submitted to ACM 20 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Model HierarchicalRL [50] Placeto [2] Baechi (m-SCT) Inception-V3 11 hrs 50 mins 1 hr 49 mins 1-10 seconds NMT (GNMT) 1 day 21 hrs 14 mins 2 days 20 hrs 40 mins 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2-48 seconds Transformer N/A N/A 1-3 seconds Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Placement Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Time to Generate a Placement for our target machine with 4 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (listed in Section 1) because of either: their standalone nature [67] (making a TensorFlow/PyTorch comparison unfair), or their limitation to Transformers [47, 63], or because they are already shown to be comparable to our performance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This also keeps our evaluation focused on comparing algorithmic approaches to RL approaches for model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Placement Time Table 3 shows both: 1) measured placement times of Baechi, and 2) calculated placement times for two learning-based techniques, namely: HierarchicalRL [50] and Placeto [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The numbers for HierarchicalRL and Placeto are normalized quantities, both derived from numbers reported in Addanki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For these two systems, we multiply the fastest step time among its reported placements, by the number of placement samples3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, HierarchicalRL’s [50] Inception-V3 placement training time is derived as a product of the reported final step time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='19 s) and the number of samples (35,800), giving 42,602 s, or 11 hrs 50 mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence, the numbers for these learning-based placers are their best-case performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In comparison, we use the worst-case placement times from Baechi, specifically from m-SCT which took the longest to generate a placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that all times in Table 3 exclude time to profile the graph, as profiling is a common baseline encountered by all the three approaches shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We find the profiling time to be low: about 10–12 s total for Inception-V3 and GNMT in Baechi-TF, and about 11–14 s for Inception-V3 and Transformer in Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, in Baechi-TF, this breaks down as 2-4 s for warmup execution, 1–3 s for graph execution for profiles, and less an 1 s for parsing profile results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 3 shows that Baechi places ML models orders of magnitude faster than the learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Inception-V3, Baechi reduces placement time, from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8 hours (using existing techniques [2, 50]), to under 10 s in both Baechi-TF and Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus Baechi is 654×–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6K× faster at placing Inception-V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For GNMT, Baechi-TF reduces placement time from several days to under 48 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus Baechi is 3392×–206K× faster at placing GNMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Transformer, Baechi-PY places it under 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Because HierarchicalRL and Placeto [2, 50] did not include Transformers in their evaluation, the corresponding entries are marked as not available in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Overall, Baechi is 654×–206K× faster at placement compared to today’s learning-based approaches [2, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 Placement with Sufficient Memory We next evaluate the effectiveness of the generated placement by measuring the step time of the placed model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', its time to execute 1 training step on an input data batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Step time is a key metric as completion time on a training set is directly proportional to step time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We first explore the scenario when each GPU has sufficient memory to run the entire model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We compare against both: i) step time on a single GPU, which might be fast because it avoids the overheads of communication, and ii) an expert-based placement scheme for placement on multiple GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3Even if one were to parallelize the learning-based placers, their resource usage would be similar to the normalized time metric we show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 21 Speedup over Single GPU Expert (4 GPUs) Model Batch Size Single GPU Expert m-TOPO m-ETF m-SCT m-ETF m-SCT m-ETF m-SCT Inception-V3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='269 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% Inception-V3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='00% (1 GPU Expert) 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='00% (1 GPU Expert) Transformer (length: 50) 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='244 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3% PyTorch 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='453 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi with Sufficient Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Average Step Times (Training) in seconds of Placed Model Graphs, and Speedup over Single GPU and Expert Placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4 GPUs (unless otherwise mentioned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The expert is a manual process and we do it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For GNMT in TensorFlow, we use the technique of Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Each LSTM layer in the encoder and decoder modules are placed on different GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The embedding layer is placed on the same GPU as the first LSTM layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The output projection layer is placed on the same GPU as the last decoder LSTM layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Inception-V3 in both TensorFlow and PyTorch, the expert is the single GPU placement, similar to HierarchicalRL [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For the Transformer model in PyTorch we use the common practice of putting the encoder on one device and the decoder on another device [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-ETF, m-SCT –VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='– Single GPU, Expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 4 shows the step times for the three algorithms in Baechi–namely m-TOPO, m-ETF, and m-SCT—as well as the single GPU and expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We show numbers for 2 batch sizes in each model, and 2 sequence lengths in GNMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We observe that for Inception-V3: 1) in TensorFlow, m-ETF and m-SCT find the same device placements as the expert, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', place all operators in a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2) In PyTorch m-ETF and m-SCT placements use three and two GPUs respectively, but have the same step time as 1-GPU expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 3) Compared to the expert, m-TOPO step time’s is higher by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3% in Baechi-TF and by 14–17% in Baechi-PY for Inception-V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This occurs because m-TOPO splits the neural network between the Inception blocks, and hence the next inception block(s) are unable to run until the previous block(s) finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In TensorFlow GNMT, first, compared to single GPU placement, m-ETF’s placements have step times that are 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9% faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The step time speedups for m-SCT over single GPU are between 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' These observations show that Baechi’s m-ETF and m-SCT are able to extract benefits of parallelism in spite of communication overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Second, in GNMT, compared to the expert, m-ETF is between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5% slower and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% faster in step times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to the expert, m-SCT is between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% slower and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9% faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In PyTorch Transformer, m-SCT and m-ETF placements are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% faster than single-GPU and expert placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' They place only the decoder’s embedding and first multi-head attention layer on a separate device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since this computation is independent of the encoder, m-SCT and m-ETF exploit the parallelism in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The rest of the decoder requires output of the encoder and is hence placed on the same device as the encoder (in contrast to the expert) to minimize communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM 22 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi with Insufficient Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Average Step Times (Training) in seconds of Placed Model Graphs (Parentheses show Slowdown compared to Sufficient Memory for the same algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Model Batch Size Memory Fraction Single GPU Expert m-TOPO m-ETF m-SCT Inception-V3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 OOM OOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='690 (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='312 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='292 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9%) TensorFlow GNMT 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 OOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='221 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='272 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='230 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='212 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) Inception-V3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 OOM OOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='275 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='250 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='7%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='254 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4%) Inception-V3 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 OOM OOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='537 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='527 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='535 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1%) PyTorch Transformer 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 OOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='262 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='240 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='241 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0%) These observations show that Baechi’s m-ETF and m-SCT are able to generate placements with step times in the same ballpark as the expert, while taking significantly less time to create a placement than the manual expert which takes minutes to hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-TOPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 4 also shows that, Baechi-TF’s m-TOPO is significantly slower than m-ETF and m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-TOPO’s step times are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8%–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4% slower than m-ETF and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8%–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3% slower than m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' After analysing m-TOPO we found that it places most of the encoder’s LSTM layers at the first two GPUs, and most of the decoder LSTM layers at the other two GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, this parallelization is offset negatively by the high data transfers between the kernel weight and the LSTM cell operators for LSTM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Similarly, with Transformer in Baechi-PY, m-TOPO’s step time is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3%–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% slower than m-ETF and m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Essentially m-TOPO fails to exploit the parallelism between the encoder and the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-SCT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Theoretical analysis in [26] shows SCT beating ETF and one would expect the same with m-SCT and m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In practice, the reverse is true—Table 4 shows that m-ETF’s step times are faster than m-SCT’s for 5 out of 6 settings in Baechi-TF (it is slower only under sequence length 40, batch size 128), and faster or equal in 3 out of all 4 settings in Baechi-PY (it is slower only under Inception-V3, batch size 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This behavior of m-SCT is because of two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' First, SCT’s optimality proof relies on the assumption that the minimum operator computation time is larger than or equal to the maximum communication time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This does not hold in our experimental machine—a 4 B GPU-GPU transfer takes 50–200 ms while, in TensorFlow, many operators execute within 1 ms, and 67% of Inception-V3’s operators take under 50 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The m-SCT LP model (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4) assumes parallel data transfers from an operator to all its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our experimental machine only allows sequential transfers (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Overall, m-SCT and m-ETF are comparable in practice, with m-ETF having a slight edge in both placement time and step time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 4Faster data transfers between GPUs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', via NVLink [55], have the potential to make m-SCT more competitive than m-ETF, but this is outside our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 23 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8 GPU0 GPU1 GPU2 GPU3 Normalized Peak Memory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0 = 30% of Max GPU memory, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4GB) Inception V3 (100% memory) Inception V3 (30% memory) GNMT-40 (100% memory) GNMT-40 (30% memory) Memory load distribution with Baechi-Tensorflow (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8 GPU0 GPU1 GPU2 GPU3 Inception V3 (100% memory) Inception V3 (30% memory) Transformer (100% memory) Transformer (30%memory) Memory load distribution with Baechi-PyTorch Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi Load Balance of Memory Usage using m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Dashed line is memory limit for each GPU (normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0 on y axis corresponds to 30% of the max GPU memory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 GB in a 8 GB GPU) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 Placement with Insufficient Memory Next, we limit per-GPU memory to a fraction of maximum available memory on the GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 5 shows results for: 1) Baechi TensorFlow with memory limited to 30%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', from 8 GB down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 GB, for: Inception-V3 with batch size of 32, and GNMT with batch size of 128 and sequence length 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' and 2) Baechi PyTorch: memory limited to 30% (Inception-V3 with batch size of 32, Transformer with batch size 64) and 40% memory limit (Inception-V3 with batch size 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A few notes follow on configuration changes in the experiments with Baechi-TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For GNMT, co-placement (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) remains enabled and we use the same configuration as Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Inception-V3 we disable co-placement as otherwise it generated a massive operator group, causing an Out of Memory error (OOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Disabling co-placement increases the number of operators to be placed from 2,620 to 7,077, and placement time from 1 s to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' No configuration changes were required for Baechi-PY experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Effect on Step Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 5 shows that the single GPU placer always suffers an OOM (Out of Memory) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The expert placer OOMs for Inception-V3 (in both TensorFlow and PyTorch), but succeeds for TensorFlow GNMT and PyTorch Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In comparison, all three variants of Baechi (m-TOPO, m-ETF, m-SCT) succeed in placing under insufficient memory under all five settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Inception-V3 in both Baechi-TF and Baechi-PY, only Baechi succeeds in placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to the sufficient memory cases (Table 4), m-ETF and m-SCT provide step times that are only 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9% worse in Baechi-TF and, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3% and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1% worse in Baechi-PY respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-TOPO in TensorFlow degrades by 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6% because of its disabled co-placement, which ballooned communication along the graph’s critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In PyTorch there is no change in m-TOPO since the algorithm does not depend on the maximum limit as long as it is more than m-TOPO’s per device cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For GNMT and Transformer, the overheads of all three Baechi algorithms and the expert are small (shown as % numbers within parentheses), meaning that with insufficient memory Baechi is nearly as fast as when memory is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Load Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Figure 7 shows the peak memory usage, normalized to the memory limit for each GPU (insufficient memory case) for Baechi-TF and Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In both Baechi-TF and Baechi-PY, for Inception-V3, Manuscript submitted to ACM 24 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 GNMT-40, m-SCT (128, 100%) GNMT-40, m-ETF (128, 100%) GNMT-40, m-SCT (256, 100%) GNMT-40, m-ETF (256, 100%) Noramalized Step Time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0 = Same as un-perturbed step time) GNMT-40 = GNMT with sequence length 40 Max, Min step-time Average step-time Step-times Sensitivity test on Baechi-Tensorflow (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='98 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 IV3, m-ETF (32, 30%) IV3, m-SCT (32, 30%) Tr, m-ETF (64, 30%) Tr, m-SCT (64, 30%) IV3, m-ETF (64, 100%) IV3, m-SCT (64, 100%) IV3 = Inception V3, Tr = Transformer Max, Min step-time Average step-time Step-times Sensitivity test on Baechi-PyTorch Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi Sensitivity to Profiling Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' All computation and communication times are perturbed randomly by up to 20% and the step time for placement generated by Baechi is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Values in X-axis parentheses are (Batch Size, Memory Fraction Available) with a 30% memory cap, a single GPU does not suffice, and that m-SCT relies on a mix of multiple GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In particular, 2 of the 4 GPUs appear to be used more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is because Inception-V3 has more barriers (sync points) than GNMT in TensorFlow, limiting Inception-V3’s ability to parallelize effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For TensorFlow GNMT and PyTorch Transformer, Baechi’s m-SCT is able to load-balance more evenly (than Inception- V3) across the GPUs, even when memory is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In fact, for both these cases we found that m-SCT generates an identical placement in both cases with sufficient and with insufficient memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This fact is also true for the expert, m- TOPO, and m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However specifically in case of GNMT with Baechi-TF, their step times are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% higher than the sufficient memory cases (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='This slowdown is because of TensorFlow runtime memory optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, when the memory usage approaches its limit, the TensorFlow runtime resorts to certain memory optimizations to decrease peak memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For the expert placement, peak memory usage for one GPU device decreases from 2 GB (83% of the memory limit) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='45 GB and thus the number of memory operations increases 6% under insufficient memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' These memory optimizations do not kick in for m-SCT, making it faster than the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Profile Sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To measure Baechi’s sensitivity to profiling errors, we perform runs where in each run all computation and communication profiles are randomly and independently perturbed by up to ±20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This should account for errors in our time measurements as well as small speed differences between the device used to profile the model and devices where the model actually runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Figure 8 shows the perturbation in step-times of the resulting placements, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' to step-time without any perturbation of the profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to the unperturbed step times, the step times with perturbed profiles remain within a fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='99× to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3× in Baechi-TF, and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='97× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='08× times in Baechi-PY Thus we conclude that m-SCT and m-ETF are resilient to reasonable levels of errors in profiled values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5 Benefit of Baechi Optimizations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 Benefit of Baechi-TF Optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 6 shows the benefit from the combined optimizations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3 in Baechi-TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use Inception-V3 with batch size 32 and GNMT with batch size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We use the m-SCT variant of Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The experimental setup has 4 GPUs with sufficient memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 25 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Benefits of Baechi-TF Optimizations (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Number of Operators to be Placed, Placement Times in seconds, and Average Step Times in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' m-SCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Model Un-Optimized Optimized Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Ops Placement (seconds) Step (seconds) Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Ops Placement (seconds) Step (seconds) Inception-V3 6884 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='302 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='269 GNMT (length: 40) 18050 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='580 542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='212 GNMT (length: 50) 22340 406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='793 706 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='267 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Benefits of communication protocol in Baechi-PY (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Step times in seconds without and with the protocol Model Algorithm Without Protocol With Protocol % Change Inception V3 (32, 30% memory) m-ETF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% m-SCT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='254 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5% Inception V3 (64, 40% memory) m-ETF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='528 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3% m-SCT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='535 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8% Transformer (64, 100% memory) m-ETF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% m-SCT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0% Overall, we observe that Baechi-TF’s combined optimizations achieve 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6×–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3× speedup in placement times, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1×–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0× speedup in step times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We discuss a few interesting aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Operator fusion (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) reduces both number of operators to be placed and thus also placement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Forward-operator-based placement (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) significantly speeds up placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely the latter optimization reduces the number of operators to be placed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='7× for Inception-V3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5×–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0× for GNMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This accelerates the placement times 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='7× for Inception-V3 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2×–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4× for GNMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Co-placement (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) is efficient because it clusters operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This reduces step times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While co-placement does not change the operator count to be placed, it decreases placement time by reducing the overhead of calculating schedulable times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 Benefit of Baechi-PY Communication Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' To evaluate the communication protocol in Baechi- PY (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2), we create a baseline plain wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In it, each node transfers the inputs from devices of its parent (if different from module’s device) by simply using blocking calls to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='to() instead of using streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Table 7 shows the comparison of step times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY’s communication protocol gives up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='5% benefit with Inception-V3, and very little benefit under Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is because, in PyTorch, both these models have a strong linear spine, which creates fewer opportunities for parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 6 Discussion and Limitations Algorithmic Approaches vs Learning-based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When we first implemented m-ETF and m-SCT, the placed models had very high step times because communication- intensive operators violated the SCT assumption (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We whittled away at this with a persistent effort at systems design and optimizations (outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1), which played a major role in bringing the step times down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Although Manuscript submitted to ACM 26 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' our exploration was efficient and we cycled new techniques and optimizations on a weekly basis, it took 1 person-year of effort to converge to what now appears in this paper for Baechi-TF, and an additional 1 person-year for Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is indicative of the difficulties associated with implementing scheduling algorithms on today’s open-source ML systems (and in a sense shows why existing learning-based approaches are so attractive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Nevertheless, our results show that the benefits of algorithmic design were worth the exploratory pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our experience also indicates reasons why developers (and companies!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=') often choose to “jump” so quickly towards using learning-based (including RL-based) solutions for scheduling problems: fast time to design (optimization of parameters and hyperparameters can be often be done as a rote task, rather than a creative task), and hence fast time to production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, this comes at the expense of latter pain points in generalizing learning-based approaches to different architectures and models (in comparison, Baechi runs as-is, given an arbitrary model and a machine profile), as well as the high times to generate a placement using learning-based approaches, which becomes a bottleneck in the exploratory design phase when the developer is iteratively building and revising their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We conclude that learning-based techniques (for any problem) should not be built in isolation from, or in lieu of, algorithmic-based approaches—but rather hand-in-hand with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Limitations of Baechi-TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The peak memory usage of TensorFlow is highly dependent on the execution order of operators [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' So Baechi would benefit most if the framework (TensorFlow or PyTorch) faithfully executed operators in the same order as specified by Baechi’s ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For Baechi-PY we enforce this via the “Reordering Problem” (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For TensorFlow while we do not enforce this ordering, and we observed in several runs of Baechi-TF that TensorFlow deviated from this order, yet memory caps were not violated for m-SCT and m-ETF in Baechi-TF runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is possible that if memory caps were tightened further (than 30%, compared to our experiments), engineering may be required for Baechi-TF to force TensorFlow to follow the ES execution order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Limitations of Baechi-PY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (i) Correctness issues with in-place operations: Inplace operations may lead to race conditions and incorrectness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Concretely, select modules in PyTorch can be made to modify the input tensors in-place, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', ReLU with in-place flag set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi-PY’s communication protocol (in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2) uses an independent tx stream to move out the tensors from a device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If the subsequent module in the compute stream is in-place, it may modify the tensor while it is being transferred out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This may lead to an incorrect tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While we did not encounter such a cases in evaluation, a simple fix is to turn off the in-place feature for the module in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This may however increase the memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (ii) Weight sharing in Transformers: Currently the Assigner (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3) does not support cases where weights are shared across multiple modules in the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', Transformers with embedder weight sharing [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (iii) Model code modifications: Some operations like concatenate and add, which take multiple inputs, must be defined as PyTorch modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Only then can the Assigner ensure that tensors being concatenated or added are on the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In most cases, this is a few lines of code change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, in Inception-V3, concatenate is used 7 times and add is used only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 7 Related work Data Parallelism (DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Data parallelism (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' DP) refers to training the same model replicas with multiple partitioned data in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This is motivated by increasing sizes of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' MALT [45] is a fault-tolerant, network-cost effective solution for data parallel ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Another common data parallelism framework is NESL [10], a first-order functional language that enables developers to put irregular-parallel program in parallel devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' OptiML [65] is a domain-specific Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 27 language (DSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Most major ML frameworks offer support for data parallelism [1, 16, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' While DP typically replicates the model on each device, ZeRO [60] eliminates this redundancy and reduces memory consumption in DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' DP is orthogonal to model parallelism, and therefore DP techniques can be integrated into Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Model Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to data parallelism, relatively fewer solutions exist for model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' DistBelief [19] and STRADS [40] require the user to manually specify device placement, while the systems in [42, 44] do not generalize to arbitrary ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' As discussed in Section 1, reinforcement-learning based approaches have been popular lately to perform placement for model parallelism, including work from Google [50, 51] and the Placeto system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ColocRL [51] trains a sequence-to- sequence model by RL to generate placements of manually grouped subsets of TensorFlow operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' HierarchicalRL [50] substitutes the human intervention for grouping operators with an ML model and jointly trains the ML models for operator grouping and device placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Placeto [2] proposes an approach that transfers learned device placement models to new ML models in order to minimize training times for the new model placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The original version of our paper [33] both inspired follow-up work [25], and also had parallel work [67], on algorithmic approaches to model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However [67] is standalone, meaning that it is not integrated with TensorFlow or PyTorch, making a fair comparison with Baechi difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Pesto [25] presents direct comparisons with Baechi (Baechi-TF)—the most important metric of placement times are similar for Pesto and Baechi, with small improvements for step time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus for practical purposes we consider Pesto to be comparable in performance to Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Works like PipeDream [27], GPipe [31], DAPPLE [22], PipeMare [79] introduce and optimize various aspects of Pipeline Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Pipeline Parallelism, the model is usually vertically split into contiguous stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Amazon Sagemaker recently introduced automating Model and Pipeline Parallelism on their platform recently, though their code is proprietary[5, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Techniques for pipeline parallelism can be integrated orthogonally into Baechi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Model Parallelism for large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Recent progress on very large language models like GPT [14, 15] have given rise to works that focus specifically on Model Parallelism for such models like Megatron-LM [63] and TeraPipe [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' However, these systems focus narrowly on Transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Alpa [82] combines intra and inter-operator parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The inter-operator parallelism is limited to vertical splits and will not, for instance, place multiple parallel branches of a model on different devices, thus making it different from model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Classical Parallel Scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Classical parallel scheduling, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', ETF [32] and SCT [26], has been widely used in task scheduling on multiple computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ETF and SCT are used as baselines by many subsequent works [20, 30, 52, 74, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' None of these address memory constraints and a finite number of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For instance, Eyraud-Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [20] investigate the execution of tree-shaped task graphs using multiple processors, but without always obeying memory restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' TensorFlow Graph Optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Existing techniques [69, 70] work only after the graph has been placed—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', to improve operations’ performance—and thus are inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=', Running Grappler (TensorFlow’s graph optimizer) generates an optimized graph protobuf, but it is unusable as it lacks certain metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s targeted problem is harder as we have to both optimize the graph and do placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 8 Conclusions We have proposed algorithmic solutions to model parallelism, useful in scenarios where devices are memory-constrained or neural networks are massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Among our three algorithms (m-ETF, m-TOPO, m-SCT), the m-SCT algorithm is Manuscript submitted to ACM 28 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' provably within a constant factor of the optimal achievable training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We have implemented these algorithms into our new Baechi system, as two systems Baechi-TF and Baechi-PY which are respectively usable in a modular manner with TensorFlow and PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Experimental results showed that, across TensorFlow and PyTorch, our approaches reduce placement time by a factor of between 654×–206000× compared to today’s state-of-the-art placement approaches which are learning-based, while increasing step time (makespan) by only up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2% compared to expert placers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When memory is constrained further, while single GPU and expert placers suffer OOM errors, Baechi’s algorithms, especially m-SCT and m-ETF, were able to place successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Compared to sufficient memory the step times suffered an increase of only up to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='8% in TensorFlow and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1% in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Further, Baechi-TF’s optimizations help reduce placement time by 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='6×–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='3×, and step time by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='1×–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='0×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We also conclude that m-SCT and m-ETF perform comparably, with m-ETF having a slight edge for slower networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The original version of our paper [33] inspired follow-up work [25] along with parallel work [67], on algorithmic approaches to model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Together, this new generation of algorithms for model parallelism offers the promise of speed, generalizability, predictability, and analyzability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' These will be invaluable as learning models, both training and inference, move closer to edge devices and human-facing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Baechi’s code is openly available at the following link: http://dprg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='uiuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='edu/downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='php Acknowledgments This work was supported in part by the following grants: NSF IIS 1909577, and NSF CNS 1908888;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' as well as by generous gifts from Capital One, Schlumberger, and Microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We thank Xiaojuan Ma for her invaluable help in reviewing the proofs in the appendix.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='com/sagemaker/latest/dg/model-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='html [6] Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [Accessed 2-July-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Amazon Web Services (AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' https://aws.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 29 [11] Keith Bonawitz, Hubert Eichner, Wolfgang Grieskamp, Dzmitry Huba, Alex Ingerman, Vladimir Ivanov, Chloé Kiddon, Jakub Konečný, Stefano Mazzocchi, Brendan McMahan, Timon Van Overveldt, David Petrou, Daniel Ramage, and Jason Roselander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Towards Federated Learning at Scale: System Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In Proceedings of Machine Learning and Systems, A.' metadata={'source': 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Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Language Models are Few-Shot Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' CoRR abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='14165 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' arXiv:2005.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Topological Sorting of Large Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ACM 5, 11 (1962), 558–562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [38] Can Karakus, Rahul Huilgol, Fei Wu, Anirudh Subramanian, Cade Daniel, Derya Çavdar, Teng Xu, Haohan Chen, Arash Rahnama, and Luis Quintela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='05972 [39] Narendra Karmarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A new Polynomial-Time Algorithm for Linear Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=" In 16th Annual ACM Symposium on Theory of Computing (STOC '84)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' ACM, 302–311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' [40] Jin Kyu Kim, Qirong Ho, Seunghak Lee, Xun Zheng, Wei Dai, Garth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Gibson, and Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' STRADS: A Distributed Framework for Scheduled Model Parallel Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=" In 11th European Conference on Computer Systems (EuroSys '16)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Alpa: Automating Inter- and Intra-Operator Parallelism for Distributed Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' CoRR abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='12023 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='12023 https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='12023 Manuscript submitted to ACM 32 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A Optimality Analysis for m-ETF We now derive an upper bound on the makespan of graph 𝐺 running on 𝑛 memory-constrained devices according to a placement generated by m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A bound for ETF (with no memory constraint on the devices) was obtained in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We extend it to the case where the devices have finite memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For any given 𝑝 devices, let 𝜔𝑝 m-etf be the m-ETF makespan and 𝜔𝑝 opt be the optimal makespan achieved using devices with infinte memory and zero device-to-device communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The makespan of m-ETF, 𝜔𝑛 m-etf is at most (2 + 𝜌)𝜔𝑅 opt, where 𝑅 is an integer < 𝑛 (computed in equation 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝐾 = 𝑛 · 𝑀 �𝑚 𝑖=1 𝑑𝑖 , where for any operator (task) 𝑖 in 𝐺, 𝑑𝑖 is the size of memory required by 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Intuitively, 𝐾 is the ratio of the total memory available from all devices to the total memory required by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus 𝐾 > 1, and for practical purposes we can assume that 𝐾 is sufficiently larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At each step, m-ETF greedily matches a ready task to an available device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Specifically, a device is said to be available if there is neither a task currently running nor has been scheduled to run on that device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' A task is said to be ready if all of its predecessors have completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝐼 and 𝐴 be the set of available devices and ready tasks at a given step respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' When a task completes, 𝐼 is updated to include the recently free device and all of the task’s children that are ready are added to 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' At each such step, a device 𝑑 is said to be memory-sufficient (MS) if the remaining free memory on 𝑑 is greater than the memory requirement of each task in 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If 𝑑 has insufficient memory for even a single task in 𝐴, we say 𝑑 is not MS thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is removed from 𝐼 and is not considered for any further placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The time (0,𝜔m-etf) can be partitioned into two distinct sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Set A containing the time-periods when all the MS devices (in that time-period) are busy and set B when at least one MS is idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Suppose B is the disjoint union of intervals (𝑏𝑙𝑖,𝑏𝑟𝑖) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e, B = (𝑏𝑙1,𝑏𝑟1) ∪ (𝑏𝑙2,𝑏𝑟2) ∪ · · · ∪ (𝑏𝑙𝑞,𝑏𝑟𝑞) where 𝑏𝑙1 < 𝑏𝑟1 < 𝑏𝑙2 < 𝑏𝑟2 · · · < 𝑏𝑙𝑞 < 𝑏𝑟𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Lemma (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We can find a chain of tasks, 𝑋 : 𝑇𝑙 → 𝑇𝑙−1 → · · · → 𝑇1 such that 𝑞 ∑︁ 𝑖=1 (𝑏𝑟1 − 𝑏𝑙1) ≤ 𝑙∑︁ 𝑗=1 𝑡(𝑇𝑗) + 𝑙−1 ∑︁ 𝑗=1 𝑐𝑗 (𝑗+1) That is, the total time period of B will be covered by computation and communication times along the chain 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will denote �𝑙−1 𝑗=1 𝑐𝑗 (𝑗+1) by 𝐶𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' For proof, we refer the reader to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='2 in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑟 be the number of devices that remain MS until the end of m-ETF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content='e at time 𝜔m-etf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' With 𝐾 > 1, we will have 𝑟 >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let ˆB be the set of all time-periods when atleast one of these 𝑟 devices is idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that ˆB ⊆ B, thus the chain 𝑋 from A will cover ˆB as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus we have: ∑︁ 𝑟 𝑡(𝜙𝑖) ≤ 𝑟 × ∑︁ 𝑇𝑗 ∈𝑋 𝑡(𝑇𝑗) + 𝑟 × 𝐶𝑋 (3) where 𝜙𝑖 is the set of times when the device 𝑑𝑖 is idle in (0,𝜔m-etf) and 𝑡(𝑇) is computation time of task 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 33 Let 𝜔𝑝 opt be the optimal makespan on 𝑝 devices with no memory limits and no communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since the makespan on any number of devices is at least as large as a chain in the graph, we have 𝜔𝑟 opt ≥ ∑︁ 𝑋 𝑡(𝑇𝑗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (𝑖), 𝜔𝑛 opt ≥ ∑︁ 𝑋 𝑡(𝑇𝑗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (𝑖𝑖) (4) Also, the net computation time of 𝐺 can be bounded as: ∑︁ 𝑇𝑗 ∈𝐺 𝑡(𝑇𝑗) ≤ 𝑟 × 𝜔𝑟 opt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (𝑖), ∑︁ 𝑇𝑗 ∈𝐺 𝑡(𝑇𝑗) ≤ 𝑛 × 𝜔𝑛 opt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (𝑖𝑖) (5) Now we bound the m-ETF makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Consider the 𝑟 devices, their idle time and the jobs running on them: 𝜔𝑛 m-etf = 1 𝑟 �∑︁ 𝑟 𝑡(𝑇𝑗) + ∑︁ 𝑟 𝑡(𝜙𝑖) � (6) ≤ 1 𝑟 �∑︁ 𝐺 𝑡(𝑇𝑗) + ∑︁ 𝑟 𝑡(𝜙𝑖) � (7) Using 3, 4(i) and 5(i), 𝜔𝑛 m-etf ≤ 1 𝑟 � 2𝑟 × 𝜔𝑟 opt + 𝑟 × 𝐶𝑋 � = 2𝜔𝑟 opt + 𝐶𝑋 Similarly, using 3, 4(ii) and 5(ii), 𝜔𝑛 m-etf ≤ 1 𝑟 � (𝑛 + 𝑟) × 𝜔𝑛 opt + 𝑟 × 𝐶𝑋 � = �𝑛 + 𝑟 𝑟 � 𝜔𝑛 opt + 𝐶𝑋 Note that 𝑟 will vary depending on the exact topological order considered for the m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' So we define 𝑅 as the minimum 𝑟 across all possible topological ordering of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus we have, 𝜔𝑛 m-etf ≤ min � 2𝜔𝑅 opt, �𝑛 + 𝑅 𝑅 � 𝜔𝑛 opt � + 𝐶𝑋 (8) Further, with 𝜌 as the ratio between maximum communication ttime and minimum computation time, we have 𝐶𝑋 ≤ 𝜌𝜔𝑛 opt (also 𝐶𝑋 ≤ 𝜌𝜔𝑅 opt ) Thus we have, 𝜔𝑛 m-etf ≤ � 1 + 𝑛 𝑅 + 𝜌 � 𝜔𝑛 opt (9) Alternatively, using the bound with 𝜔𝑅 opt, 𝜔𝑛 m-etf ≤ (2 + 𝜌)𝜔𝑅 opt (10) Here we note that bound in Equation 10 is of the same form as the original bound on ETF given in [32], where 𝑅 replaces 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus the makespan of m-ETF, like ETF, is within a constant factor of the optimal Finally, 𝑅 can be computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let the largest memory requirement of any task in 𝐺 be 𝐽 × 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since the devices become non-MS when they can not place any of the available task in A , a memory of only (1 − 𝐽)𝑀 is use-able at each device in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Thus by greedily filling in the tasks onto devices, we get: Manuscript submitted to ACM 34 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 𝑅 ≥ 𝑛 − � �𝑚 𝑖=1 𝑑𝑖 (1 − 𝐽)𝑀 � = 𝑛 − 𝑛 (1 − 𝐽)𝐾 Rounding it up, we have, 𝑅 = � 𝑛 � 1 − 1 (1 − 𝐽)𝐾 �� (11) □ B Optimality Analysis of m-SCT We now formally prove that m-SCT’s approximation ratio to optimal is an additive constant away from SCT’s approxi- mation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since SCT itself was known to be within a constant factor of optimal [26], our result means that m-SCT is also within a constant factor of optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Recall that we assume 𝜌 - the ration of maximum communication time to minimum computation time (defined in Table: 1) - is less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will use similar notation to our analysis for m-ETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝐾 = 𝑛 · 𝑀 �𝑚 𝑖=1 𝑑𝑖 , where for any operator (task) 𝑖 in 𝐺, 𝑑𝑖 is the size of memory required by 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will define 𝐽 as the ratio between largest memory requirement from a single task and 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Formally, 𝐽 = max𝑖 ∈[𝑚] 𝑑𝑖 𝑀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑠𝑖 be the start time of task 𝑖 in m-SCT, and 𝑠∞ 𝑖 be the start time of task i in the infinite device SCT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑢𝑗 be the time where a task 𝑗 becomes urgent, which is exactly the earliest time when task 𝑗 can start on any device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Formally, 𝑢𝑗 = 𝑚𝑎𝑥𝑖→𝑗 ∈𝐸(𝐺)𝑠𝑖 + 𝑝𝑖 + 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Similar to m-ETF, we will say a device 𝑑 is memory sufficient (abbreviated as MS) at time 𝑇 if and only if remaining free memory on 𝑑 is greater than the memory requirement of each task in 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Finally, we will use 𝑟 to denote the number of devices that are MS throughout the scheduling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will now analyse the approximation ratio of m-SCT with three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' First we will show that not many tasks are impacted by devices going out of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Next we will show that any MS device must be idle only for a limited time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Our proof for this step follows a similar outline to the proof of Theorem 3 in [26], but our proof is significantly shorter due to a condensed case analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Finally, we will bound the makespace of m-SCT by summing up the idle and busy time on MS devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' There are at most 𝑛 − 𝑟 task pairs (𝑖, 𝑗) such that 𝑗 is 𝑖’s favourite child, however when 𝑗 is scheduled, the device 𝑑 where 𝑖 is scheduled on does not have sufficient memory for task 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since 𝑖 is scheduled on device 𝑑, 𝑑 must be memory sufficient when 𝑖 is scheduled, but is no longer memory sufficient sometime after 𝑖 is scheduled (since 𝑑 does not have sufficient memory for task 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since there are in total 𝑛 devices and 𝑟 devices are always memory sufficient throughout m-SCT, there must only be 𝑛 − 𝑟 events where a device transition from being memory sufficient to not memory sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' □ Lemma 3 (Variant of Lemma 6 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Given two time units 𝑠′ ≤ 𝑠 such that 𝑠 − 𝑠′ ≤ 𝑐𝑚𝑎𝑥, let 𝑖 be a task such that 𝑠𝑖 ≤ 𝑠′ ≤ 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑚𝑎𝑥, then any busy or awake device at 𝑠′ is free for at most max(𝑠′ − 𝑠𝑖,𝑐𝑚𝑎𝑥) time during [𝑠𝑖,𝑠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (1) If a device 𝑑 is busy at 𝑠′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let task 𝑎 be the task that is running at time 𝑠′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' There are two possibilities: If task 𝑎 started before 𝑠𝑖, then device 𝑑 is busy for at least 𝑠′ −𝑠𝑖 time, thus free for at most 𝑠 −𝑠′ ≤ 𝑐𝑚𝑎𝑥 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' On the other hand, if task 𝑎 started at some time 𝑠∗ where 𝑠𝑖 ≤ 𝑠∗ ≤ 𝑠′, then either task 𝑎 is still being executed at 𝑠, or task 𝑎 has completed at 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In the first case device 𝑑 is free for at most 𝑠∗ −𝑠𝑖 ≤ 𝑠′ −𝑠𝑖 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' In the second Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 35 case device 𝑑 is busy for at least 𝑘𝑎 ≥ 𝑐𝑚𝑎𝑥 ≥ 𝑠 − 𝑠′ time, and thus is free for at most 𝑠 − 𝑠𝑖 − (𝑠 − 𝑠′) = 𝑠′ − 𝑠𝑖 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We conclude that device 𝑑 must be busy for at least min{𝑠′ − 𝑠𝑖,𝑠 − 𝑠′} time during [𝑠𝑖,𝑠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (2) If a device is awake at 𝑠′, let 𝑎 be the last task on 𝑑 before 𝑠′ and let 𝑠∗ be when 𝑎 finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then we know by the nature of our algorithm that some task 𝑏 will start on 𝑑 no later than 𝑠∗ + 𝑐𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since 𝑠 − 𝑠′ ≤ 𝑐𝑚𝑎𝑥, we know that 𝑏 is not yet finished at time 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Therefore either task 𝑎 starts after 𝑠𝑖 (which means the device is busy for at least 𝑘𝑎 ≥ 𝑐𝑚𝑎𝑥 time and free for at most 𝑠 − 𝑠𝑖 − 𝑐𝑚𝑎𝑥 ≤ 𝑠′ − 𝑠𝑖 time), or the device is vacant for at most 𝑐𝑚𝑎𝑥 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Assume that task 𝑗’s favourite parent𝑖∗’s device is MS during time period [𝑠𝑖,𝑠𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then there exists a predecessor 𝑖 of 𝑗 such that the total amount of idle time during [𝑠𝑖,𝑠𝑗] on any device 𝑑 that is MS throughout the period is at most 𝑠∞ 𝑗 − 𝑠∞ 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Note that since task 𝑗’s favourite predecessor 𝑖∗’s device is MS during [𝑠𝑖,𝑠𝑗], it is possible to schedule 𝑗 on the same device as 𝑖∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' This fact will be used in the case analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑖 be a predecessor of 𝑗 such that 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗 is maximized (namely, 𝑢𝑗 = 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Notice also that after 𝑗 becomes urgent at 𝑢𝑗 and before 𝑗 is scheduled, all memory sufficient devices must be busy (otherwise 𝑗 would have been scheduled on a device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence for any 𝑇 < 𝑠𝑗, the total vacant time for an MS device during [𝑇,𝑠𝑗] is equal to its total vacant time during [𝑇,𝑢𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Now we will discuss three different scenarios and prove that in each scenario, an MS device is vacant for at most 𝑠∞ 𝑗 − 𝑠∞ 𝑖 time during [𝑠𝑖,𝑠𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (1) When 𝑗 is not the favorite child of 𝑖, we know that in the infinite device SCT algorithm, 𝑖 and 𝑗 are scheduled on different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence 𝑠∞ 𝑗 ≥ 𝑠∞ 𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' On the other hand, in m-SCT, after 𝑗 becomes urgent (𝑗 becomes urgent at time 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗) and before 𝑗 is scheduled, any MS device must be busy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Therefore the amount of vacant time on each MS device during [𝑠𝑖,𝑠𝑗] must be at most 𝑘𝑖 + 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (2) When 𝑗 is the favorite child of 𝑖, but 𝑖’s device is not awake when task 𝑖 ends, we know that the time 𝑗 can be ready on𝑖’s device is the same as 𝑗’s urgent time, which means there is another task𝑤 such that𝑠𝑤+𝑘𝑤+𝑐𝑤𝑗 = 𝑠𝑖+𝑘𝑖+𝑐𝑖𝑗, but 𝑗 is not 𝑤’s favorite child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We can now use exactly the same argument as in the first case to prove that vacant time on any MS device during [𝑠𝑖,𝑠𝑗] must be at most 𝑘𝑖 + 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' (3) When 𝑗 is the favorite child of 𝑖, and 𝑖’s device is awake when task 𝑖 ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Denote the time 𝑗 becomes ready on 𝑖’s device as 𝑟𝑒𝑎𝑑𝑦(𝑗), there must exist some 𝑗’s predecessor 𝑦 ≠ 𝑖 such that 𝑠𝑦 + 𝑘𝑦 + 𝑐𝑦𝑗 = 𝑟𝑒𝑎𝑑𝑦(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since 𝑖’s device is awake when task 𝑖 ends, task 𝑗 will be scheduled on 𝑖’s device if it is still idle by 𝑟𝑒𝑎𝑑𝑦(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence a task 𝑤 (which is either 𝑗 or an urgent task) has to be scheduled on the device of 𝑖 at or before 𝑟𝑒𝑎𝑑𝑦(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will now consider the predecessor successor pair (𝑦, 𝑗), and prove that during [𝑠𝑦,𝑢𝑗] the vacant time on any MS machine is at most 𝑠∞ 𝑗 − 𝑠∞ 𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If 𝑤 = 𝑗, note that 𝑗 is not 𝑦’s favorite child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence in the infinite device SCT, 𝑗 and 𝑦 are not on the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We hence conclude that 𝑠𝑗 − 𝑠𝑦 = 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 = 𝑘𝑦 + 𝑐𝑦𝑗 ≤ 𝑠∞ 𝑗 − 𝑠∞ 𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' If 𝑤 ≠ 𝑗, then 𝑤 must be urgent (the only tasks that are allowed to be scheduled on an awake machine is the favorite child and urgent tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence at the start time of 𝑤, it must be the case that all MS devices are either busy or awake (because if there is a free MS device, k would have been scheduled on it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By Lemma 3, any busy or awake device at the start time of 𝑤 can only be vacant for at most 𝑚𝑎𝑥(𝑠𝑤 − 𝑠𝑦,𝑐𝑚𝑎𝑥) time during [𝑠𝑦,𝑢𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Now we will upper bound 𝑚𝑎𝑥(𝑠𝑤 −𝑠𝑦,𝑐𝑚𝑎𝑥) using the facts 1) 𝑤 happens before 𝑟𝑒𝑎𝑑𝑦(𝑗), but after Manuscript submitted to ACM 36 Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' task 𝑖 is completed (namely, after 𝑠𝑖 + 𝑘𝑖) and 2) 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 ≥ 𝑘𝑦 ≥ 𝑐𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' 𝑚𝑎𝑥(𝑠𝑤 − 𝑠𝑦,𝑐𝑚𝑎𝑥) ≤ 𝑚𝑎𝑥(𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦,𝑐𝑚𝑎𝑥) = 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Since in the infinite device SCT, 𝑗 and 𝑦 are not on the same device, we now conclude that the total vacant time on an MS device must be at most 𝑟𝑒𝑎𝑑𝑦(𝑗) − 𝑠𝑦 = 𝑘𝑦 + 𝑐𝑦𝑗 ≤ 𝑠∞ 𝑗 − 𝑠∞ 𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Assume that task 𝑗’s favourite parent 𝑖∗’s device is not MS during time period [𝑠𝑖,𝑠𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then there exists a predecessor 𝑖 of 𝑗 such that the total amount of idle time during [𝑠𝑖,𝑠𝑗] on any device 𝑑 that is MS throughout the period is at most 𝑠∞ 𝑗 − 𝑠∞ 𝑖 + 𝑐𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' As argued in Lemma 4, any MS device must be busy after 𝑢𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑖 be a predecessor of 𝑗 such that 𝑠𝑖 +𝑘𝑖 +𝑐𝑖𝑗 is maximized (namely, 𝑢𝑗 = 𝑠𝑖 + 𝑘𝑖 + 𝑐𝑖𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Then 𝑢𝑗 − 𝑠𝑖 = 𝑘𝑖 + 𝑐𝑖𝑗 ≤ 𝑠∞ 𝑗 − 𝑠∞ 𝑖 + 𝑐𝑖𝑗 (because even with infinite number of devices, task 𝑖 must be fully executed before task 𝑗 starts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' □ Let 𝑅 be the minimum 𝑟 across all possible graph configurations with memory availability parameter 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will use 𝜔𝑝 m-sct and 𝜔𝑝 sct to denote the makespan of m-SCT (with memory limit) and SCT (without memory limit) with 𝑝 devices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Analogously, we will use 𝜔𝑝 m-opt and 𝜔𝑝 opt be the optimal makespan on 𝑝 devices with memory limit and with no memory limit respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We will use 𝛼 to denote the approximation of the infinite device SCT (against the optimal makespan with infinite devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' The makespan of m-SCT is at most ( 𝑝 𝑅 + 𝛼) · 𝜔𝑝 opt + (𝑛−𝑅) 𝑅 𝑐𝑚𝑎𝑥 for any 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝐷𝑀𝑆 be the set of all devices that are MS throughout the m-SCT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' We know that |𝐷𝑀𝑆 | = 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' It is clear that the total amount of computation time spent on devices in 𝐷𝑀𝑆 is at most the sum of computational time for all tasks, which is at most 𝜔𝑟 opt · 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Now we will count the amount of time a device 𝑑 ∈ 𝐷𝑀𝑆 is idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' WLOG, let 𝑇1 be the task that finishes last in m-SCT and let 𝑇𝑙 → 𝑇𝑙−1 → · · · → 𝑇1 be the chain of task in 𝐺 ending at 𝑇1 such that 𝑇𝑙 is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Before the start time 𝑠𝑙 of 𝑇𝑙, all MS devices must be busy, because 𝑇𝑙 is urgent from time 0 and would have been scheduled on a device as soon as it becomes idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Let 𝑛𝑑 be the number of task pairs (𝑖, 𝑗) such that 𝑖 is 𝑗’s favourite parent but 𝑖’s parent is not MS when task 𝑗 starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' By Lemma 2, 4 and 5 we know that during [𝑠𝑙,𝜔𝑝 m-sct] = [𝑠𝑙,𝑠1 + 𝑘1] (𝑠1, 𝑘1 are the start time and computation time of 𝑇1 respectively) , the amount of time 𝑑 is idle is at most �� � 𝑙−1 ∑︁ 𝑗=1 � 𝑠∞ 𝑗 − 𝑠∞ 𝑗+1 ��� � + 𝑛𝑑 · 𝑐𝑚𝑎𝑥 ≤ 𝜔∞ sct + 𝑛𝑑 · 𝑐𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Summing these all up for all devices in 𝐷𝑀𝑆 we get that the total idle time across all devices in 𝐷𝑀𝑆 is at most 𝑟 · 𝜔∞ sct + (𝑛 − 𝑟) · 𝑐𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM Baechi: Fast Device Placement of Machine Learning Graphs 37 We now conclude that 𝑟 · 𝜔𝑝 m-sct ≤ 𝑟 · 𝜔𝑟 opt + 𝑟 · 𝜔∞ sct + (𝑛 − 𝑟) · 𝑐𝑚𝑎𝑥 ⇒ 𝜔𝑝 m-sct ≤ 𝜔𝑟 opt + 𝜔∞ sct + 𝑛 − 𝑟 𝑟 𝑐𝑚𝑎𝑥 ≤ 𝜔𝑟 opt + 𝛼 · 𝜔∞ opt + 𝑛 − 𝑟 𝑟 𝑐𝑚𝑎𝑥 ≤ 𝜔𝑅 opt + 𝛼 · 𝜔∞ opt + 𝑛 − 𝑅 𝑅 𝑐𝑚𝑎𝑥 (because 𝑅 ≤ 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Lastly, observe that (without memory limit), the optimal makespan with 𝑅 devices is at most 𝑝 𝑅 times the optimal makespan with 𝑝 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Also, 𝜔∞ opt ≤ 𝜔𝑅 opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Hence 𝜔𝑛 m-sct ≤ ( 𝑝 𝑅 + 𝛼) · 𝜔𝑝 opt + (𝑛−𝑅) 𝑅 𝑐𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' One could minimize the RHS over all 𝑝 to get the best upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' □ Using the same analysis as for m-ETF, we know that 𝑅 = � 𝑛 � 1 − 1 (1 − 𝐽)𝐾 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} +page_content=' Manuscript submitted to ACM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFAT4oBgHgl3EQfyx73/content/2301.08695v1.pdf'} diff --git a/P9E3T4oBgHgl3EQfyAsA/vector_store/index.faiss b/P9E3T4oBgHgl3EQfyAsA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..44f84ab1af7beeb7647fbfcfa288cfb59c3a7ef8 --- /dev/null +++ b/P9E3T4oBgHgl3EQfyAsA/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2cf24b4be588e2d66034f3b04bdd870846eb2c4dcf7634277231bdf760fdc0a +size 2555949 diff --git a/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/2301.05151v1.pdf.txt b/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/2301.05151v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..acc4454a36f5b6e1494986ea66531e8593333c4f --- /dev/null +++ b/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/2301.05151v1.pdf.txt @@ -0,0 +1,1842 @@ +arXiv:2301.05151v1 [math.RT] 12 Jan 2023 +A local-global principle for unipotent +characters +Damiano Rossi +Abstract +We obtain an adaptation of Dade’s Conjecture and Späth’s Character Triple Conjecture to +unipotent characters of simple, simply connected finite reductive groups of type A, B and C. +In particular, this gives a precise formula for counting the number of unipotent characters of +each defect d in any Brauer ℓ-block B in terms of local invariants associated to e-local struc- +tures. This provides a geometric version of the local-global principle in representation theory +of finite groups. A key ingredient in our proof is the construction of certain parametrisations of +unipotent generalised Harish-Chandra series that are compatible with isomorphisms of charac- +ter triples. +Contents +1 +Introduction +2 +1.1 +Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Notation and background material +5 +2.1 +Characters and blocks of finite groups +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.2 +Finite reductive groups and unipotent characters . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +e-Harish-Chandra theory for unipotent characters . . . . . . . . . . . . . . . . . . . . . +7 +2.4 +Pseudo-unipotent characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3 +Compatibility with isomorphisms of character triples +10 +3.1 +Equivariance and maximal extendibility . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3.2 +Construction of GF -block isomorphisms of character triples . . . . . . . . . . . . . . . +14 +3.3 +Proof of Theorem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4 +Consequences of Theorem C +18 +4.1 +Parametrisation of pseudo-unipotent characters of Levi subgroups . . . . . . . . . . . +19 +4.2 +Above e-Harish-Chandra series +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +2010 Mathematical Subject Classification: 20C20, 20C33. +Key words and phrases: Dade’s Conjecture, Character Triple Conjecture, finite reductive groups, unipotent characters. +This work is partially supportedby the EPSRC grant EP/T004592/1 and was written during a research visit of the author at +the Universitá degli Studi di Firenze. The author would like to thank Silvio Dolfi and all the members of the algebra group +in the Department of Mathematics for their hospitality and, in particular, Carolina Vallejo for some comments concerning +the local-global principle. Moreover, the author would like to thank Lucas Ruhstorfer for some helpful conversation on +the paper [Bro-Ruh]. +1 + +5 +Towards Theorem A and Theorem B +22 +5.1 +Preliminaries on e-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +5.2 +Proof of Theorem A +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +5.3 +Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +1 +Introduction +The local-global conjectures are currently some of the most interesting and challenging problems in +representation theory of finite groups. Among others, these include the McKay Conjecture [McK72], +the Alperin—McKay Conjecture [Alp76] and Alperin’s Weight Conjecture [Alp87] all of which can +be deduced by a deeper statement known as Dade’s Conjecture [Dad92], [Dad94], [Dad97]. The +latter also implies the celebrated Brauer’s Height Zero Conjecture introduced in [Bra56] and whose +proof has recently been completed in [MNSFT22] and [Ruh22a] while relying on a combined effort +of many other authors. +In this paper, we are particularly interested in Dade’s Conjecture which, for every prime number +ℓ, suggests a precise formula for counting the number of irreducible characters of a finite group, +with a given ℓ-defect and belonging to a given Brauer ℓ-block, in terms of the ℓ-local structure of +the group itself. This conjecture has been further extended in [Spä17] where the Character Triple +Conjecture was formulated by introducing a compatibility with N-block isomorphisms of character +triples, hereinafter denoted by ∼N, as defined in [Spä17, Definition 3.6]. This notion plays a funda- +mental role in many aspects of group representation theory and, as we will see later, gives us a way +to control the representation theory of local subgroups. Furthermore, it was exploited to reduce +Dade’s Conjecture to finite quasi-simple groups as explained in [Spä17, Theorem 1.3]. +Our aim is to adapt and prove the two conjectures described in the previous paragraph to the case of +unipotent characters of finite reductive groups. The approach considered here is inspired by ideas +introduced by the author in [Ros22c] and provides further evidence for the conjectures formulated in +that paper [Ros22c, Conjecture C and Conjecture D]. In particular, the ℓ-local structures considered +above are replaced by more suitable e-local structures arising from the geometry of the underlying +algebraic group that are compatible with the framework of Deligne–Lusztig theory. Therefore, our +results also suggest the existence of an e-local-global principle for the representation theory of finite +reductive groups. +More precisely, let G be a simple, simply connected group of type A, B or C which is defined +over an algebraically closed field of positive characteristic p and let F ∶ G → G be a Frobenius +endomorphism endowing G, as a variety, with an Fq-structure for some power q of p. We denote by +GF the finite reductive group consisting of the Fq-rational points on G. Furthermore, we fix an odd +prime ℓ different from p and denote by e the multiplicative order of q modulo ℓ. We let Le(G,F) +denote the set of e-chains of (G,F) of the form σ = {G = L0 > L1 > ⋅⋅⋅ > Ln} where each Li +is an e-split Levi subgroup of (G,F). The final term of the e-chain σ is denoted by L(σ) = Ln, +while ∣σ∣ ∶= n is the length of σ. Observe that the latter induces a partition of the set Le(G,F) into +the sets Le(G,F)± consisting of those e-chains σ that satisfy (−1)∣σ∣ = ±1. Furthermore, notice +that GF acts by conjugation on the set Le(G,F) and indicate by GF +σ the stabiliser of the e-chain +σ. It follows directly from the definition that this action preserves the length of e-chains and, in +particular, it restricts to an action of GF on the set Le(G,F)>0 of e-chains of positive length. +2 + +Now, to each non-negative integer d and Brauer ℓ-block B of the finite group GF , we associate a +set Ld +u(B)± consisting of quadruples (σ,M,µ,ϑ) where σ is an e-chain belonging to Le(G,F)±, +(M,µ) is a unipotent e-cuspidal pair of (L(σ),F) such that M does not coincide with G, and ϑ is an +irreducible character of the e-chain stabiliser GF +σ belonging to the character set Uchd(Bσ,(M,µ)) +defined by the choice of d, B, σ and (M,µ) as described in Definition 5.5. Once again, the group GF +acts by conjugation on Ld +u(B)± and we indicate the corresponding set of GF -orbits by Ld +u(B)±/GF . +Moreover, for every such orbit ω, we denote by ω● the corresponding GF -orbit of pairs (σ,ϑ) such +that (σ,M,µ,ϑ) ∈ ω for some unipotent e-cuspidal pair (M,µ). +With the above notation, we are now able to state our first main result. For simplicity, in the next +theorem we assume that the prime ℓ does not divide the greatest common divisor (q ± 1,n + 1) +whenever (G,F) is of type An(±q) and where An(−q) denotes 2An(q) as usual. Observe however +that this assumption can be removed as explained in Remark 5.7 (see Theorem 5.9 for the more +general statement). +Theorem A. For every Brauer ℓ-block B of GF and every non-negative integer d, there exists an +AutF(GF )B-equivariant bijection +Λ ∶ Ld +u(B)+/GF → Ld +u(B)−/GF +such that +(Xσ,ϑ,GF +σ ,ϑ) ∼GF (Xρ,χ,GF +ρ ,χ) +for every ω ∈ Ld +u(B)+/GF, any (σ,ϑ) ∈ ω●, any (ρ,χ) ∈ Λ(ω)● and where X ∶= GF ⋊ AutF(GF ) +and AutF(GF ) is the group of automorphisms described in Section 3.1. +The above theorem provides an adaptation of Späth’s Character Triple Conjecture to the framework +of Deligne–Lusztig theory for the unipotent characters of finite reductive groups. Theorem A also +offers further evidence for the validity of [Ros22c, Conjecture D], in fact the set Ld +u(B)± introduced +above is a subset of the set of quadruples Ld(B)± considered in [Ros22c, Conjecture D] which +is identified by only selecting unipotent e-cuspidal pairs (M,µ) among those appearing in such +quadruples. +Next, we obtain a formula for counting the number of unipotent characters of ℓ-defect d in the +Brauer ℓ-block B in terms of local invariants associated to e-local structures. For each e-chain σ of +(G,F) with positive length, we define kd +u(Bσ) to be the number of characters belonging to one of +the character sets Uchd(Bσ,(M,µ)) for some unipotent e-cuspidal pair (M,µ) of (L(σ),F) up to +GF +σ -conjugation (see also (5.10)). Furthermore, let kd +u(B) and kd +c,u(B) be the number of irreducible +characters with ℓ-defect d and belonging to the Brauer ℓ-block B that are unipotent and unipotent +e-cuspidal respectively. Then, by using the bijection given by Theorem A we can determine the +difference kd +u(B) − kd +c,u(B) in terms of an alternating sum involving the terms kd +u(Bσ) arising +from the e-local structure GF +σ . +3 + +Theorem B. For every Brauer ℓ-block B of GF and every non-negative integer d, we have +kd +u(B) − kd +c,u(B) = ∑ +σ +(−1)∣σ∣+1kd +u(Bσ) +where σ runs over a set of representatives for the action of GF on Le(G,F)>0. +We point out that the restriction on the prime ℓ made for simplification before Theorem A only +concerns the condition on isomorphisms of character triples and hence does not affect Theorem B. +As before, this result provides an adaptation of Dade’s Conjecture to the framework of Deligne– +Lusztig theory for the unipotent characters of finite reductive groups and gives new evidence in +favour of [Ros22c, Conjecture C]. The necessity for the introduction of the corrective term kd +c,u(B) +in the equality of Theorem B can be understood as an analogue to the exclusion of the case of blocks +with central defect in the statement of Dade’s Conjecture or, depending on the formulation under +consideration, of the case where d = 0. We refer the reader to the more detailed discussion given in +the paragraph following Definition 5.2. Finally, we mention that Theorem B also provides evidence +for a positive answer to a question recently posed by Broué [Bro22a]. +It is particularly interesting to notice that, to the author’s knowledge, Theorem B cannot be obtained +directly using techniques available at the present time, but only as a consequence of the existence +of GF -block isomorphisms of character triples as those considered in Theorem A. In fact, while +Deligne–Lusztig theory allows us to control the representation theory of finite reductive groups, it +is not sufficient to control the representation theory of e-chain stabilisersGF +σ . However, observe that +the stabiliser GF +σ contains the finite reductive group L(σ)F as a normal subgroup. Therefore, we +can first use Deligne–Lusztig theory to study the characters of L(σ)F and then apply Clifford theory +via GF -block isomorphisms of character triples to control the characters of GF +σ (see Proposition 4.5 +and Proposition 5.6 for further details). +In order to achieve the latter step, we need to make Deligne–Lusztig theory and, more precisely, e- +Harish-Chandra theory for unipotent characters compatible with GF -block isomorphisms of char- +acter triples. This ideas was first suggested by the author in [Ros22c, Parametrisation B] and further +studied in [Ros22d]. Our next result, which is a key ingredient in the proofs of Theorem A and +Theorem B, establishes this conjectured parametrisation in the unipotent case under the assump- +tion specified above. This can also be seen as an extension of the parametrisation introduced by +Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] to the language of GF-block isomorphisms +of character triples. +Theorem C. For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)- +equivariant bijection +ΩG +(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) +that preserves the ℓ-defect of characters and such that +(Xχ,GF ,χ) ∼GF (NXχ(L),NG(L)F ,ΩG +(L,λ)(χ)) +for every χ ∈ E(GF ,(L,λ)) and where X ∶= GF ⋊ AutF(GF ). +The proof of Theorem C, and therefore of Theorem A and Theorem B, partially relies on certain +4 + +conditions on the extendibility of characters of e-split Levi subgroups that were first introduced to +settle the inductive conditions for the McKay Conjecture and the Alperin–McKay Conjecture, and +then further studied in the context of Parametrisation B of [Ros22c] (see the exact statement given +in [Ros22d, Definition 5.2]). These conditions were obtain, under certain assumptions, for groups +of type A, B and C in the papers [BS20], [Bro22b] and [Bro-Ruh] respectively. Nonetheless, a +version of these results is expected to hold in general and hence we believe that the above theorems, +obtained here for types A, B and C with respect to an odd prime ℓ, will extend to the general case +as well. +1.1 +Structure of the paper +The paper is organised as follows. In Section 2 we introduce the necessary notation and recall the +main definitions and results used throughout the paper. Furthermore, in Section 2.4 we introduce +the notion of pseudo-unipotent character (see Definition 2.2) and prove a result on the regularity +of blocks covering those containing such characters. Next, in Section 3 we start working towards +a proof of Theorem C. First, in Section 3.1 we consider certain equivariance properties that can +be established in the presence of extendibility conditions for characters of e-split Levi subgroups. +Here, we also present a candidate for the bijection ΩG +(L,λ) required by Theorem C. Next, in Sec- +tion 3.2 we construct the required GF-block isomorphisms of character triples. Using these results, +we can then prove Theorem C in Section 3.3. The following step is to extend the parametrisation +of unipotent e-Harish-Chandra series in the group G, as given by Theorem C, to a parametrisa- +tion of pseudo-unipotent e-Harish-Chandra series in F-stable Levi subgroups K of (G,F). This +is done in Theorem 4.4. Once this is established, in Section 4.2 we exploit the theory of GF -block +isomorphisms to obtain bijections above e-Harish-Chandra series that are required to control the +representation theory of the e-chain stabilisers GF +σ . A more detailed analysis of the characters of +GF +σ is carried out in Section 5.1. In particular, we obtain a parametrisation of the character sets +Uchd(Bσ,(M,µ)) in Proposition 5.6. Finally, in Section 5.2 and Section 5.3 we apply these results +to prove Theorem A and Theorem B respectively. +2 +Notation and background material +2.1 +Characters and blocks of finite groups +We recall some standard notation from representation theory of finite groups as can be found in +[Isa76] and [Nav98], for instance. Let Irr(G) the set of ordinary irreducible characters. If N ⊴ G +and ϑ ∈ Irr(N), then we denote by Irr(G ∣ ϑ) the set of irreducible characters of G that lie above +ϑ. More generally, if S is a subset of irreducible characters of N, then we denote by Irr(G ∣ S) the +union of the sets Irr(G ∣ ϑ) for ϑ ∈ S, that is, the set of irreducible characters of G that lie above +some character in the set S. +Next, we denote by Gϑ the stabiliser of the irreducible character ϑ ∈ Irr(N) under the conjugacy +action of G and say that ϑ is G-invariant if G = Gϑ. In this case, we say that (G,N,ϑ) is a character +triple. These objects provide important information in the study of Clifford theory and play a crucial +role in many aspects of the local-global conjectures. Of paramount importance is the introduction of +certain binary relations on the set of character triples. We refer the reader to [Nav18, Chapter 5 and +10] and [Spä18] for a more detailed introduction to these ideas and for the necessary background on +5 + +projective representations. The binary relation considered here was introduced in [Spä17, Definition +3.6] and is known as N-block isomorphism of character triples, denoted by ∼N. This equivalence +relation has further been studied in [Ros22a]. +In order to construct N-block isomorphisms of character triples, it is often useful to prove certain +results on the extendibility of characters. Here, we introduce the notion of maximal extendibility (see +[MS16, Definition 3.5]) that will be considered in the following sections. Let N ⊴ G be finite groups +and consider S a subset of irreducible characters of N. Then, we say that maximal extendibility +holds for the set S with respect to the inclusion N ⊴ G if every character ϑ ∈ S extends to its +stabiliser Gϑ. More precisely, we can specify an extension map +Λ ∶ S → +∐ +N≤H≤G +Irr(H) +(2.1) +that sends each character ϑ ∈ S to an extension Λ(ϑ) of ϑ to the stabiliser Gϑ. +Next, we consider modular representation theory with respect to a fixed prime number ℓ. For χ ∈ +Irr(G), there exist unique non-negative integers d(χ), called the ℓ-defect of χ, such that ℓd(χ) = +∣G∣ℓ/χ(1)ℓ and where for an integer n we denote by nℓ the largest power of ℓ that divides n. For +any d ≥ 0, let Irrd(G) be the set of irreducible characters χ of G that satisfy d(χ) = d and denote by +kd(G) its cardinality. Associated to the prime ℓ, we also have the set of Brauer ℓ-blocks of G. Each +block is uniquely determined by the central functions λB (see [Nav98, p. 49]). For every χ ∈ Irr(G), +we denote by bl(χ) the unique block that satisfies χ ∈ Irr(bl(χ)). Furthermore, if H ≤ G and b is +a block of H, then bG denotes the block of G obtained via Brauer’s induction (when it is defined). +If B is a block of G and d ≥ 0, then let Irrd(B) be the set of irreducible characters belonging to the +block B and having defect d. The cardinality of Irrd(B) is denoted by kd(B). +We conclude this introductory section with an analogue of [Isa76, Problem 5.3] for blocks that will +be used in the sequel. +Lemma 2.1. Let H ≤ G be finite groups and consider blocks b of H and B of G. If ζ is a linear +character of G, then: +(i) there are blocks b ⋅ ζH of H and B ⋅ ζ of G satisfying +Irr(b ⋅ ζH) = {ψζH ∣ ψ ∈ Irr(b)} +and +Irr(B ⋅ ζ) = {χζ ∣ χ ∈ Irr(B)}; +(ii) If bG = B, then (b ⋅ ζH)G = B ⋅ ζ. +Proof. The first point is [Riz18, Lemma 2.1]. Next, let g ∈ G and denote by ClG(g) the G-conjugacy +class of g and by ClG(g)+ the corresponding conjugacy class sum in the group algebra. Since the +intersection ClG(g) ∩ H is a union of H-conjugacy classes, we can find h1,... ,hn ∈ ClG(g) ∩ H +such that +ClG(g) ∩ H = +n +∐ +i=1 +ClH(hi) +and where n is zero if ClG(g) ∩ H is empty. In particular, observe that ζ(hi) = ζ(g) since λ is a +6 + +class function of G. Now, using the notation of [Nav98, p.87] we obtain +λB⋅ζ (ClG(g)+) = λB (ClG(g)+)ζ(g) += λG +b (ClG(g)+)ζ(g) += +n +∑ +i=1 +λb (ClH(hi)+)ζ(g) += +n +∑ +i=1 +λb (ClH(hi)+)ζH(hi) += +n +∑ +i=1 +λb⋅ζH (ClH(hi)+) = λG +b⋅ζH (ClG(g)) +where for every algebraic integer α of C we denote by α its reduction modulo a maximal ideal +containing the prime ℓ (see [Nav98, Chapter 2]). This shows that B ⋅ ζ = (b ⋅ ζH)G and we are +done. +2.2 +Finite reductive groups and unipotent characters +Let G be a connected reductive group defined over an algebraic closure of a field of positive char- +acteristic p different from ℓ and consider a Frobenius endomorphism F ∶ G → G associated with +an Fq-structure for a power q of p. The set of Fq-rational points on the variety G is denoted by GF +and is called a finite reductive group. By abuse of notation we also refer to the pair (G,F) as a finite +reductive group. +Let L be a Levi subgroup of a parabolic subgroup P of G and assume that L (but not necessarily +P) is F-stable. Using ℓ-adic cohomology, Deligne–Lusztig [DL76] and Lusztig [Lus76] defined a +Z-linear map +RG +L≤P ∶ ZIrr(LF) → ZIrr(GF ) +with adjoint +∗RG +L≤P ∶ ZIrr(GF ) → ZIrr(LF) . +The exact definition can be found in [CE04, Section 8.3]. These maps are known to be independent +of the choice of the parabolic subgroup P in almost all cases (see [BM11] and [Tay18]) and, in +particular, in those considered in this paper. Therefore, we will always omit P and denote RG +L≤P +simply by RG +L . Next, using Deligne–Lusztig induction we define the unipotent characters of GF . +These are the irreducible characters χ of GF that appear as an irreducible constituent of the virtual +character RG +T(1T) for some F-stable maximal torus T of G. The set of unipotent characters of GF +is denoted by Uch(GF ) and its cardinality by ku(GF ). Similarly, if B is a block of GF and d a +non-negative integer, then kd +u(B) denotes the cardinality of the intersection Uch(GF ) ∩ Irrd(B). +2.3 +e-Harish-Chandra theory for unipotent characters +Denote by e the multiplicative order of q modulo ℓ, if ℓ is odd, or modulo 4, if ℓ = 2. In this section, +we collect the main results of e-Harish-Chandra theory for unipotent characters. This was first in- +troduced by Fong and Srinivasan [FS86] for classical groups and then further developed by Broué, +Malle and Michel [BMM93] for unipotent characters. The compatibility of this theory with Brauer +ℓ-blocks was described by Cabanes and Enguehard in [CE94] for good primes and completed by +7 + +Enguehard [Eng00] for bad primes. These results also provide a description of the characters be- +longing to unipotent blocks (see [CE94, Theorem (iii)]). Another description of these characters was +provided by the author in [Ros22c] under certain resctrictions on the prime ℓ (see also [Ros22c, Re- +mark 4.14] for a comparison between the two descriptions). We refer the reader to the monographs +[CE04] and [GM20] for a more complete account of this beautiful theory. +The theory of Φe-subgroups that constitutes the foundation of e-Harish-Chandra theory was intro- +duced in [BM92]. Following their terminology, we say that an F-stable torus S of G is a Φe-torus if +its order polynomial is a power of the e-th cyclotomic polynomial, that is, if P(S,F ) = Φn +e for some +integer n and where Φe denotes the e-th cyclotomic polynomial (see [CE04, Definition 13.3]). Then, +we say that a Levi subgroup L of G is an e-split Levi subgroup if there exists a Φe-torus S such +that L = CG(S). More precisely, we say that L is an e-split Levi subgroup of (G,F) to emphasise +the role of the Frobenius endomorphism F. Observe that, for any torus T, there exists a unique +maximal Φe-torus of T denoted by TΦe (see [CE04, Proposition 13.5 3.4]). Then, it can be shown +that an F-stable Levi subgroup L of G is e-split if and only if L = CG(Z○(L)Φe) (see, for instance, +[GM20, Proposition 3.5.5]). +Next, recall that (L,λ) is a unipotent e-cuspidal pair of (G,F) if L is an e-split Levi subgroup +of (G,F) and λ ∈ Irr(LF ) satisfies ∗RL +M(λ) = 0 for every e-split Levi subgroup M < L. A +character λ with the property above is said to be a unipotent e-cuspidal character of LF . We denote +by CPu(G,F) the set of unipotent e-cuspidal pairs of (G,F) and by kc,u(GF ) the number of +unipotent e-cuspidal characters of GF . Moreover, we define the e-Harish-Chandra series associate +to the e-cuspidal pair (L,λ) to be the set of irreducible constituents of the virtual character RG +L (λ), +denoted by E(GF ,(L,λ)). +Unipotent characters where parametrised by Broué, Malle and Michel [BMM93, Theorem 3.2] by us- +ing e-Harish-Chandra theory. Their description can be divided into two parts. First, each unipotent +character lies in a unique e-Harish-Chandra series, that is, +Uch (GF ) = ∐ +(L,λ) +E (GF ,(L,λ)) +where (L,λ) runs over a set of representatives for the action of GF on the set of unipotent e- +cuspidal pairs of (G,F) as explained in [BMM93, Theorem 3.2 (1)]. This is a well known fact +and will be used throughout the paper without further reference. As a consequence of the partition +above, it now remains to parametrise the unipotent e-Harish-Chandra series. If (L,λ) is a unipotent +e-cuspidal pair, we denote by WG(L,λ)F ∶= NG(L)F +λ /LF the corresponding relative Weyl group. +Then, [BMM93, Theorem 3.2 (2)] parametrises the characters in an e-Harish-Chandra series in terms +of the characters in the relative Weyl group by showing the existence of a bijection +Irr(WG(L,λ)F ) → E(GF ,(L,λ)). +(2.2) +In Section 3 we reformulate (2.2) in order to obtain Theorem C. +Unipotent e-Harish-Chandra series are also used to parametrise the so-called unipotent blocks, that +is, those blocks that contain unipotent characters. This is the main result of [CE94]. More precisely, +if ℓ is odd and good for G, with ℓ ≠ 3 if 3D4 is an irreducible rational component of (G,F), then +for every ℓ-block B of GF there exists a unipotent e-cuspidal pair (L,λ), with (L,λ) unique up to +8 + +GF -conjugation, such that all the irreducible constituents of RG +L (λ) belongs to the block B. In this +case, we write B = bGF (L,λ) and we also have +Uch(GF ) ∩ Irr(bGF (L,λ)) = E(GF ,(L,λ)). +Moreover, [CE94, Proposition 3.3 (ii) and Proposition 4.2] imply that bl(λ)GF = B. +2.4 +Pseudo-unipotent characters +We denote by (G∗,F ∗) a group in duality with (G,F) with respect to a choice of an F-stable +maximal torus T of G and an F ∗-stable maximal torus T∗ of G∗. If τ ∶ Gsc → [G,G] is a simply +connected covering (see [GM20, Remark 1.5.13]), then there exists an isomorphisms between the +abelian groups +Z(G∗)F ∗ → Irr(GF /τ(Gsc)) +z ↦ ˆzG +according to [CE04, (8.19)]. Notice that, if L is an F-stable Levi subgroup of G, then its dual L∗ is +an F ∗-stable Levi subgroup of G∗ and we have Z(G∗)F ∗ ≤ Z(L∗)F ∗. In particular, every element +z ∈ Z(G∗)F ∗ defines a linear characters of ˆzL and restriction of characters yields the equality +(ˆzG)LF = ˆzL. +In the next definition, we consider charactersthat are obtained by multiplying these linear characters +with unipotent characters. +Definition 2.2. Let (K,F) be a finite reductive group and consider a Levi subgroup of L ≤ K +and an irreducible character θ ∈ Irr(LF). We say that θ is (K,F)-pseudo-unipotent if there exists +an element z ∈ Z(K∗)F ∗ such that θˆzL is unipotent. Moreover, for every unipotent character +λ ∈ Uch(LF ), we denote by psK(λ) the set of (K,F)-pseudo-unipotent characters of LF of the +form λˆzL for some z ∈ Z(K∗)F ∗. Moreover, we denote by psK(LF ) the set of all (K,F)-pseudo +unipotent characters of LF. When the group K coincides with L, we denote the set of characters +psL(LF ) simply by ps(LF ). +In accordance with the terminology introduced above, we say that an e-Harish-Chandra series of +(K,F) is pseudo-unipotent if it is of the form E(KF ,(L,ν)) for some ν ∈ psK(λ) and where +(L,λ) is a unipotent e-cuspidal pair of (K,F). In this case, we also say that (L,ν) is a pseudo- +unipotent e-cuspidal pair. We define the union of all the series associated to characters in psK(λ) +by E(KF ,(L,psK(λ))). Since +RK +L (λˆzL) = RK +L (λ)ˆzK +for every z ∈ Z(K∗)F ∗ by [CE04, (8.20)], we deduce that the elements of the pseudo-unipotent +e-Harish-Chandra series E(KF ,(L,λˆz)) are exactly the irreducible characters of the form ϕˆzK +for some unipotent character ϕ ∈ E(KF ,(L,λ)). Moreover, we point out that λ is the unique +unipotent character in the set psK(λ) according to [CE04, Proposition 8.26]. Similarly, the unipotent +characters in the set E(KF ,(L,psK(λ))) are those in the series E(KF ,(L,λ)). +Our next lemma, shows that blocks covering pseudo-unipotent characters are regular as defined in +[Nav98, p.210]. +9 + +Lemma 2.3. Let L be an F-stable Levi subgroup of G and suppose that ℓ is odd and good for G. +For every LF ≤ H ≤ NG(L)F and every character ϑ ∈ Irr(H) lying above some pseudo-unipotent +character in ps(LF ), the block bl(ϑ) is LF-regular. In particular, the Brauer induced block bl(ϑ)H is +defined and is the unique block of H covering bl(ϑ). +Proof. Let ϕ ∈ Uch(LF ) and z ∈ Z(L∗)F ∗ such that ϕˆzL lies below the character ϑ and chose +a unipotent e-cuspidal pair (M,µ) of L such that ϕ ∈ E(LF ,(M,µ)). In particular, bl(ϕ) = +bLF (M,µ) according to [CE94]. If Q ∶= Z(M)F +ℓ , then MF = CGF (Q) according to [CE94, Propo- +sition 3.3 (ii)]. Moreover, observe that [CE94, Proposition 4.2] implies that bl(ϕ) = bLF (M,µ) = +bl(µ)LF while [Riz18, Lemma 2.1] implies that bl(ϕ) and bl(ϕˆzL) have the same defect groups. +Now, applying [Nav98, Lemma 4.13 and Theorem 9.26], we can find defect groups Dϑ, Dϕ and Dµ +of bl(ϑ), bl(ϕ) and bl(µ) respectively with the property that Dµ ≤ Dϕ ≤ Dϑ. Since Q ≤ Oℓ(MF ) ≤ +Dµ by [Nav98, Theorem 4.8], we deduce that Q ≤ Dϑ and hence CH(Dϑ) ≤ CH(Q) = MF ≤ LF. +By [Nav98, Lemma 9.20] we conclude that the block bl(ϑ) is LF -regular. The second part of the +lemma now follows from [Nav98, Theorem 9.19]. +3 +Compatibility with isomorphisms of character triples +The aim of this section is to show how the bijection (2.2) can be made compatible with isomorphisms +of character triples and with the action of automorphisms. This property was first suggested by the +author in [Ros22c, Parametrisation B] and further studied in [Ros22d]. Our Theorem C gives a +solution of this conjectured result for unipotent e-Harish-Chandra series and groups of type A, B +and C. Before proceeding further, we show how the parametrisation (2.2) can be reformulated in +a more convenient form. For this, let (L,λ) be a unipotent e-cuspidal pair of (G,F) and assume +that ̂λ is an extension of λ to the stabiliser NG(L)F +λ . Then, by applying Gallagher’s theorem [Isa76, +Corollary 6.17] and the Clifford correspondence [Isa76, Theorem 6.11] we obtain a bijection +Irr(WG(L,λ)F ) → Irr(NG(L)F ∣ λ) +η ↦ (̂λη) +NG(L)F +and therefore (2.2) holds if an only if there exists a bijection +E(GF ,(L,λ)) → Irr(NG(L)F ∣ λ) . +(3.1) +This new reformulation will allow us to introduce the aforementioned compatibility with isomor- +phisms of character triple isomorphisms. +3.1 +Equivariance and maximal extendibility +In this section, we consider some equivariance properties for the parametrisation (3.1) which are +related to maximal extendibility (see (2.1)) of unipotent characters. +As in the previous sections, consider a connected reductive group G with a Frobenius endomor- +phism F ∶ G → G defining an Fq-structure on G. We denote by AutF(GF ) the set of those auto- +morphisms of GF obtained by restricting some bijective morphism of algebraic groups σ ∶ G → G +that commutes with F to the set of Fq-rational points GF . Notice that the restriction of such a +10 + +morphism σ to GF , which by abuse of notation we denote again by σ, is an automorphism of the +finite group GF . We refer the reader to [CS13, Section 2.4] for further details. In particular, observe +that any morphism σ with the properties above is determined by its restriction to GF up to a power +of F and hence it follows that AutF(GF ) acts on the set of F-stable closed connected subgroups of +G. Then, given an F-stable closed connected subgroup H of G, we can define the set AutF(GF )H +consisting of those automorphisms σ as above that stabilise the algebraic group H. +Now, let ℓ be a prime number not dividing q and denote by e the order of q modulo ℓ or q modulo 4 +if ℓ = 2. In order to control the action of automorphism on unipotent e-Harish-Chandra series, we +exploit a result of Cabanes and Späth. More precisely, in [CS13, Theorem 3.4] it was shown that the +parametrisation given by Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] commutes with the +action of those automorphisms in the set AutF(GF ). Notice that the statement of [CS13, Theorem +3.4] only considers unipotent e-cuspidal pairs (L,λ) where L is a minimal e-split Levi subgroups +(which is enough for the purpose of dealing with the McKay Conjecture). However, their proof +works for the general case as well. +Proposition 3.1. For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)- +equivariant bijection +IG +(L,λ) ∶ Irr(WG(L,λ)F ) → E (GF ,(L,λ)) +such that +IG +(L,λ)(η)(1)ℓ = ∣GF ∶ NG(L,λ)F ∣ℓ ⋅ λ(1)ℓ ⋅ η(1)ℓ +for every η ∈ Irr(WG(L,λ)F ). +Proof. This follows from the proof of [CS13, Theorem 3.4]. See also [Ros22d, Theorem 3.4]. +As explained at the beginning of this section, if the character λ extends to the stabiliser NG(L)F +λ , +then we can use the bijection (2.2) to obtain (3.1). A similar argument can be used to include the +equivariance property described above and obtain an equivariant version of (3.1). Observe that, by +the discussion on automorphisms above, it follows that the group AutF(GF ) acts on the set of e- +cuspidal pairs (L,λ) and therefore we can define the stabiliser AutF(GF )(L,λ). Furthermore, recall +that we denote by d(χ) the ℓ-defect of an irreducible character χ. +Corollary 3.2. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an ex- +tension λ◇ ∈ Irr(NG(L)F +λ ) which is additionally AutF(GF )(L,λ)-invariant. Then there exists an +AutF(GF )(L,λ)-equivariant bijection +ΩG +(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) +such that +d(χ) = d(ΩG +(L,λ)(χ)) +for every χ ∈ E(GF ,(L,λ)). +11 + +Proof. Consider the bijection IG +(L,λ) given by Proposition 3.1 and define the map +ΩG +(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) +IG +(L,λ)(η) ↦ (λ◇η)NG(L)F +for every η ∈ Irr(WG(L,λ)F ) and where λ◇ is the extension of λ to NG(L)F +λ given in the statement. +This is a well defined bijection by the Clifford correspondence [Isa76, Theorem 6.11] and Gallagher’s +theorem [Isa76, Corollary 6.17]. Moreover, for every α ∈ AutF(GF ) such that (L,λ)α = (L,λ) and +every η ∈ Irr(WG(L,λ)F ) we have +((λ◇η)NG(L)F ) +α += ((λ◇η)α) +NG(L)F += (λ◇ηα)NG(L)F +because α stabilises λ◇. On the other hand +IG +(L,λ)(η)α = IG +(L,λ) (ηα) +by the properties of IG +(L,λ) and hence we conclude that ΩG +(L,λ) is AutF(GF )(L,λ)-equivariant. Fur- +thermore, if we consider η ∈ Irr(WG(L,λ)F ) and define the characters χ ∶= IG +(L,λ)(η) and ψ ∶= +(λ◇η)NG(L)F , then the degree formula from Proposition 3.1 implies that +ℓd(χ) = +∣GF ∣ℓ +χ(1)ℓ += +∣NG(L,λ)F ∣ℓ +λ(1)ℓ ⋅ η(1)ℓ += +∣NG(L)F ∣ℓ +ψ(1)ℓ += ℓd(ψ) +and hence we deduce that d(χ) = d(ψ) as required. +Next, we consider a regular embedding G ≤ ̃G as defined in [CE04, (15.1)]. Then, ̃G is a connected +reductive group with connected centre and whose derived subgroup coincides with that of G, that +is, [̃G, ̃G] = [G,G]. In particular, observe that ̃G = Z(̃G)G, that G is normal in ̃G and that +the quotient ̃G/G is an abelian group. Moreover, for every Levi subgroup L of G, we deduce +that ̃L ∶= Z(̃G)L is a Levi subgroup of ̃G and that L ≤ ̃L is again a regular embedding. These +observations will be used throughout this paper without further reference. +We also recall that, according to [DM91, Proposition 13.20], restriction of characters yields a bi- +jection between the unipotent characters of ̃GF and those of GF . In particular, every unipotent +character of GF is ̃GF -invariant. Using this observation, we can compare the relative Weyl groups +in ̃GF with those in GF . +Lemma 3.3. Let (L,λ)be a unipotent e-cuspidal pair of (G,F), set ̃L = LZ(̃G) and consider a unipo- +tent extension ̃λ of λ to ̃LF. Then, N ̃ +G(L)F +λ = N ̃ +G(L)F +̃λ and we have W ̃ +G(̃L,̃λ)F ≃ WG(L,λ)F . +Proof. Since ̃λ extends λ, it is clear that the stabiliser N ̃ +G(L)F +̃λ is contained in N ̃ +G(L)F +λ . On the +other hand, let x ∈ N ̃ +G(L)F +λ and observe that ̃λx is a unipotent character of ̃LF that restricts to +λx = λ. Then, [DM91, Proposition 13.20] implies that ̃λx = ̃λ and therefore x ∈ N ̃ +G(L)F +̃λ . From this, +we also conclude that N ̃ +G(L)F +̃λ = ̃LFNG(L)F +λ and therefore that W ̃ +G(̃L,̃λ)F ≃ WG(L,λ)F . +12 + +As a consequence of the lemma above, we show that when λ extends to its stabiliser NG(L)F +λ , +then every irreducible character of NG(L) that lies above λ is N ̃ +G(L)F -invariant and extends to +N ̃ +G(L)F . +Corollary 3.4. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an extension +λ◇ ∈ Irr(NG(L)F +λ ). Then every character of NG(L)F lying above λ extends to N ̃ +G(L)F . +Proof. To start, we fix a unipotent extension ̃λ of λ to ̃LF and recall that N ̃ +G(L)F +λ = N ̃ +G(L)F +̃λ +according to Lemma 3.3. Then, applying [Spä10, Lemma 4.1 (a)] we deduce that there exists an +extension ̃λ◇ of λ◇ to N ̃ +G(L)F +λ that also extends ̃λ. Consider now an irreducible character ψ of +NG(L)F lying above λ. By Gallagher’s theorem [Isa76, Corollary 6.17] and the Clifford correspon- +dence [Isa76, Theorem 6.11], it follows that there exists an irreducible character η of the relative +Weyl group WG(L,λ)F such that ψ is induced from the irreducible character ψ0 ∶= ηλ◇. More- +over, by using Lemma 3.3, we have W ̃ +G(̃L,̃λ)F ≃ WG(L,λ)F . Then, η, viewed as a character of +NG(L)F +λ , admits an extension, say ̃η, to N ̃ +G(L)F +λ . Now, define ̃ψ0 ∶= ̃η̃λ◇ and observe that ̃ψ0 lies +above ̃λ. By the Clifford correspondence, it follows that the character ̃ψ of N ̃ +G(L)F induced from +̃ψ0 is irreducible and therefore, applying [Isa76, Problem 5.2], we conclude that ̃ψ extends ψ. The +proof is now complete. +We can now construct a parametrisation of unipotent e-Harish-Chandra series in the group ̃GF +which agrees with the bijection ΩG +(L,λ) from Corollary 3.2 via restriction of characters. +Proposition 3.5. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an +extension λ◇ ∈ Irr(NG(L)F +λ ) which is additionally AutF(GF )(L,λ)-invariant. If ̃λ is a unipotent +extension of λ to ̃LF , then there exists a bijection ̃Ω ̃ +G +(̃L,̃λ) making the following diagram commute +E (̃GF ,(̃L,̃λ)) +Irr(N ̃ +G(L)F ∣ ̃λ) +E (GF ,(L,λ)) +Irr(NG(L)F ∣ λ) +̃Ω ̃ +G +(̃L,̃λ) +Res +̃ +GF +GF +Res +N ̃ +G(L)F +NG(L)F +ΩG +(L,λ) +and where ΩG +(L,λ) is the bijection given by Corollary 3.2. +Proof. First, observe that λ has an extension ̃λ to ̃LF according to [DM91, Proposition 13.20]. More- +over, restrictions from ̃GF to GF induces a bijection from the set E(̃GF ,(̃L,̃λ)) to E(GF ,(L,λ)) +according to [CE94, Proposition 3.1]. Next, consider a character ψ ∈ Irr(NG(L)F ) lying above λ +and observe that ψ admits an extension ̃ψ0 ∈ Irr(N ̃ +G(L)F ) by Corollary 3.4. Let ̃λ0 be an irre- +ducible constituent of the restriction ̃ψ0,̃LF and notice that ̃ +λ0 is an extension of λ since ̃LF /LF +is abelian. Now, Gallagher’s theorem [Isa76, Corollary 6.17] implies that there exists a linear char- +acter ν ∈ Irr(̃LF /LF) such that ̃λ0ν = ̃λ. Since N ̃ +G(L)F /NG(L)F ≃ ̃LF/LF we can identify ν +with its extension to N ̃ +G(L)F . Then, it follows that the character ̃ψ ∶= ̃ψ0ν is an extension of ψ to +N ̃ +G(L)F lying above ̃λ. Then the assignment ψ ↦ ̃ψ defines a bijection between Irr(NG(L)F ∣ λ) +13 + +and Irr(N ̃ +G(L)F ∣ ̃λ) whose inverse is given by restriction of characters. We can now define +̃Ω +̃ +G +(̃L,̃λ) (̃χ) ∶= ̃ψ +for every ̃χ ∈ E(̃GF ,(̃L,̃λ)) and ̃ψ ∈ Irr(N ̃ +G(L)F ∣ ̃λ) whenever ΩG +(L,λ)(̃χGF ) = ̃ψNG(L)F . +3.2 +Construction of GF-block isomorphisms of character triples +From now on, we assume that G is simple, simply connected and of type A, B or C. Furthermore, +we suppose that ℓ is odd and denote by e the order of q modulo ℓ. +We now give a more explicit construction of the group of automorphism AutF(GF ). Fix a max- +imally split torus T0 contained in an F-stable Borel subgroup B0 of G. This choice corresponds +to a set of graph automorphisms γ ∶ G → G and a field endomorphism F0 ∶ G → G. More pre- +cisely, if we consider the set of simple roots ∆ ⊆ Φ(G,T0) corresponding to the choice T0 ⊆ B0, +then we have an automorphism γ ∶ G → G given by γ(xα(t)) ∶= xγ(α)(t) for every t ∈ Ga and +α ∈ ±∆ and where γ is a symmetry of the Dynkin diagram of ∆, while F0(xα(t)) ∶= xα(tp) for +every t ∈ Ga and α ∈ Φ(G,T0). Here, we denote by xα ∶ Ga → G a one-parameter subgroup +corresponding to α ∈ Φ(G,T0). We define the subgroup A of AutF(GF ) generated by the graph +and field automorphisms described above. +In addition, we choose our regular embedding G ≤ ̃G to be defined in such a way that the graph and +field automorphisms extends to ̃G (see, for instance, [MS16, Section 2B]). In particular, the group +A acts via automorphisms on ̃GF and we can form the external semidirect product ̃GF ⋊ A which +acts on GF . It turns out that ̃GF ⋊A and AutF(GF ) induce the same set of automorphisms on the +finite group GF (see, for instance, [GLS98, Section 2.5]). +Throughout this section, we consider a fixed unipotent e-cuspidal pair (L,λ) of (G,F) and a unipo- +tent extension ̃λ of λ to ̃LF (whose existence is ensured by [DM91, Proposition 13.20]) where, as +always, we define ̃L ∶= LZ(̃G). In the next lemma, we show that the hypothesis of Corollary 3.2 is +satisfied under our assumptions. +Lemma 3.6. There exists an extension λ◇ of λ to NG(L)F +λ that is (̃GF A)(L,λ)-invariant. +Proof. Using [BS20, Theorem 4.3 (i)], [Bro22b, Theorem 1.2 (a)] and the results of [Bro-Ruh], we +obtain an extension λ◇ of λ to NG(L)F +λ which is (GF A)(L,λ)-invariant. Since (̃GF A)(L,λ) = +̃L(GF A)(L,λ) it suffices to show that λ◇ is ̃LF-invariant. However, the latter assertion follows +immediately from the fact that λ◇ extends to N ̃ +G(L)F +λ according to Lemma 3.3 and [Spä10, Lemma +4.1 (a)]. +As an immediate consequence of the lemma above, we deduce that every character of NG(L)F +lying above λ extends to N ̃ +G(L)F . This can be considered as a local analogue of [DM91, Proposition +13.20]. +Lemma 3.7. Every irreducible character of NG(L)F lying above λ extends to N ̃ +G(L)F . +Proof. This follows from Corollary 3.4 whose hypothesis is satisfied by Lemma 3.6. +14 + +We point out that, under our assumptions, every irreducible character of NG(L)F lying above λ +extends to its stabiliser in N ̃ +G(L)F because the quotient N ̃ +G(L)F /NG(L)F is cyclic according to +[GM20, Proposition 1.7.5]. However, in the lemma above we are also showing, using independent +methods, that each such character is N ̃ +G(L)F -invariant. +Using Lemma 3.6, we can now define bijections Ω ∶= ΩG +(L,λ) and ̃Ω ̃ +G +(̃L,̃λ) as described in Corol- +lary 3.2 and Proposition 3.5 respectively. In what follows, we consider the sets of characters G ∶= +E(GF ,(L,λ)), L ∶= Irr(NG(L)F ∣ λ), ̃G ∶= E(̃GF ,(̃L,̃λ)) and ̃L ∶= Irr(N ̃ +G(L)F ∣ ̃λ). Our next +aim is to show that the parametrisation Ω is compatible with GF -block isomorphisms of charac- +ter triples. We start by checking the group theoretic properties required for the existence of such +isomorphisms (see [Spä17, Remark 3.7 (i)]). +Lemma 3.8. For every χ ∈ G and ψ ∶= Ω(χ) ∈ L we have (̃GF A)L,χ = (̃GF A)L,ψ and ̃GFAχ = +GF (̃GF A)L,ψ. +Proof. We argue as in the proof of [Ros22c, Lemma 4.2]. To start, we observe that (̃GF A)(L,λ),χ = +(̃GF A)(L,λ),ψ since the map Ω is (̃GF A)(L,λ)-equivariant. Set U(χ) ∶= (̃GF A)L,χ and U(ψ) ∶= +(̃GF A)L,ψ. First, consider x ∈ U(χ) and observe that according to [BMM93, Theorem 3.2 (1)] there +exists y ∈ NG(L)F such that (L,λ)xy = (L,λ). In particular, xy ∈ (̃GF A)(L,λ),χ = (̃GF A)(L,λ),ψ +and hence x ∈ U(ψ) since ψy = ψ. This shows that U(χ) ≤ U(ψ). On the other hand, suppose +that x ∈ U(ψ). By Clifford’s theorem there exists y ∈ NG(L)F such that λxy = λ and so xy ∈ +(̃GF A)L,ψ = (̃GF A)L,χ. Since χy = χ we deduce that x ∈ U(χ) and hence U(χ) = U(ψ). To +conclude, it is enough to show that ̃GFAχ = GF U(χ). First, notice that GF U(χ) ≤ ̃GF Aχ since +χ is ̃GF -invariant. On the other hand, for x ∈ ̃GF Aχ we know that (L,λ)x is GF -conjugate to +(L,λ) thanks to [BMM93, Theorem 3.2 (1)]. Therefore, we obtain x ∈ GF U(χ) and as explained +above this concludes the proof. +We now apply Lemma 3.8 to show that the map ̃Ω satisfies some useful equivariance properties. +Before doing so, we need to introduce some notation. For this purpose, consider a pair (G∗,F ∗) +dual to (G,F) and a pair (̃G∗,F ∗) dual to (̃G,F). Let i∗ ∶ ̃G∗ → G∗ be the surjection induced by +duality from the inclusion G ≤ ̃G and observe that Ker(i∗) = Z(̃G∗) since G is simply connected +(see [CE04, Section 15.1]). As shown in [CE04, (15.2)], there exists an isomorphism +Ker(i∗)F → Irr(̃GF /GF) +(3.2) +z ↦ ̂z̃ +G +Furthermore, if L is an F-stable Levi subgroup of G and z ∈ Ker(i∗), then we define ̂z̃L to be the +restriction of ̂z̃ +G to ̃LF and ̂zN ̃ +G(L) to be the restriction of ̂z̃ +G to N ̃ +G(L)F . We set K ∶= Ker(i∗) and +obtain an action of the group K on the characters of ̃GF , ̃LF and N ̃ +G(L)F as defined in [Ros22d, +Definition 2.1]. Moreover, we consider the external semidirect product (̃GF A)⋉K given by defining +zx as the unique element of K corresponding to the character (̂z̃ +G)x of the quotient ̃GF /GF via +the isomorphism specified in (3.2), whenever x ∈ ̃GF A and z ∈ K. Then, for every F-stable Levi +subgroup L of G, we obtain an action of (̃GF A)L ⋉ K on the irreducible characters of ̃LF and +N ̃ +G(L)F . We denote by ((̃GF A)L ⋉ K)̃λ the stabiliser of ̃λ ∈ Irr(̃LF ). In particular, it follows +that ((̃GF A)L ⋉ K)̃λ acts on the sets of characters ̃G and ̃L. Next, we show that the bijection ̃Ω is +compatible with this action. +15 + +Lemma 3.9. The bijection ̃Ω is (N ̃ +G(L)F (̃GF A)(L,λ) ⋊ K)̃λ-equivariant. +Proof. Let ̃χ ∈ ̃G and ̃ψ ∈ ̃L. By the definition of ̃Ω, we have ̃Ω(̃χ) = ̃ψ if and only if Ω(χ) = ψ where +χ ∶= ̃χGF and ψ ∶= ̃ψNG(L)F . Now, if we consider g ∈ N ̃ +G(L)F , x ∈ (̃GF A)(L,λ) and z ∈ K such +that (gx,z) stabilises ̃λ, then we obtain +̃Ω(̃χ(gx,z)) = ̃ψ(gx,z) +if and only if +Ω((̃χ(gx,z)) +GF ) = (̃ψ(gx,z)) +NG(L)F . +(3.3) +However, since the restriction of ̃χ(gx,z) to GF coincides with χx and the restriction of ̃ψ(gx,z) to +NG(L)F coincides with ψx, we deduce that the equality in (3.3) holds by the equivariant properties +of Ω as described in Corollary 3.2. +One of the main ingredients for the construction of the projective representations needed to obtain +GF -block isomorphisms of character triples is given by the following two lemmas on maximal +extendibility. +Lemma 3.10. Maximal extendibility holds for G with respect to the inclusion GF ⊴ GF A, that is, +every character χ ∈ G extends to GF Aχ. +Proof. If G is of type B or C then the result follows from [Isa76, Corollary 11.22] since A is cyclic. +Then, we can assume that G is of type A in which case the result follows from [CS17, Theorem 4.1] +(see also [Mal08, Theorem 2.4]). +The local version of the lemma above is a consequence of the results obtained in [BS20]. +Lemma 3.11. Maximal extendibility holds for L with respect to the inclusion NG(L)F ⊴ (GF A)L, +that is, every character ψ ∈ L extends to (GF A)L,ψ. +Proof. As in the proof of Lemma 3.10, it is enough to prove the result in the case where G is of type +A. In fact, if G is of type B or C, then the quotient (GF A)(L,ψ)/NG(L)F is cyclic because it it a +subquotient of A. Now, if G is of type A the result follows from [BS20, Theorem 1.2]. +Finally, we can start constructing isomorphisms of character triples for the bijection Ω. As a first +step, we obtain a weaker isomorphism, know as GF -central isomorphism of character triples and +denoted by ∼c +GF , whose requirements are given by [Spä17, Remark 3.7 (i)-(iii)] and replacing the +condition on defect groups by imposing that CG(N) ≤ H1 ∩ H2 with the notations used there. We +refer the reader to [Ros22b, Definition 3.3.4] for a precise definition. +Proposition 3.12. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have +(̃GFAχ,GF ,χ) ∼c +GF ((̃GF A)L,ψ,NG(L)F ,ψ) . +16 + +Proof. We start by constructing projective representations associated with χ and ψ. According to +Proposition 3.5 we can find a unipotent extension ̃χ ∈ ̃G of χ to ̃GF . Furthermore, by Lemma 3.10 +there exists an extension χ′ of χ to GF Aχ. Let ̃Dglo be a representation of ̃GF affording ̃χ and D′ +glo +a representation of GF Aχ affording χ′. Now, [Spä12, Lemma 2.11] implies that +Pglo ∶ (̃GF A)χ → GLχ(1)(C) +defined by Pglo(x1x2) ∶= ̃ +Dglo(x1)D′ +glo(x2) for every x1 ∈ ̃GF and x2 ∈ GF Aχ is a projective +representation associated with χ. Next, observe that ̃ψ ∶= ̃Ω(̃χ) ∈ ̃L is an extension of ψ to N ̃ +G(L)F +and consider an extension ψ′ of ψ to (GF A)L,ψ given by Lemma 3.11. Let ̃Dloc be a representation +of N ̃ +G(L)F affording ̃ψ and D′ +loc a representation of (GF A)L,ψ affording ψ′. Once again, [Spä12, +Lemma 2.11] shows that the map +Ploc ∶ (̃GF A)L,ψ → GLψ(1)(C) +given by Ploc(x1x2) ∶= ̃Dloc(x1)D′ +loc(x2) for every x1 ∈ N ̃ +G(L)F and x2 ∈ (GF A)L,ψ is a pro- +jective representation associated with ψ. We denote by αglo and αloc the factor set of Pglo and +Ploc respectively. As explained in the proof of [Ros22d, Theorem 4.3], in order to prove that αglo +coincides with αloc via the isomorphism ̃GF Aχ/GF ≃ (̃GF A)L,ψ/NG(L)F , it suffices to show +that +(µglo +x )N ̃ +G(L)F = µloc +x +(3.4) +for every x ∈ (GF A)L,χ and where µglo +x +∈ Irr(̃GF /GF ) and µloc +x +∈ Irr(N ̃ +G(L)F /NG(L)F ) are +determined by Gallagher’s theorem (see [Isa76, Corollary 6.17]) via the equalities ̃χ = µglo +x ̃χx and ̃ψ = +µloc +x ̃ψx respectively. Because (GF A)L,χ = NG(L)F (GF A)(L,λ),χ, we may assume that x stabilises +λ. Let z ∈ K such that µglo +x += ˆz̃ +G and observe that (x,z) is an element of (GF A)(L,λ),χ ⋊ K that +stabilises ̃χ. Then, applying [BMM93, Theorem 3.2 (1)], we deduce that ̃λ and ̃λ(x,z) are N ̃ +G(L)F - +conjugate and we may choose g ∈ N ̃ +G(L)F such that ̃λ = (̃λ(x,z))g = ̃λ(xg,z). In other words +(xg,z) ∈ (N ̃ +G(L)F (̃GF A)(L,λ) ⋊ K)̃λ +and thus Lemma 3.9 implies that the equality ̃χ = ̃χ(xg,z) holds if and only if ̃ψ = ̃ψ(xg,z). From this, +we immediately deduce the equality required in (3.4). +Next, denote by ζglo and ζloc the scalar functions associated to Pglo and Ploc respectively. To con- +clude the proof, it remains to show that ζglo and ζloc coincide on C(̃ +GF A)χ(GF ) = Z(̃GF ). As in the +proof of [Ros22d, Theorem 4.3], it is enough to show that the restrictions of ̃χ and ̃ψ to Z(̃GF ) are +multiples of a common irreducible constituent. This follows from the fact that unipotent characters +contain the center in their kernel. In fact, on one hand, 1Z(̃ +GF ) is the unique irreducible constituent +of ̃χZ(̃ +GF ) because ̃χ is unipotent. On the other hand, ̃ψ lies above ̃λ and, since Z(̃GF ) ≤ Z(̃LF ) +and ̃λ is unipotent, we deduce that 1Z(̃ +GF ) is the unique irreducible constituent of ̃ψZ(̃ +GF ). This +completes the proof. +We conclude this section by verifying the remaining condition [Spä17, Remark 3.7 (iv)] and obtain +the required GF -block isomorphisms of character triples for the map Ω. +17 + +Proposition 3.13. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have +(̃GFAχ,GF ,χ) ∼GF ((̃GF A)L,ψ,NG(L)F ,ψ) . +Proof. By Proposition 3.12 it is enough to check the block theoretic requirement given by [Spä17, +Remark 3.7 (ii) and (iv)]. First, observe that under our assumption [CE94, Proposition 3.3 (ii)] shows +that LF = CGF (E) where E ∶= Z(L)F +ℓ . In particular, NJ(L) = NJ(E) for every GF ≤ J ≤ +̃GF . Furthermore, for every block C0 of NJ(L) and every defect group D of C0 we have E ≤ +Oℓ(NJ(L)) ≤ D and hence C ̃ +GF (D) ≤ N ̃ +G(L)F . Now, [KS15, Theorem B] implies that for every +block C of N ̃ +G(L)F covering C0, the induced blocks B ∶= C ̃GF and B0 ∶= CJ +0 are well-defined and +B covers B0. +Let ̃χ ∈ ̃G be an extension of χ and set ̃ψ ∶= ̃Ω(̃χ). By Lemma 2.3 the block of ̃C of ̃ψ coincides +with the induced block bl(̃λ)N ̃ +G(L)F . Furthermore, by [CE94, Proposition 4.2] we know that the +block ̃B of ̃χ coincides with b ̃ +GF (̃L,̃λ) = bl(̃λ)̃ +GF . Then, by the transitivity of block induction we +get ̃ +B = ̃C ̃ +GF . Consider now GF ≤ J ≤ ̃GF as in the previous paragraph and notice that bl(̃χJ) +is the unique block of J covered by ̃B. Now, since bl(̃ψNJ (L)) is covered by ̃C, we deduce that +bl(̃ψNJ (L))J is covered by ̃B and therefore +bl(̃χJ) = bl( ̃ψNJ (L)) +J . +(3.5) +As explained in the proof of [Ros22d, Theorem 4.8] we can now use (3.5) together with Proposition +3.12 to conclude the proof via an application of [Spä17, Theorem 4.1 (i)]. +3.3 +Proof of Theorem C +Proof of Theorem C. The hypothesis of Corollary 3.2 is satisfied under our restrictions on G accord- +ing to Lemma 3.6 and therefore we obtain an AutF(GF )(L,λ)-equivariant bijection +ΩG +(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) +that preserves the ℓ-defect of characters. Next, observe that the groups ̃GF A and X ∶= GF ⋊ +AutF(GF ) induce the same automorphisms on GF according to the description given in [GLS98, +Section 2.5]. Then, by applying [Spä17, Theorem 5.3] and Proposition 3.13, we conclude that +(Xχ,GF ,χ) ∼GF (NX(L)ψ,NG(L)F ,ψ) +for every χ ∈ E(GF ,(L,λ)) and where ψ ∶= ΩG +(L,λ)(χ) and the proof is now complete. +4 +Consequences of Theorem C +In this section, we collect some consequences of Theorem C. First, we extend the parametrisation +obtained in Theorem C from unipotent e-Harish-Chandra series of the simple group G to pseudo- +unipotent (see Definition 2.2) e-Harish-Chandra series of the Levi subgroups of G. More precisely, +for every F-stable Levi subgroup K of G, we construct a parametrisation of the e-Harish-Chandra +series associated to e-cuspidal pairs of the form (L,λ) for some (K,F)-pseudo-unipotent character +λ ∈ psK(LF ). In a second step, we construct character bijections above this parametrisation by ex- +ploiting results on isomorphisms of character triples (see Corollary 4.6). This will allow us to control +the characters of e-chain stabilisers lying above pseudo-unipotent characters (see Proposition 5.6). +18 + +4.1 +Parametrisation of pseudo-unipotent characters of Levi subgroups +Let K be an F-stable Levi subgroup of G and set K0 ∶= [K,K]. Observe that since the group G +is simply connected, the subgroup K0 is also simply connected according to [MT11, Proposition +12.14]. In addition, under our assumption on the type of G, we deduce that the simple components +of K0 can only be of some of the types A, B or C. +Proposition 4.1. For every unipotent e-cuspidal pair (L0,λ0) of (K0,F) there exists a defect pre- +serving AutF(KF +0 )(L0,λ0)-equivariant bijection +ΩK0 +(L0,λ0) ∶ E (KF +0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0) +such that +(Yϑ,KF +0 ,ϑ) ∼KF +0 (NYϑ(L0),NK0(L0)F ,ΩK0 +(L0,λ0)(ϑ)) +for every ϑ ∈ E(KF +0 ,(L0,λ0)) and where Y ∶= KF +0 ⋊ AutF(KF +0 ). +Proof. Notice that K0 is the direct product of simple algebraic groups K1,... ,Kn and that the action +of F permutes the simple components Ki. Denote the direct product of the simple components in +each F-orbit by Hj for j = 1,... ,t. The (Hj,F) are the irreducible rational components of (K,F) +and we have KF +0 = HF +1 × ⋯ × HF +t . Similarly, if we define the intersections Mj ∶= L0 ∩ Hj, then +we have a decomposition LF +0 = MF +1 × ⋯ × MF +t . In particular, we can write λ0 = µ1 × ⋯ × µt +with µj ∈ Irr(MF +j ). In this case, notice that (Mj,µj) is a unipotent e-cuspidal pair of (Hj,F). +Next, suppose that Hj = Hj,1 × ⋅⋅⋅ × Hj,mj and observe that HF +j ≃ HF mj +j,1 . By the discussion at the +beginning of this section we know that Hj,1 is a simple, simply connected group of type A, B or C +and hence it satisfies the assumptions of Theorem C. Then, via the isomorphism HF +j ≃ HF mj +j,1 , we +obtain an AutF(HF +j )(Mj,µj)-equivarint bijection +ΩHj +(Mj,µj) ∶ E (HF +j ,(Mj,µj)) → Irr(NHj(Mj)F ∣µj) +that preserves the defect of characters and such that +(Yj,ϑ,HF +j ,ϑ) ∼HF +j (NYj,ϑ(Mj),NHj(Mj)F ,ΩHj +(Mj,µj)(ϑ)) +(4.1) +for every ϑ ∈ E(HF +j ,(Mj,µj)) and where Yj ∶= HF +j ⋊ AutF(HF +j ). Since the characters in the sets +E(KF +0 ,(L0,λ0)) and Irr(NK0(L0)F ∣ λ0) are direct products of characters belonging to the sets +E(HF +j ,(Mj,µj)) and Irr(NHj(Mj)F ∣ µj) respectively, we obtain a bijection +ΩK0 +(L0,λ0) ∶ E (KF +0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0) +by setting +ΩK0 +(L0,λ0) (ϑ1 × ⋅⋅⋅ × ϑt) ∶= ΩH1 +(M1,µ1)(ϑ1) × ⋅⋅⋅ × ΩHt +(Mt,µt)(ϑt) +for every ϑj ∈ E(HF +j ,(Mj,µj)). Finally, arguing as in the proof of [Ros22c, Proposition 6.5], we de- +duce that the bijection ΩK0 +(L0,λ0) preserves the defect of characters, is AutF(KF +0 )(L0,λ0)-equivariant, +and, using (4.1), it induces the KF +0 -block isomorphisms of character triples required in the state- +ment. +19 + +In our next result, we replace the automorphism group Y ∶= KF +0 ⋊ AutF(KF +0 ) with the group of +automorphisms of GF stabilising K, that is, X ∶= (GF ⋊ AutF(GF ))K. To do so, we apply the +so-called Butterfly theorem [Spä17, Theorem 5.3] which basically states that, for any finite group G, +the notion of G-block isomorphism of character triples only depends on the automorphisms induced +on G. +Corollary 4.2. Let (L0,λ0) be a unipotent e-cuspidal pair of (K0,F). The map ΩK0 +(L0,λ0) given by +Proposition 4.1 is AutF(GF )K,(L0,λ0)-equivariant and satisfies +(Xϑ,KF +0 ,ϑ) ∼KF +0 (NXϑ(L0),NK0(L0)F ,ΩK0 +(L0,λ0)(ϑ)) +(4.2) +for every ϑ ∈ E(KF +0 ,(L0,λ0)) and where X ∶= (GF ⋊ AutF(GF ))K. +Proof. First, observe that AutF(GF )K is contained in AutF(KF +0 ) because K0 is an F-stable char- +acteristic subgroup of K. In particular, we deduce that the map ΩK0 +(L0,λ0) is AutF(GF )K,(L0,λ0)- +equivariant. Next, to obtain (4.2), we apply [Spä17, Lemma 3.8 and Theorem 5.3] to the isomorphism +of character triples given by Proposition 4.1 as explained in the proof of [Ros22c, Corollary 6.8]. +Isomorphisms of character triples play a fundamental role in representation theory of finite groups +and in the study of the local-global conjectures. One of the most important consequences of the +existence of isomorphisms of character triples is the possibility to lift character bijections. For in- +stance, the main result of [NS14], shows how to apply this technique to construct bijections above +characters of height zero in the context of the Alperin–McKay Conjecture [NS14, Theorem B]. The +main consequence of this result, which follows from an argument introduced by Murai [Mur12], is +a reduction theorem for the celebrated Brauer’s Height Zero Conjecture [NS14, Theorem A]. This +strategy ultimately lead to the solution of Brauer’s conjecture [Ruh22a] and [MNSFT22]. For other +applications of isomorphisms of character triples see [Tur17], [NSV20], [Ros22a], [Ruh22b] [Ros23] +and [MR22, Proposition 1.1]. +In our next result, we exploit this idea in order to lift the bijections given by Proposition 4.1 to +the Levi subgroup K. Consequently, we extend the parametrisation of unipotent e-Harish-Chandra +series given by Theorem C for the simple group G to a parametrisation of e-Harish-Chandra series +associated to (K,F)-pseudo-unipotent characters for every F-stable Levi subgroup K of G. First, +we need a preliminary lemma. +Lemma 4.3. Let (L,λ) be a unipotent e-cuspidal pair of (K,F) and define X ∶= (GF ⋊AutF(GF ))K. +If KF ≤ H ≤ NG(L)F and Q is an ℓ-radical subgroup of NH(L), then CX(Q) ≤ NX(L). +Proof. Let E ∶= Z(L)F +ℓ and observe that L = C○ +G(E) according to [CE94, Proposition 3.3 (ii)]. Now, +since Oℓ(NH(L)) is the smallest ℓ-radical subgroup of NH(L) [Dad92, Proposition 1.4], we deduce +that E ≤ Oℓ(NH(L)) ≤ Q and it follows that CX(Q) ≤ CX(E) ≤ NX(L) as wanted. +20 + +Theorem 4.4. For every unipotent e-cuspidal pair (L,λ) of (K,F) there exists a defect preserving +AutF(GF )K,(L,λ)-equivariant bijection +ΩK +(L,λ) ∶ E (KF ,(L,psK(λ))) → Irr(NK(L)F ∣ psK(λ)) +such that +(Xχ,KF ,χ) ∼KF (NXχ(L),NK(L)F ,ΩK +(L,λ)(χ)) +for every χ ∈ E(KF ,(L,psK(λ))) and where X ∶= (GF ⋊ AutF(GF ))K. +Proof. Recall that K0 = [K,K] and define L0 ∶= L ∩ K0 and λ0 the restriction of λ to LF +0 . Observe +that (L0,λ0) is a unipotent e-cuspidal pair of (K0,F). Let z ∈ Z(K∗)F ∗ and consider a character +χ belonging to E(KF ,(L,λˆzL)). Since the restriction of λˆzL to LF +0 coincides with λ0, [GM20, +Corollary 3.3.25] implies that χ lies above some character in E(KF +0 (L0,λ0)). On the other hand, +suppose that χ ∈ Irr(KF ) lies above some χ0 ∈ E(KF +0 ,(L0,λ0)). By [CE94, Proposition 3.1] the +character χ0 has an extension χ′ ∈ E(KF ,(L,λ)) and hence, using Gallagher’s theorem [Isa76, +Corollary 6.17] and [CE04, (8.19)], we can find z ∈ Z(K∗)F ∗ such that χ = χ′ˆzK. Since χ′ˆzK is a +character of E(KF ,(L,λˆzL)) according to [CE04, (8.20)], we conclude that +E (KF,(L,psK(λ))) = Irr(KF ∣ E (KF +0 ,(L0,λ0))) . +(4.3) +Next, suppose that ψ ∈ Irr(NK(L)F ∣ λˆzL). In this case, ψ lies above the restriction of λˆzL to LF +0 +which coincides with λ0. In particular, there exists some ϕ ∈ Irr(NK0(L0)F ∣ λ0) such that ψ lies +above ϕ. On the other, if χ lies above such a character ϕ ∈ Irr(NK0(L0)F ∣ λ0), then it lies above +λ0 and therefore we can find z ∈ Z(K∗)F ∗ such that ψ ∈ Irr(NK(L)F ∣ λˆzL). This shows that +Irr(NK(L)F ∣ psK(λ)) = Irr(NK(L)F ∣ Irr(NK0(L0)F ∣ λ0)). +(4.4) +Finally, consider the map ΩK0 +(L0,λ0) given by Proposition 4.1. Then, the result follows from (4.3) +and (4.4) by applying [Ros22c, Proposition 6.1 and Remark 6.2] as explained in the proof of [Ros22c, +Corollary 6.10] and using the KF -block isomorphisms of character triples obtained in Corollary 4.2. +Here, we consider A ∶= GF ⋊ AutF(GF ), A0 ∶= NA(L), K ∶= KF +0 , K0 = NK0(L)F = NK0(L0)F , +G ∶= GF , X ∶= (GF ⋊ AutF(GF ))K, S ∶= E(KF +0 ,(L0,λ0)), S0 ∶= Irr(NK0(L0)F ∣ λ0), V ∶= +(GF ⋊ AutF(GF ))K,S and U ∶= (GF ⋊ AutF(GF ))K,L,Y0. Observe that the condition on defect +groups required by [Ros22c, Proposition 6.1] is satisfied by Lemma 4.3. +4.2 +Above e-Harish-Chandra series +We now further extend Theorem C by lifting the character bijections from Theorem 4.4 with respect +to normal inclusions. +Proposition 4.5. Consider the setup of Theorem 4.4 and let KF ≤ H ≤ NG(K)F . Then, there exists +a defect preserving AutF(GF )H,K,(L,λ)-equivariant bijection +ΩK,H +(L,λ) ∶ Irr(H ∣E (KF,(L,psK(λ)))) → Irr(NH(L)∣psK(λ)) +such that +(NX(H)χ,H,χ) ∼H (NX(H,L)χ,NH(L),ψ) +for every χ ∈ Irr(H ∣ E(KF ,(L,psK(λ)))) and where X ∶= (GF ⋊ AutF(GF ))K. +21 + +Proof. We apply [Ros22c, Proposition 6.1] to the bijection given by Theorem 4.4. We consider A ∶= +GF ⋊ AutF(GF ), G ∶= GF, K ∶= KF , A0 ∶= NA(L), X ∶= NA(K), S ∶= E(KF ,(L,psK(λ))), +S0 ∶= Irr(NK(L)F ∣ psK(λ)), U ∶= X0,λ, V ∶= XS and J ∶= H. Notice that the conditions (i)-(iii) of +[Ros22c, Proposition 6.1] are satisfied by [BMM93, Theorem 3.2 (1)]. Furthermore, the requirements +about defect groups are satisfied by Lemma 4.3. Therefore, as explained in [Ros22c, Proposition +6.11], we obtain the claimed result by applying [Ros22c, Proposition 6.1 and Remark 6.2]. +Before proceeding further, we point out an interesting analogy with another important character +correspondence. The Glauberman correspondence plays a fundamental role in the study of the local- +global counting conjectures and lies at the heart of most reduction theorems. In its most basic form, +it states that for every finite ℓ-group L acting on a finite ℓ′-group K, there exists a bijection +fL ∶ IrrL(K) → Irr(NK(L)) +between the set of L-invariant characters of K and the characters of the normaliser NK(L) (see, +for instance, [Nav18, Section 2.3]). A very deep result due to Dade [Dad80] and recently reproved by +Turull [Tur08], shows that, if K and L are subgroups of a finite group G and KP ≤ H ≤ KNG(L), +then the Glauberman correspondence fL can be lifted to a character correspondence for H, that is, +there exists a bijection +f H +L ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ fL(χ)) +(4.5) +for every χ ∈ IrrL(K). On the other hand, the parametrisation of unipotent e-Harish-Chandra +series obtained by Broué, Malle and Michel [BMM93, Theorem 3.2] lies at the centre of the proofs +of the local-global counting conjectures for finite reductive groups. It is interesting to note that our +methods yield a character bijection above e-Harish-Chandra series which is analogous to (4.5) in +the context of the Glauberman correspondence. This is an immediate consequence of Proposition +4.5. +Corollary 4.6. Consider the setup of Theorem 4.4 and let KF ≤ H ≤ NG(K)F . Then, there exists a +bijection +ΨH +χ ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ ΩK +(L,λ)(χ)) +for every χ ∈ E(KF ,(L,psK(λ))). +Proof. This follows immediately from the proof of Proposition 4.5 by following the construction +made in [Ros22c, Proposition 6.1]. +5 +Towards Theorem A and Theorem B +Finally, we apply the results obtained in the previous sections to prove Theorem A which is our +main result. Then, we obtain Theorem B as a corollary by applying the e-Harish-Chandra theory +for unipotent characters developed by Broué, Malle and Michel [BMM93] and by Cabanes and En- +guehard [CE94]. Before doing so, we introduce the relevant notation and prove some preliminary +results. +22 + +5.1 +Preliminaries on e-chains +Our first aim is to define e-local structures for finite reductive groups that play a role analogue to that +of ℓ-chains in the context of Dade’s Conjecture and the Character Triple Conjecture. The connection +between the set of e-chains and that of ℓ-chains has already been studied in [Ros22c, Section 7.2]. +These results provide a way to obtain Dade’s Conjecture and the Character Triple Conjecture as a +consequence of [Ros22c, Conjecture C and Conjecture D]. The possibility to use different types of +chains is crucial in the study of Dade’s Conjecture and has been introduced by Knörr and Robinson +[KR89]. Their results were insipred by previous studies conducted by many authors including Brown +[Bro75] and Quillen [Qui78] who analised the homotopy theory of associated simplicial complexes. +Definition 5.1. We denote by Le(G,F) the set of e-chains of the finite reductive group (G,F), +that is, chains of the form +σ = {G = L0 > L1 > ⋅⋅⋅ > Ln} +where n is a non-negative integer and each Li is an e-split Levi subgroup of (G,F). We denote by +∣σ∣ ∶= n the length of the e-chain σ and by L(σ) its last term. Furthermore, we define Le(G,F)>0 +to be the set of e-chains having length strictly larger than 0. +Observe that the notion of length defined above, induces a partition of the set Le(G,F) into e- +chains of even and odd length. More precisely, we denote by Le(G,F)± the subset of those e-chains +σ ∈ Le(G,F) that satisfy (−1)∣σ∣ = ±1. +In what follows, given an e-chain σ and an e-split Levi subgroup M of (L(σ),F), we denote by +σ+M the e-chain obtainedby adding M at the end of σ. We also allow the possibility that M = L(σ), +in which case we have σ + L(σ) = σ. Vice versa, we denote by σ − L(σ) the e-chain obtained by +removing the last term L(σ) from σ. In this way we obtain (σ + M) − L(σ + M) = σ where as +usual L(σ + M) denotes the final term of the e-chain σ + M. Here, we use the convention that +σ0 − L(σ0) = σ0 = σ0 + G where σ0 = {G} is the trivial e-chain. +Next, consider the action of GF on the set of e-chains Le(G,F) induced by conjugation: for every +g ∈ GF and σ = {Li}i, we define +σg ∶= {G = L0 > Lg +1 > ⋅⋅⋅ > Lg +n}. +It follows from this definition that the stabiliser GF +σ coincides with the intersection of the normalis- +ers NG(Li)F for i = 1,... ,n. Similarly, we can define an action of AutF(GF ) on Le(G,F) and +give an analogous description of the chains stabilisers AutF(GF )σ. In particular, notice that the last +term of the chain satisfies L(σ)F ⊴ GF +σ . Using this observation, we can use the results of Section +4.2 to control the characters of GF +σ that lie above pseudo-unipotent series of L(σ). +Definition 5.2. For every e-chain σ ∈ Le(G,F) we denote by CPu(σ) the set of unipotent e- +cuspidal pairs (M,µ) ∈ CPu(L(σ),F) that satisfy M < G. Furthermore, for any such pair (M,µ) ∈ +CPu(σ), we define the character set +Uch(GF +σ ,(M,µ)) ∶= +⎧⎪⎪⎨⎪⎪⎩ +Irr(GF +σ ∣ E (L(σ)F ,(M,psL(σ)(µ)))) +L(σ) > M +(5.1) +Irr(GF +σ ∣ E (L(σ)F ,(M,psL(σ−L(σ))(µ)))) +L(σ) = M +(5.2) +23 + +The need to distinguish the cases (5.1) and (5.2) will become apparent in the proofs of Proposition +5.6 and Theorem 5.9 below. Observe that in the definition above, we are excluding the degenerate +case where G = L(σ) = M and therefore the chain σ − L(σ) in the case (5.2) is always defined. To +understand the reason why we are excluding this case, we can consider an analogy with Dade’s Con- +jecture. For every finite group G, recall that k(G) denotes the number of its irreducible characters +and that, for any non-negative integer d, the symbol kd(G) denotes the number of those irreducible +characters of ℓ-defect d. The local-global counting conjectures provide a way to determine the global +invariants kd(G) in terms of ℓ-local structures. This idea was made precise by Isaacs and Navarro +[IN20]. According to their definitions, the block-free version of Dade’s Conjecture can be stated by +saying that the functions kd are chain local for every d > 0. Consequently, and because a sum of +chain local functions is chain local, we deduce that the difference k − k0 = ∑d>0 kd is a chain local +function. On the other hand, using the fact that groups admitting a character of ℓ-defect zero have +trivial ℓ-core, it is easy to see that k0 is not chain local. The exclusion of the case G = L(σ) = M can +be explained by interpreting these observations in the context of unipotent characters. Recall that +ku(GF ) and kc,u(GF ) denote the number of unipotent characters of GF and unipotent e-cuspidal +characters of GF respectively. If ℓ does not divide the order of Z(GF ), then [CE94] implies that the +unipotent e-cuspidal characters of GF have defect zero. Therefore, as in the case of Dade’s Con- +jecture, the global invariant we want to determine e-locally is the difference ku(GF ) − kc,u(GF ). +Finally, notice that kc,u(GF ) is exactly the number of unipotent e-cuspidal pairs (M,µ) of L(σ) +satisfying G = L(σ) = M. +In the following lemma, we show that if the set Uch(GF +σ ,(M,µ)) is non-empty then (M,µ) is +uniquely defined up to GF +σ -conjugation. +Lemma 5.3. Let σ ∈ Le(G,F) and consider two unipotent e-cuspidal pairs (M,µ) and (K,κ) +in CPu(σ). If the sets Uch(GF +σ ,(M,µ)) and Uch(GF +σ ,(K,κ)) have non-trivial intersection, then +(M,µ) and (K,κ) are GF +σ -conjugate. +Proof. Suppose that ϑ is a character belonging to Uch(GF +σ ,(M,µ)) and Uch(GF +σ ,(K,κ)). If we +set L ∶= L(σ), then we can find elements s,t ∈ Z(L∗)F ∗ and characters ϕ ∈ E(LF ,(M,µ)) and +ψ ∈ E(LF ,(K,κ)) such that ϑ lies above ϕˆsL and ψˆtL. By Clifford’s theorem, we deduce that +ϕˆsL = (ψˆtL)g for some g ∈ GF +σ . Furthermore, since ˆs is a linear character, we obtain that ϕ = +ψg(ˆtL)g(ˆsL)−1. Since both ϕ and ψg are unipotent characters of LF , using [CE04, Proposition +8.26] we deduce that (ˆtL)g(ˆsL)−1 = 1L and therefore ϕ = ψg. But then, [BMM93, Theorem 3.2(1)] +shows that (M,µ) and (K,κ)g are LF-conjugate and the result follows. +Next, we describe the block theory associated to characters in the sets introduced in Definition 5.2. +Lemma 5.4. Let σ ∈ Le(G,F) and consider a unipotent e-cuspidal pair (M,µ) ∈ CPu(σ) and a +character ϑ ∈ Uch(GF +σ ,(M,µ)). Then: +(i) the block bl(ϑ) is L(σ)F -regular; +(ii) if the character ϑ lies above a given ϕˆzL(σ) ∈ E(L(σ)F ,(M,µˆzM)) for some z ∈ Z(L(σ)∗)F ∗, +then we have +bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F +and +bl(ϑ) = bl(ϕˆzL(σ))GF +σ = bl(µˆzM)GF +σ +24 + +(iii) the induced block bl(ϑ)GF is defined. +Proof. The first point follows from Lemma 2.3 by choosing L = L(σ) and H = GF +σ . Furthermore, +in the case of (5.2) observe that L(σ) ≤ L(σ − L(σ)) and hence Z(L(σ − L(σ))∗) ≤ Z(L(σ)∗). +Therefore, we can always find ϕ and z as in the statement of (ii). Since ϕ is an irreducible constituent +of the virtual character RL(σ) +M +(µ), it follows from [CE94, Proposition 4.2] (whose assumptions are +satisfied by [CE94, Proposition 3.3 (ii)]) that bl(ϕ) = bL(σ)F (M,µ) = bl(µ)L(σ)F . Then, since ˆzM +is the restriction of the linear character ˆzL(σ) to MF , we deduce from Lemma 2.1 that +bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F . +Now, [Nav98, Theorem 9.19] implies that +bl(ϑ) = bl(ϕˆzL(σ)) +GF +σ +and the second point follows by the transitivity of block induction. Finally, set Q ∶= Z(M)F +ℓ and +observe that QCGF (Q) = MF ≤ NGF (Q) by [CE94, Proposition 3.3(ii)]. Then, [Nav98, Theorem +4.14] implies that bl(µˆzM)GF is well defined and so is bl(ϑ)GF by (ii) and transitivity of block +induction. This concludes the proof. +Using the lemma above, we can now define the following character set. This yields the e-local object +through which we can determine the number of unipotent characters in a given block of B of GF +and with a given defect d ≥ 0 (see Section 5.3). +Definition 5.5. Let B be a block of GF and d a non-negative integer. For every e-chain σ ∈ +Le(G,F) and unipotent e-cuspidal pair (L,λ) ∈ CPu(σ) we define the character set +Uchd (Bσ,(M,µ)) ∶= {ϑ ∈ Uch (GF +σ ,(M,µ)) ∣ d(ϑ) = d,bl(ϑ)GF = B}. +where bl(ϑ)GF is defined according to Lemma 5.4 (iii). Furthermore, we denote the cardinality of +this set by +kd +u (Bσ,(M,µ)) ∶= ∣Uchd (Bσ,(M,µ))∣. +To conclude this section, we show that Proposition 4.5 can be used to parametrise the character sets +from Definition 5.5. +Proposition 5.6. Let B be a block of G and d a non-negative integer. If σ ∈ Le(G,F) and (M,µ) is +a unipotent e-cuspidal pair in CPu(σ) then there exists an AutF(GF )B,σ,(M,µ)-equivariant bijection +ΩB,d +σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bσ+M,(M,µ)) +such that +(Xσ,ϑ,GF +σ ,ϑ) ∼GFσ (Xσ+M,ϑ,GF +σ+M,ΩB,d +σ,(M,µ)(ϑ)) +for every ϑ ∈ Uchd(Bσ,(M,µ)) and where X ∶= GF ⋊ AutF(GF ). +25 + +Proof. First, observe that if M coincides with the last term L(σ) of the chain σ, then we have +σ + M = σ which implies Uchd(Bσ,(M,µ)) = Uchd(Bσ+M,(M,µ)). In this case the result holds +by defining ΩB,d +σ,(M,µ) as the identity. Therefore, we can assume that M < L(σ) and define ρ ∶= σ+M. +Now, according to (5.1) we have +Uch (GF +σ ,(M,µ)) = Irr(GF +σ ∣ E (L(σ)F ,(M,psL(σ)(µ)))) . +(5.3) +On the other hand, noticing that M coincides with the last term L(ρ) of the chain ρ and that +ρ−L(ρ) = σ, we obtain the equality E(L(ρ)F ,(M,psL(ρ−L(ρ))(µ))) = psL(σ)(µ). Then, observing +that GF +ρ = NGFσ (M), we can apply (5.2) to obtain the equality +Uch (GF +ρ ,(M,µ)) = Irr(NGFσ (M) ∣ psL(σ)(µ))) . +(5.4) +Next, we apply Proposition 4.5 by choosing the groups in that statement to be H = GF +σ , K = +L(σ) and (L,λ) = (M,µ). By (5.3) and (5.4), we deduce that there exists an AutF(GF )σ,(M,µ)- +equivariant bijection +ΩL(σ),GF +σ +(M,µ) +∶ Uch(GF +σ ,(M,µ)) → Uch(GF +ρ ,(M,µ)). +(5.5) +Moreover, using the H-block isomorphisms given by Proposition 4.5 together with [Spä17, Lemma +3.8 (b)], we deduce that +(Xσ,ϑ,GF +σ ,ϑ) ∼GFσ (Xρ,ϑ,GF +ρ ,ΩL(σ),GF +σ +(M,µ) +(ϑ)) +(5.6) +for every ϑ ∈ Uchd(GF +σ ,(M,µ)). To conclude, observe first that ΩL(σ),GF +σ +(M,µ) +sends characters of +defect d to characters of defect d. Moreover, by the transitivity of block induction and using (5.6), +we deduce that +bl(ϑ)GF = bl(ΩL(σ),GF +σ +(M,µ) +(ϑ)) +GF +. +This shows that the bijection from (5.5) sends characters in the set Uchd(Bσ,(M,µ)) to charac- +ters in the set Uchd(Bσ+M,(M,µ)) and therefore it restricts to a bijection, denoted by ΩB,d +σ,(M,µ), +satisfying the properties required in the statement. This completes the proof. +We conclude this section with a remark on the isomorphisms of character triples obtained in Propo- +sition 5.6. +Remark 5.7. Suppose that ℓ does not divide q ± 1 if G is of type A(±q). In this case, every e-split +Levi subgroup L of G satisfies L = C○ +G(Z(L)F +ℓ ) according to [CE04, Proposition 13.19]. This fact +can be used to show that the GF +σ -block isomorphisms of character triples given by Proposition 5.6 +can be extended to GF -block isomorphisms of character triples. First, we claim that +CGF Xσ,ϑ(D) ≤ Xσ,ϑ +(5.7) +for every irreducible character ϑ of GF +σ and every ℓ-radical subgroup D of GF +σ+M. Define Qi ∶= +Z○(Li)F +ℓ for every e-split Levi subgroup Li appearing in the chain σ. Then, using the fact that D is +26 + +ℓ-radical, we obtain the inclusions Qi ≤ Oℓ(GF +σ ) ≤ D. Therefore, every element x ∈ GF Xσ,ϑ that +centralises D centralises also each Qi and hence normalises each Li. It follows that +CGF Xσ,ϑ(D) ≤ (GF Xσ,ϑ)σ = Xσ,ϑ +as required by (5.7). We can now apply [Ros22a, Lemma 2.11] to the GF +σ -block isomorphisms given +by Proposition 5.6 to show that +(Xσ,ϑ,GF +σ ,ϑ) ∼GF (Xσ+M,ϑ,GF +σ+M,ΩB,d +σ,(M,µ)(ϑ)) +for every ϑ ∈ Uchd(Bσ,(M,µ)). +5.2 +Proof of Theorem A +We are finally ready to prove our main theorem which provides a bijection for unipotent characters +in the spirit of the Character Triple Conjecture [Spä17, Conjecture 6.3]. In this section, we prove a +slightly stronger result that provides further information on the type of e-chains and isomorphisms +of character triples. In the following definition we introduce the analogue of the set Cd(B)± con- +sidered in the Character Triple Conjecture as defined in [Spä17, p. 1097]. +Definition 5.8. Let B be a block of GF and consider a non-negative integer d. We define the set +Ld +u(B)± = {(σ,M,µ,ϑ) ∣ σ ∈ Le(G,F)±,(M,µ) ∈ CPu(σ),ϑ ∈ Uchd (Bσ,(M,µ))} . +The conjugacy action of GF induces an action of GF on Ld +u(B)± defined by (σ,M,µ,ϑ)g ∶= +(σg,Mg,µg,ϑg) for every element g ∈ GF and (σ,M,µ,ϑ) ∈ Ld +u(B)±. We denote by Ld +u(B)±/GF +the corresponding set of GF -orbits of tuples. Moreover, for every such orbit ω, we denote by ω● the +corresponding GF -orbit of pairs (σ,ϑ) such that (σ,M,µ,ϑ) ∈ ω for some (M,µ) ∈ CPu(σ). In +other words, if we indicate by (σ,M,µ,ϑ) the GF-orbit of (σ,M,µ,ϑ), then (σ,M,µ,ϑ) +● is the +GF -orbit of the pairs (σg,ϑg). +In a similar way, if AutF(GF )B denotes the set of those automorphisms α ∈ AutF(GF ) that sta- +bilise B, then we can define (σ,M,µ,ϑ)α ∶= (σα,Mα,µα,ϑα) for every α ∈ AutF(GF )B and +(σ,M,µ,ϑ) ∈ Ld +u(B). In this way, we obtain an action of the group AutF(GF )B on the set Ld +u(B)± +and on the corresponding set of orbits Ld +u(B)±/GF . +Theorem 5.9. For every block B of GF and every non-negative integer d, there exists an AutF(GF )B- +equivariant bijection +Λ ∶ Ld +u(B)+/GF → Ld +u(B)−/GF . +Moreover, for every ω ∈ Ld +u(B)+/GF, any (σ,ϑ) ∈ ω● and any (ρ,χ) ∈ Λ(ω)● we have +∣σ∣ = ∣ρ∣ ± 1 +and +(Xσ,ϑ,GF +σ ,ϑ) ∼J (Xρ,χ,GF +ρ ,χ) +with J = GF +σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF +ρ , if ∣σ∣ = ∣ρ∣ + 1, and where X ∶= GF ⋊ AutF(GF ). +27 + +Proof. Define A ∶= AutF(GF ) and observe that X = GF ⋊ A. In a first step, we construct an +equivariant bijection between triples of the form (σ,M,µ). More precisely, let S denote the set of +such triples (σ,M,µ) with σ ∈ Le(G,F) and (M,µ) ∈ CPu(σ). We define a map +∆ ∶ S → S +by setting +∆((σ,M,µ)) ∶= +⎧⎪⎪⎨⎪⎪⎩ +(σ + M,M,µ) , +L(σ) > M +(σ − M,M,µ) , +L(σ) = M. +Observe that the chain σ−M is always defined since M < G by the definition of CPu(σ). Moreover, +it is clear from the definition above that the map ∆ is A-equivariant and satisfies ∆2 = Id. Therefore, +observing that ∣σ ± M∣ = ∣σ∣ ± 1, we conclude that ∆ restricts to an A-equivariant bijection +∆ ∶ S+ → S− +where S± denotes the set of those triples (σ,M,µ) of S that satisfy σ ∈ Le(G,F)±. Furthermore, +notice once again that if ∆((σ,M,µ)) = (ρ,K,κ), then +∣σ∣ = ∣ρ∣ ± 1. +(5.8) +Now, fix an AB-transversal T+ in S+ and observe that the image of T+ under the map ∆, denoted by +T−, is an AB-transversal in S because of the equivariance property of ∆. Consider (σ,M,µ) ∈ T+ +and write ∆((σ,M,µ)) = (ρ,M,µ). In what follows, we may assume without loss of generality +that L(σ) > M and that ρ = σ + M, otherwise we repeat the arguments verbatim by replacing +(σ,M,µ) with (ρ,M,µ). By Proposition 5.6 we obtain an AB,σ,(M,µ)-equivariant bijection +ΩB,d +σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bρ,(M,µ)) +such that +(Xσ,ϑ,GF +σ ,ϑ) ∼GFσ (Xρ,χ,GF +ρ ,χ) +(5.9) +for every ϑ ∈ Uchd(Bσ,(M,µ)) and where χ is the image of ϑ. Consequently, if U(σ,M,µ) ++ +is an +AB,(σ,M,µ)-transversal in the character set Uchd(Bσ,(M,µ)), then its image, denoted by U(ρ,M,µ) +− +, +under the bijection above is an AB,(ρ,M,µ)-transversal in the character set Uchd(Bρ,(M,µ)) be- +cause AB,(σ,M,µ) = AB,(ρ,M,µ). +Now, by the discussion in the previous paragraph and using Lemma 5.3, we conclude that the sets +of GF -orbits +L+ ∶= {(σ,M,µ,ϑ) ∣ (σ,M,µ) ∈ T+,ϑ ∈ U(σ,M,µ) ++ +} +and +L− ∶= {(ρ,M,µ,χ) ∣ (ρ,M,µ) ∈ T−,χ ∈ U(ρ,M,µ) +− +} +are AB-transversals in the sets Ld +u(B)+/GF and Ld +u(B)−/GF respectively. Finally, we can define +the bijection Λ by setting +Λ((σ,M,µ,ϑ) +x) ∶= (ρ,M,µ,χ)x +28 + +for every x ∈ AB and every (σ,M,µ,ϑ) ∈ L+ and (ρ,M,µ,χ) ∈ L− satisfying ∆(σ,M,µ) = +(ρ,M,µ) and such that +χ = +⎧⎪⎪⎪⎨⎪⎪⎪⎩ +ΩB,d +σ,(M,µ)(ϑ), +ρ = σ + M +(ΩB,d +ρ,(M,µ)) +−1 +(ϑ), +ρ = σ − M. +Using (5.8) and (5.9) together with the definition of Λ, we conclude that the properties required in +the statement are satisfied and the proof is now complete. +Now, as a consequence of Theorem 5.9 and Remark 5.7, we can finally prove Theorem A. +Proof of Theorem A. Assume that ℓ does not divide q ± 1 whenever (G,F) is of type A(±q). Con- +sider the bijection Λ from Theorem 5.9 and chose ω ∈ Ld +u(B)+/GF, (σ,ϑ) ∈ ω● and (ρ,χ) ∈ Λ(ω)●. +Then, we have +(Xσ,ϑ,GF +σ ,ϑ) ∼J (Xρ,χ,GF +ρ ,χ) +with J = GF +σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF +ρ , if ∣σ∣ = ∣ρ∣ + 1. In either cases, applying Remark 5.7, we +deduce that +(Xσ,ϑ,GF +σ ,ϑ) ∼GF (Xρ,χ,GF +ρ ,χ) +as required by Theorem A. +5.3 +Proof of Theorem B +Our final goal is to obtain a counting argument for unipotent characters as a consequence of The- +orem 5.9. Recall that Dade’s Conjecture provides a way to determine the number of characters in +a given ℓ-block B and with a given defect d in terms of ℓ-local structures. Theorem B provides an +adaptation of this idea to the unipotent characters of finite reductive groups by means of e-local +structures compatible with e-Harish-Chandra theory (see Definition 5.5). For every σ ∈ Le(G,F) +we define +kd +u(Bσ) ∶= +∑ +(M,µ) +kd +u(Bσ,(M,µ)) +(5.10) +where (M,µ) runs over a set of representatives for the action of GF +σ on CPu(σ). Moreover, recall +that kd +u(B) and kd +c,u(B) denote the number of irreducible characters belonging to the block B and +with defect d that are unipotent and unipotent e-cuspidal respectively. +Proof of Theorem B. To start, we determine the cardinality of the sets of GF-orbits Ld +u(B)±/GF . +By applying Lemma 5.3, we obtain +∣Ld +u(B)±/GF ∣ = +∑ +σ,(M,µ) +kd +u(Bσ,(M,µ)) = ∑ +σ +kd +u(Bσ) +(5.11) +where σ runs over a set of representatives, say L±, for the action of GF on Le(G,F)± and (M,µ) +runs over a set of representatives for the action of GF +σ on CPu(σ). Next, we isolate the contribution +given by the trivial chain σ0 ∶= {G} ∈ Le(G,F)+ to the sum in (5.11). In this case, we have +L(σ0) = G and hence psL(σ)(µ) = {µ} for every (M,µ) ∈ CPu(σ0) because the center Z(G∗)F ∗ +29 + +is trivial under our assumptions. Consequently, using Definition 5.2 and Definition 5.5, we deduce +that +kd +u(Bσ0) = +∑ +(M,µ) +kd +u(Bσ0,(M,µ)) +(5.12) += +∑ +(M,µ) +∣Irrd(B) ∩ E(GF ,(M,µ))∣ += kd +u(B) − kd +c,u(B) +where the last equality holds by [BMM93, Theorem 3.2 (1)] and recalling that every pair (M,µ) ∈ +CPu(σ0) satisfies M < G = L(σ0). 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DINI, VIALE MORGAGNI 67/A, FIRENZE, +ITALY +Email address: damiano.rossi00@gmail.com +33 + diff --git a/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/load_file.txt b/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88ee70826a0763ae05f794c1a81f576eaae1565e --- /dev/null +++ b/PtE4T4oBgHgl3EQfkg1D/content/tmp_files/load_file.txt @@ -0,0 +1,1759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf,len=1758 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='05151v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='RT] 12 Jan 2023 A local-global principle for unipotent characters Damiano Rossi Abstract We obtain an adaptation of Dade’s Conjecture and Späth’s Character Triple Conjecture to unipotent characters of simple, simply connected finite reductive groups of type A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, this gives a precise formula for counting the number of unipotent characters of each defect d in any Brauer ℓ-block B in terms of local invariants associated to e-local struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This provides a geometric version of the local-global principle in representation theory of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' A key ingredient in our proof is the construction of certain parametrisations of unipotent generalised Harish-Chandra series that are compatible with isomorphisms of charac- ter triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Structure of the paper .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 21 2010 Mathematical Subject Classification: 20C20, 20C33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Key words and phrases: Dade’s Conjecture, Character Triple Conjecture, finite reductive groups, unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This work is partially supportedby the EPSRC grant EP/T004592/1 and was written during a research visit of the author at the Universitá degli Studi di Firenze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The author would like to thank Silvio Dolfi and all the members of the algebra group in the Department of Mathematics for their hospitality and, in particular, Carolina Vallejo for some comments concerning the local-global principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, the author would like to thank Lucas Ruhstorfer for some helpful conversation on the paper [Bro-Ruh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 1 5 Towards Theorem A and Theorem B 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Preliminaries on e-chains .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 29 1 Introduction The local-global conjectures are currently some of the most interesting and challenging problems in representation theory of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Among others, these include the McKay Conjecture [McK72], the Alperin—McKay Conjecture [Alp76] and Alperin’s Weight Conjecture [Alp87] all of which can be deduced by a deeper statement known as Dade’s Conjecture [Dad92], [Dad94], [Dad97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The latter also implies the celebrated Brauer’s Height Zero Conjecture introduced in [Bra56] and whose proof has recently been completed in [MNSFT22] and [Ruh22a] while relying on a combined effort of many other authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this paper, we are particularly interested in Dade’s Conjecture which, for every prime number ℓ, suggests a precise formula for counting the number of irreducible characters of a finite group, with a given ℓ-defect and belonging to a given Brauer ℓ-block, in terms of the ℓ-local structure of the group itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This conjecture has been further extended in [Spä17] where the Character Triple Conjecture was formulated by introducing a compatibility with N-block isomorphisms of character triples, hereinafter denoted by ∼N, as defined in [Spä17, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This notion plays a funda- mental role in many aspects of group representation theory and, as we will see later, gives us a way to control the representation theory of local subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, it was exploited to reduce Dade’s Conjecture to finite quasi-simple groups as explained in [Spä17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Our aim is to adapt and prove the two conjectures described in the previous paragraph to the case of unipotent characters of finite reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The approach considered here is inspired by ideas introduced by the author in [Ros22c] and provides further evidence for the conjectures formulated in that paper [Ros22c, Conjecture C and Conjecture D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, the ℓ-local structures considered above are replaced by more suitable e-local structures arising from the geometry of the underlying algebraic group that are compatible with the framework of Deligne–Lusztig theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, our results also suggest the existence of an e-local-global principle for the representation theory of finite reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, let G be a simple, simply connected group of type A, B or C which is defined over an algebraically closed field of positive characteristic p and let F ∶ G → G be a Frobenius endomorphism endowing G, as a variety, with an Fq-structure for some power q of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by GF the finite reductive group consisting of the Fq-rational points on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, we fix an odd prime ℓ different from p and denote by e the multiplicative order of q modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We let Le(G,F) denote the set of e-chains of (G,F) of the form σ = {G = L0 > L1 > ⋅⋅⋅ > Ln} where each Li is an e-split Levi subgroup of (G,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The final term of the e-chain σ is denoted by L(σ) = Ln, while ∣σ∣ ∶= n is the length of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that the latter induces a partition of the set Le(G,F) into the sets Le(G,F)± consisting of those e-chains σ that satisfy (−1)∣σ∣ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, notice that GF acts by conjugation on the set Le(G,F) and indicate by GF σ the stabiliser of the e-chain σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It follows directly from the definition that this action preserves the length of e-chains and, in particular, it restricts to an action of GF on the set Le(G,F)>0 of e-chains of positive length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2 Now, to each non-negative integer d and Brauer ℓ-block B of the finite group GF , we associate a set Ld u(B)± consisting of quadruples (σ,M,µ,ϑ) where σ is an e-chain belonging to Le(G,F)±, (M,µ) is a unipotent e-cuspidal pair of (L(σ),F) such that M does not coincide with G, and ϑ is an irreducible character of the e-chain stabiliser GF σ belonging to the character set Uchd(Bσ,(M,µ)) defined by the choice of d, B, σ and (M,µ) as described in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Once again, the group GF acts by conjugation on Ld u(B)± and we indicate the corresponding set of GF -orbits by Ld u(B)±/GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every such orbit ω, we denote by ω● the corresponding GF -orbit of pairs (σ,ϑ) such that (σ,M,µ,ϑ) ∈ ω for some unipotent e-cuspidal pair (M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' With the above notation, we are now able to state our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For simplicity, in the next theorem we assume that the prime ℓ does not divide the greatest common divisor (q ± 1,n + 1) whenever (G,F) is of type An(±q) and where An(−q) denotes 2An(q) as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe however that this assumption can be removed as explained in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7 (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 for the more general statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every Brauer ℓ-block B of GF and every non-negative integer d, there exists an AutF(GF )B-equivariant bijection Λ ∶ Ld u(B)+/GF → Ld u(B)−/GF such that (Xσ,ϑ,GF σ ,ϑ) ∼GF (Xρ,χ,GF ρ ,χ) for every ω ∈ Ld u(B)+/GF, any (σ,ϑ) ∈ ω●, any (ρ,χ) ∈ Λ(ω)● and where X ∶= GF ⋊ AutF(GF ) and AutF(GF ) is the group of automorphisms described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The above theorem provides an adaptation of Späth’s Character Triple Conjecture to the framework of Deligne–Lusztig theory for the unipotent characters of finite reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Theorem A also offers further evidence for the validity of [Ros22c, Conjecture D], in fact the set Ld u(B)± introduced above is a subset of the set of quadruples Ld(B)± considered in [Ros22c, Conjecture D] which is identified by only selecting unipotent e-cuspidal pairs (M,µ) among those appearing in such quadruples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we obtain a formula for counting the number of unipotent characters of ℓ-defect d in the Brauer ℓ-block B in terms of local invariants associated to e-local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For each e-chain σ of (G,F) with positive length, we define kd u(Bσ) to be the number of characters belonging to one of the character sets Uchd(Bσ,(M,µ)) for some unipotent e-cuspidal pair (M,µ) of (L(σ),F) up to GF σ -conjugation (see also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, let kd u(B) and kd c,u(B) be the number of irreducible characters with ℓ-defect d and belonging to the Brauer ℓ-block B that are unipotent and unipotent e-cuspidal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, by using the bijection given by Theorem A we can determine the difference kd u(B) − kd c,u(B) in terms of an alternating sum involving the terms kd u(Bσ) arising from the e-local structure GF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 3 Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every Brauer ℓ-block B of GF and every non-negative integer d, we have kd u(B) − kd c,u(B) = ∑ σ (−1)∣σ∣+1kd u(Bσ) where σ runs over a set of representatives for the action of GF on Le(G,F)>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We point out that the restriction on the prime ℓ made for simplification before Theorem A only concerns the condition on isomorphisms of character triples and hence does not affect Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As before, this result provides an adaptation of Dade’s Conjecture to the framework of Deligne– Lusztig theory for the unipotent characters of finite reductive groups and gives new evidence in favour of [Ros22c, Conjecture C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The necessity for the introduction of the corrective term kd c,u(B) in the equality of Theorem B can be understood as an analogue to the exclusion of the case of blocks with central defect in the statement of Dade’s Conjecture or, depending on the formulation under consideration, of the case where d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We refer the reader to the more detailed discussion given in the paragraph following Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, we mention that Theorem B also provides evidence for a positive answer to a question recently posed by Broué [Bro22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It is particularly interesting to notice that, to the author’s knowledge, Theorem B cannot be obtained directly using techniques available at the present time, but only as a consequence of the existence of GF -block isomorphisms of character triples as those considered in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In fact, while Deligne–Lusztig theory allows us to control the representation theory of finite reductive groups, it is not sufficient to control the representation theory of e-chain stabilisersGF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' However, observe that the stabiliser GF σ contains the finite reductive group L(σ)F as a normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, we can first use Deligne–Lusztig theory to study the characters of L(σ)F and then apply Clifford theory via GF -block isomorphisms of character triples to control the characters of GF σ (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In order to achieve the latter step, we need to make Deligne–Lusztig theory and, more precisely, e- Harish-Chandra theory for unipotent characters compatible with GF -block isomorphisms of char- acter triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This ideas was first suggested by the author in [Ros22c, Parametrisation B] and further studied in [Ros22d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Our next result, which is a key ingredient in the proofs of Theorem A and Theorem B, establishes this conjectured parametrisation in the unipotent case under the assump- tion specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This can also be seen as an extension of the parametrisation introduced by Broué, Malle and Michel in [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (2)] to the language of GF-block isomorphisms of character triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)- equivariant bijection ΩG (L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) that preserves the ℓ-defect of characters and such that (Xχ,GF ,χ) ∼GF (NXχ(L),NG(L)F ,ΩG (L,λ)(χ)) for every χ ∈ E(GF ,(L,λ)) and where X ∶= GF ⋊ AutF(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The proof of Theorem C, and therefore of Theorem A and Theorem B, partially relies on certain 4 conditions on the extendibility of characters of e-split Levi subgroups that were first introduced to settle the inductive conditions for the McKay Conjecture and the Alperin–McKay Conjecture, and then further studied in the context of Parametrisation B of [Ros22c] (see the exact statement given in [Ros22d, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These conditions were obtain, under certain assumptions, for groups of type A, B and C in the papers [BS20], [Bro22b] and [Bro-Ruh] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Nonetheless, a version of these results is expected to hold in general and hence we believe that the above theorems, obtained here for types A, B and C with respect to an odd prime ℓ, will extend to the general case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Structure of the paper The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In Section 2 we introduce the necessary notation and recall the main definitions and results used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 we introduce the notion of pseudo-unipotent character (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) and prove a result on the regularity of blocks covering those containing such characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, in Section 3 we start working towards a proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 we consider certain equivariance properties that can be established in the presence of extendibility conditions for characters of e-split Levi subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Here, we also present a candidate for the bijection ΩG (L,λ) required by Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 we construct the required GF-block isomorphisms of character triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using these results, we can then prove Theorem C in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The following step is to extend the parametrisation of unipotent e-Harish-Chandra series in the group G, as given by Theorem C, to a parametrisa- tion of pseudo-unipotent e-Harish-Chandra series in F-stable Levi subgroups K of (G,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This is done in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Once this is established, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 we exploit the theory of GF -block isomorphisms to obtain bijections above e-Harish-Chandra series that are required to control the representation theory of the e-chain stabilisers GF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' A more detailed analysis of the characters of GF σ is carried out in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, we obtain a parametrisation of the character sets Uchd(Bσ,(M,µ)) in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 we apply these results to prove Theorem A and Theorem B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2 Notation and background material 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Characters and blocks of finite groups We recall some standard notation from representation theory of finite groups as can be found in [Isa76] and [Nav98], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let Irr(G) the set of ordinary irreducible characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If N ⊴ G and ϑ ∈ Irr(N), then we denote by Irr(G ∣ ϑ) the set of irreducible characters of G that lie above ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More generally, if S is a subset of irreducible characters of N, then we denote by Irr(G ∣ S) the union of the sets Irr(G ∣ ϑ) for ϑ ∈ S, that is, the set of irreducible characters of G that lie above some character in the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we denote by Gϑ the stabiliser of the irreducible character ϑ ∈ Irr(N) under the conjugacy action of G and say that ϑ is G-invariant if G = Gϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, we say that (G,N,ϑ) is a character triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These objects provide important information in the study of Clifford theory and play a crucial role in many aspects of the local-global conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Of paramount importance is the introduction of certain binary relations on the set of character triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We refer the reader to [Nav18, Chapter 5 and 10] and [Spä18] for a more detailed introduction to these ideas and for the necessary background on 5 projective representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The binary relation considered here was introduced in [Spä17, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6] and is known as N-block isomorphism of character triples, denoted by ∼N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This equivalence relation has further been studied in [Ros22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In order to construct N-block isomorphisms of character triples, it is often useful to prove certain results on the extendibility of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Here, we introduce the notion of maximal extendibility (see [MS16, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5]) that will be considered in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let N ⊴ G be finite groups and consider S a subset of irreducible characters of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, we say that maximal extendibility holds for the set S with respect to the inclusion N ⊴ G if every character ϑ ∈ S extends to its stabiliser Gϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, we can specify an extension map Λ ∶ S → ∐ N≤H≤G Irr(H) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) that sends each character ϑ ∈ S to an extension Λ(ϑ) of ϑ to the stabiliser Gϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we consider modular representation theory with respect to a fixed prime number ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For χ ∈ Irr(G), there exist unique non-negative integers d(χ), called the ℓ-defect of χ, such that ℓd(χ) = ∣G∣ℓ/χ(1)ℓ and where for an integer n we denote by nℓ the largest power of ℓ that divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For any d ≥ 0, let Irrd(G) be the set of irreducible characters χ of G that satisfy d(χ) = d and denote by kd(G) its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Associated to the prime ℓ, we also have the set of Brauer ℓ-blocks of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Each block is uniquely determined by the central functions λB (see [Nav98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every χ ∈ Irr(G), we denote by bl(χ) the unique block that satisfies χ ∈ Irr(bl(χ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, if H ≤ G and b is a block of H, then bG denotes the block of G obtained via Brauer’s induction (when it is defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If B is a block of G and d ≥ 0, then let Irrd(B) be the set of irreducible characters belonging to the block B and having defect d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The cardinality of Irrd(B) is denoted by kd(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We conclude this introductory section with an analogue of [Isa76, Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3] for blocks that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let H ≤ G be finite groups and consider blocks b of H and B of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If ζ is a linear character of G, then: (i) there are blocks b ⋅ ζH of H and B ⋅ ζ of G satisfying Irr(b ⋅ ζH) = {ψζH ∣ ψ ∈ Irr(b)} and Irr(B ⋅ ζ) = {χζ ∣ χ ∈ Irr(B)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (ii) If bG = B, then (b ⋅ ζH)G = B ⋅ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The first point is [Riz18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, let g ∈ G and denote by ClG(g) the G-conjugacy class of g and by ClG(g)+ the corresponding conjugacy class sum in the group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since the intersection ClG(g) ∩ H is a union of H-conjugacy classes, we can find h1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' ,hn ∈ ClG(g) ∩ H such that ClG(g) ∩ H = n ∐ i=1 ClH(hi) and where n is zero if ClG(g) ∩ H is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, observe that ζ(hi) = ζ(g) since λ is a 6 class function of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, using the notation of [Nav98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='87] we obtain λB⋅ζ (ClG(g)+) = λB (ClG(g)+)ζ(g) = λG b (ClG(g)+)ζ(g) = n ∑ i=1 λb (ClH(hi)+)ζ(g) = n ∑ i=1 λb (ClH(hi)+)ζH(hi) = n ∑ i=1 λb⋅ζH (ClH(hi)+) = λG b⋅ζH (ClG(g)) where for every algebraic integer α of C we denote by α its reduction modulo a maximal ideal containing the prime ℓ (see [Nav98, Chapter 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This shows that B ⋅ ζ = (b ⋅ ζH)G and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 Finite reductive groups and unipotent characters Let G be a connected reductive group defined over an algebraic closure of a field of positive char- acteristic p different from ℓ and consider a Frobenius endomorphism F ∶ G → G associated with an Fq-structure for a power q of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The set of Fq-rational points on the variety G is denoted by GF and is called a finite reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By abuse of notation we also refer to the pair (G,F) as a finite reductive group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let L be a Levi subgroup of a parabolic subgroup P of G and assume that L (but not necessarily P) is F-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using ℓ-adic cohomology, Deligne–Lusztig [DL76] and Lusztig [Lus76] defined a Z-linear map RG L≤P ∶ ZIrr(LF) → ZIrr(GF ) with adjoint ∗RG L≤P ∶ ZIrr(GF ) → ZIrr(LF) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The exact definition can be found in [CE04, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These maps are known to be independent of the choice of the parabolic subgroup P in almost all cases (see [BM11] and [Tay18]) and, in particular, in those considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, we will always omit P and denote RG L≤P simply by RG L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, using Deligne–Lusztig induction we define the unipotent characters of GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These are the irreducible characters χ of GF that appear as an irreducible constituent of the virtual character RG T(1T) for some F-stable maximal torus T of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The set of unipotent characters of GF is denoted by Uch(GF ) and its cardinality by ku(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Similarly, if B is a block of GF and d a non-negative integer, then kd u(B) denotes the cardinality of the intersection Uch(GF ) ∩ Irrd(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 e-Harish-Chandra theory for unipotent characters Denote by e the multiplicative order of q modulo ℓ, if ℓ is odd, or modulo 4, if ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this section, we collect the main results of e-Harish-Chandra theory for unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This was first in- troduced by Fong and Srinivasan [FS86] for classical groups and then further developed by Broué, Malle and Michel [BMM93] for unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The compatibility of this theory with Brauer ℓ-blocks was described by Cabanes and Enguehard in [CE94] for good primes and completed by 7 Enguehard [Eng00] for bad primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These results also provide a description of the characters be- longing to unipotent blocks (see [CE94, Theorem (iii)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Another description of these characters was provided by the author in [Ros22c] under certain resctrictions on the prime ℓ (see also [Ros22c, Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='14] for a comparison between the two descriptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We refer the reader to the monographs [CE04] and [GM20] for a more complete account of this beautiful theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The theory of Φe-subgroups that constitutes the foundation of e-Harish-Chandra theory was intro- duced in [BM92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Following their terminology, we say that an F-stable torus S of G is a Φe-torus if its order polynomial is a power of the e-th cyclotomic polynomial, that is, if P(S,F ) = Φn e for some integer n and where Φe denotes the e-th cyclotomic polynomial (see [CE04, Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, we say that a Levi subgroup L of G is an e-split Levi subgroup if there exists a Φe-torus S such that L = CG(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, we say that L is an e-split Levi subgroup of (G,F) to emphasise the role of the Frobenius endomorphism F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that, for any torus T, there exists a unique maximal Φe-torus of T denoted by TΦe (see [CE04, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, it can be shown that an F-stable Levi subgroup L of G is e-split if and only if L = CG(Z○(L)Φe) (see, for instance, [GM20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, recall that (L,λ) is a unipotent e-cuspidal pair of (G,F) if L is an e-split Levi subgroup of (G,F) and λ ∈ Irr(LF ) satisfies ∗RL M(λ) = 0 for every e-split Levi subgroup M < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' A character λ with the property above is said to be a unipotent e-cuspidal character of LF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by CPu(G,F) the set of unipotent e-cuspidal pairs of (G,F) and by kc,u(GF ) the number of unipotent e-cuspidal characters of GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, we define the e-Harish-Chandra series associate to the e-cuspidal pair (L,λ) to be the set of irreducible constituents of the virtual character RG L (λ), denoted by E(GF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Unipotent characters where parametrised by Broué, Malle and Michel [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] by us- ing e-Harish-Chandra theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Their description can be divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, each unipotent character lies in a unique e-Harish-Chandra series, that is, Uch (GF ) = ∐ (L,λ) E (GF ,(L,λ)) where (L,λ) runs over a set of representatives for the action of GF on the set of unipotent e- cuspidal pairs of (G,F) as explained in [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This is a well known fact and will be used throughout the paper without further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As a consequence of the partition above, it now remains to parametrise the unipotent e-Harish-Chandra series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If (L,λ) is a unipotent e-cuspidal pair, we denote by WG(L,λ)F ∶= NG(L)F λ /LF the corresponding relative Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (2)] parametrises the characters in an e-Harish-Chandra series in terms of the characters in the relative Weyl group by showing the existence of a bijection Irr(WG(L,λ)F ) → E(GF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) In Section 3 we reformulate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) in order to obtain Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Unipotent e-Harish-Chandra series are also used to parametrise the so-called unipotent blocks, that is, those blocks that contain unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This is the main result of [CE94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, if ℓ is odd and good for G, with ℓ ≠ 3 if 3D4 is an irreducible rational component of (G,F), then for every ℓ-block B of GF there exists a unipotent e-cuspidal pair (L,λ), with (L,λ) unique up to 8 GF -conjugation, such that all the irreducible constituents of RG L (λ) belongs to the block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, we write B = bGF (L,λ) and we also have Uch(GF ) ∩ Irr(bGF (L,λ)) = E(GF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (ii) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] imply that bl(λ)GF = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 Pseudo-unipotent characters We denote by (G∗,F ∗) a group in duality with (G,F) with respect to a choice of an F-stable maximal torus T of G and an F ∗-stable maximal torus T∗ of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If τ ∶ Gsc → [G,G] is a simply connected covering (see [GM20, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13]), then there exists an isomorphisms between the abelian groups Z(G∗)F ∗ → Irr(GF /τ(Gsc)) z ↦ ˆzG according to [CE04, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='19)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Notice that, if L is an F-stable Levi subgroup of G, then its dual L∗ is an F ∗-stable Levi subgroup of G∗ and we have Z(G∗)F ∗ ≤ Z(L∗)F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, every element z ∈ Z(G∗)F ∗ defines a linear characters of ˆzL and restriction of characters yields the equality (ˆzG)LF = ˆzL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In the next definition, we consider charactersthat are obtained by multiplying these linear characters with unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (K,F) be a finite reductive group and consider a Levi subgroup of L ≤ K and an irreducible character θ ∈ Irr(LF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We say that θ is (K,F)-pseudo-unipotent if there exists an element z ∈ Z(K∗)F ∗ such that θˆzL is unipotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every unipotent character λ ∈ Uch(LF ), we denote by psK(λ) the set of (K,F)-pseudo-unipotent characters of LF of the form λˆzL for some z ∈ Z(K∗)F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, we denote by psK(LF ) the set of all (K,F)-pseudo unipotent characters of LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' When the group K coincides with L, we denote the set of characters psL(LF ) simply by ps(LF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In accordance with the terminology introduced above, we say that an e-Harish-Chandra series of (K,F) is pseudo-unipotent if it is of the form E(KF ,(L,ν)) for some ν ∈ psK(λ) and where (L,λ) is a unipotent e-cuspidal pair of (K,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, we also say that (L,ν) is a pseudo- unipotent e-cuspidal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We define the union of all the series associated to characters in psK(λ) by E(KF ,(L,psK(λ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since RK L (λˆzL) = RK L (λ)ˆzK for every z ∈ Z(K∗)F ∗ by [CE04, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20)], we deduce that the elements of the pseudo-unipotent e-Harish-Chandra series E(KF ,(L,λˆz)) are exactly the irreducible characters of the form ϕˆzK for some unipotent character ϕ ∈ E(KF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, we point out that λ is the unique unipotent character in the set psK(λ) according to [CE04, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Similarly, the unipotent characters in the set E(KF ,(L,psK(λ))) are those in the series E(KF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Our next lemma, shows that blocks covering pseudo-unipotent characters are regular as defined in [Nav98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 9 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let L be an F-stable Levi subgroup of G and suppose that ℓ is odd and good for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every LF ≤ H ≤ NG(L)F and every character ϑ ∈ Irr(H) lying above some pseudo-unipotent character in ps(LF ), the block bl(ϑ) is LF-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, the Brauer induced block bl(ϑ)H is defined and is the unique block of H covering bl(ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ϕ ∈ Uch(LF ) and z ∈ Z(L∗)F ∗ such that ϕˆzL lies below the character ϑ and chose a unipotent e-cuspidal pair (M,µ) of L such that ϕ ∈ E(LF ,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, bl(ϕ) = bLF (M,µ) according to [CE94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If Q ∶= Z(M)F ℓ , then MF = CGF (Q) according to [CE94, Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, observe that [CE94, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] implies that bl(ϕ) = bLF (M,µ) = bl(µ)LF while [Riz18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] implies that bl(ϕ) and bl(ϕˆzL) have the same defect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, applying [Nav98, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13 and Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='26], we can find defect groups Dϑ, Dϕ and Dµ of bl(ϑ), bl(ϕ) and bl(µ) respectively with the property that Dµ ≤ Dϕ ≤ Dϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since Q ≤ Oℓ(MF ) ≤ Dµ by [Nav98, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8], we deduce that Q ≤ Dϑ and hence CH(Dϑ) ≤ CH(Q) = MF ≤ LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By [Nav98, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20] we conclude that the block bl(ϑ) is LF -regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The second part of the lemma now follows from [Nav98, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 3 Compatibility with isomorphisms of character triples The aim of this section is to show how the bijection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) can be made compatible with isomorphisms of character triples and with the action of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This property was first suggested by the author in [Ros22c, Parametrisation B] and further studied in [Ros22d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Our Theorem C gives a solution of this conjectured result for unipotent e-Harish-Chandra series and groups of type A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Before proceeding further, we show how the parametrisation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) can be reformulated in a more convenient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For this, let (L,λ) be a unipotent e-cuspidal pair of (G,F) and assume that ̂λ is an extension of λ to the stabiliser NG(L)F λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, by applying Gallagher’s theorem [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17] and the Clifford correspondence [Isa76, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11] we obtain a bijection Irr(WG(L,λ)F ) → Irr(NG(L)F ∣ λ) η ↦ (̂λη) NG(L)F and therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) holds if an only if there exists a bijection E(GF ,(L,λ)) → Irr(NG(L)F ∣ λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) This new reformulation will allow us to introduce the aforementioned compatibility with isomor- phisms of character triple isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Equivariance and maximal extendibility In this section, we consider some equivariance properties for the parametrisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) which are related to maximal extendibility (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1)) of unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As in the previous sections, consider a connected reductive group G with a Frobenius endomor- phism F ∶ G → G defining an Fq-structure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by AutF(GF ) the set of those auto- morphisms of GF obtained by restricting some bijective morphism of algebraic groups σ ∶ G → G that commutes with F to the set of Fq-rational points GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Notice that the restriction of such a 10 morphism σ to GF , which by abuse of notation we denote again by σ, is an automorphism of the finite group GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We refer the reader to [CS13, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, observe that any morphism σ with the properties above is determined by its restriction to GF up to a power of F and hence it follows that AutF(GF ) acts on the set of F-stable closed connected subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, given an F-stable closed connected subgroup H of G, we can define the set AutF(GF )H consisting of those automorphisms σ as above that stabilise the algebraic group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, let ℓ be a prime number not dividing q and denote by e the order of q modulo ℓ or q modulo 4 if ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In order to control the action of automorphism on unipotent e-Harish-Chandra series, we exploit a result of Cabanes and Späth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, in [CS13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4] it was shown that the parametrisation given by Broué, Malle and Michel in [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (2)] commutes with the action of those automorphisms in the set AutF(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Notice that the statement of [CS13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4] only considers unipotent e-cuspidal pairs (L,λ) where L is a minimal e-split Levi subgroups (which is enough for the purpose of dealing with the McKay Conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' However, their proof works for the general case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)- equivariant bijection IG (L,λ) ∶ Irr(WG(L,λ)F ) → E (GF ,(L,λ)) such that IG (L,λ)(η)(1)ℓ = ∣GF ∶ NG(L,λ)F ∣ℓ ⋅ λ(1)ℓ ⋅ η(1)ℓ for every η ∈ Irr(WG(L,λ)F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This follows from the proof of [CS13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' See also [Ros22d, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As explained at the beginning of this section, if the character λ extends to the stabiliser NG(L)F λ , then we can use the bijection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) to obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' A similar argument can be used to include the equivariance property described above and obtain an equivariant version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that, by the discussion on automorphisms above, it follows that the group AutF(GF ) acts on the set of e- cuspidal pairs (L,λ) and therefore we can define the stabiliser AutF(GF )(L,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, recall that we denote by d(χ) the ℓ-defect of an irreducible character χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an ex- tension λ◇ ∈ Irr(NG(L)F λ ) which is additionally AutF(GF )(L,λ)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then there exists an AutF(GF )(L,λ)-equivariant bijection ΩG (L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) such that d(χ) = d(ΩG (L,λ)(χ)) for every χ ∈ E(GF ,(L,λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider the bijection IG (L,λ) given by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 and define the map ΩG (L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) IG (L,λ)(η) ↦ (λ◇η)NG(L)F for every η ∈ Irr(WG(L,λ)F ) and where λ◇ is the extension of λ to NG(L)F λ given in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This is a well defined bijection by the Clifford correspondence [Isa76, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11] and Gallagher’s theorem [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every α ∈ AutF(GF ) such that (L,λ)α = (L,λ) and every η ∈ Irr(WG(L,λ)F ) we have ((λ◇η)NG(L)F ) α = ((λ◇η)α) NG(L)F = (λ◇ηα)NG(L)F because α stabilises λ◇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand IG (L,λ)(η)α = IG (L,λ) (ηα) by the properties of IG (L,λ) and hence we conclude that ΩG (L,λ) is AutF(GF )(L,λ)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Fur- thermore, if we consider η ∈ Irr(WG(L,λ)F ) and define the characters χ ∶= IG (L,λ)(η) and ψ ∶= (λ◇η)NG(L)F , then the degree formula from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 implies that ℓd(χ) = ∣GF ∣ℓ χ(1)ℓ = ∣NG(L,λ)F ∣ℓ λ(1)ℓ ⋅ η(1)ℓ = ∣NG(L)F ∣ℓ ψ(1)ℓ = ℓd(ψ) and hence we deduce that d(χ) = d(ψ) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we consider a regular embedding G ≤ ̃G as defined in [CE04, (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, ̃G is a connected reductive group with connected centre and whose derived subgroup coincides with that of G, that is, [̃G, ̃G] = [G,G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, observe that ̃G = Z(̃G)G, that G is normal in ̃G and that the quotient ̃G/G is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every Levi subgroup L of G, we deduce that ̃L ∶= Z(̃G)L is a Levi subgroup of ̃G and that L ≤ ̃L is again a regular embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These observations will be used throughout this paper without further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We also recall that, according to [DM91, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20], restriction of characters yields a bi- jection between the unipotent characters of ̃GF and those of GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, every unipotent character of GF is ̃GF -invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using this observation, we can compare the relative Weyl groups in ̃GF with those in GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L,λ)be a unipotent e-cuspidal pair of (G,F), set ̃L = LZ(̃G) and consider a unipo- tent extension ̃λ of λ to ̃LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, N ̃ G(L)F λ = N ̃ G(L)F ̃λ and we have W ̃ G(̃L,̃λ)F ≃ WG(L,λ)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since ̃λ extends λ, it is clear that the stabiliser N ̃ G(L)F ̃λ is contained in N ̃ G(L)F λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, let x ∈ N ̃ G(L)F λ and observe that ̃λx is a unipotent character of ̃LF that restricts to λx = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, [DM91, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20] implies that ̃λx = ̃λ and therefore x ∈ N ̃ G(L)F ̃λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' From this, we also conclude that N ̃ G(L)F ̃λ = ̃LFNG(L)F λ and therefore that W ̃ G(̃L,̃λ)F ≃ WG(L,λ)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 12 As a consequence of the lemma above, we show that when λ extends to its stabiliser NG(L)F λ , then every irreducible character of NG(L) that lies above λ is N ̃ G(L)F -invariant and extends to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an extension λ◇ ∈ Irr(NG(L)F λ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then every character of NG(L)F lying above λ extends to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To start, we fix a unipotent extension ̃λ of λ to ̃LF and recall that N ̃ G(L)F λ = N ̃ G(L)F ̃λ according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, applying [Spä10, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 (a)] we deduce that there exists an extension ̃λ◇ of λ◇ to N ̃ G(L)F λ that also extends ̃λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider now an irreducible character ψ of NG(L)F lying above λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Gallagher’s theorem [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17] and the Clifford correspon- dence [Isa76, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11], it follows that there exists an irreducible character η of the relative Weyl group WG(L,λ)F such that ψ is induced from the irreducible character ψ0 ∶= ηλ◇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More- over, by using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3, we have W ̃ G(̃L,̃λ)F ≃ WG(L,λ)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, η, viewed as a character of NG(L)F λ , admits an extension, say ̃η, to N ̃ G(L)F λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, define ̃ψ0 ∶= ̃η̃λ◇ and observe that ̃ψ0 lies above ̃λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By the Clifford correspondence, it follows that the character ̃ψ of N ̃ G(L)F induced from ̃ψ0 is irreducible and therefore, applying [Isa76, Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2], we conclude that ̃ψ extends ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We can now construct a parametrisation of unipotent e-Harish-Chandra series in the group ̃GF which agrees with the bijection ΩG (L,λ) from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 via restriction of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an extension λ◇ ∈ Irr(NG(L)F λ ) which is additionally AutF(GF )(L,λ)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If ̃λ is a unipotent extension of λ to ̃LF , then there exists a bijection ̃Ω ̃ G (̃L,̃λ) making the following diagram commute E (̃GF ,(̃L,̃λ)) Irr(N ̃ G(L)F ∣ ̃λ) E (GF ,(L,λ)) Irr(NG(L)F ∣ λ) ̃Ω ̃ G (̃L,̃λ) Res ̃ GF GF Res N ̃ G(L)F NG(L)F ΩG (L,λ) and where ΩG (L,λ) is the bijection given by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, observe that λ has an extension ̃λ to ̃LF according to [DM91, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More- over, restrictions from ̃GF to GF induces a bijection from the set E(̃GF ,(̃L,̃λ)) to E(GF ,(L,λ)) according to [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, consider a character ψ ∈ Irr(NG(L)F ) lying above λ and observe that ψ admits an extension ̃ψ0 ∈ Irr(N ̃ G(L)F ) by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ̃λ0 be an irre- ducible constituent of the restriction ̃ψ0,̃LF and notice that ̃ λ0 is an extension of λ since ̃LF /LF is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, Gallagher’s theorem [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17] implies that there exists a linear char- acter ν ∈ Irr(̃LF /LF) such that ̃λ0ν = ̃λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since N ̃ G(L)F /NG(L)F ≃ ̃LF/LF we can identify ν with its extension to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, it follows that the character ̃ψ ∶= ̃ψ0ν is an extension of ψ to N ̃ G(L)F lying above ̃λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then the assignment ψ ↦ ̃ψ defines a bijection between Irr(NG(L)F ∣ λ) 13 and Irr(N ̃ G(L)F ∣ ̃λ) whose inverse is given by restriction of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We can now define ̃Ω ̃ G (̃L,̃λ) (̃χ) ∶= ̃ψ for every ̃χ ∈ E(̃GF ,(̃L,̃λ)) and ̃ψ ∈ Irr(N ̃ G(L)F ∣ ̃λ) whenever ΩG (L,λ)(̃χGF ) = ̃ψNG(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 Construction of GF-block isomorphisms of character triples From now on, we assume that G is simple, simply connected and of type A, B or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, we suppose that ℓ is odd and denote by e the order of q modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We now give a more explicit construction of the group of automorphism AutF(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Fix a max- imally split torus T0 contained in an F-stable Borel subgroup B0 of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This choice corresponds to a set of graph automorphisms γ ∶ G → G and a field endomorphism F0 ∶ G → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More pre- cisely, if we consider the set of simple roots ∆ ⊆ Φ(G,T0) corresponding to the choice T0 ⊆ B0, then we have an automorphism γ ∶ G → G given by γ(xα(t)) ∶= xγ(α)(t) for every t ∈ Ga and α ∈ ±∆ and where γ is a symmetry of the Dynkin diagram of ∆, while F0(xα(t)) ∶= xα(tp) for every t ∈ Ga and α ∈ Φ(G,T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Here, we denote by xα ∶ Ga → G a one-parameter subgroup corresponding to α ∈ Φ(G,T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We define the subgroup A of AutF(GF ) generated by the graph and field automorphisms described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In addition, we choose our regular embedding G ≤ ̃G to be defined in such a way that the graph and field automorphisms extends to ̃G (see, for instance, [MS16, Section 2B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, the group A acts via automorphisms on ̃GF and we can form the external semidirect product ̃GF ⋊ A which acts on GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It turns out that ̃GF ⋊A and AutF(GF ) induce the same set of automorphisms on the finite group GF (see, for instance, [GLS98, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Throughout this section, we consider a fixed unipotent e-cuspidal pair (L,λ) of (G,F) and a unipo- tent extension ̃λ of λ to ̃LF (whose existence is ensured by [DM91, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20]) where, as always, we define ̃L ∶= LZ(̃G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In the next lemma, we show that the hypothesis of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 is satisfied under our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' There exists an extension λ◇ of λ to NG(L)F λ that is (̃GF A)(L,λ)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using [BS20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (i)], [Bro22b, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (a)] and the results of [Bro-Ruh], we obtain an extension λ◇ of λ to NG(L)F λ which is (GF A)(L,λ)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since (̃GF A)(L,λ) = ̃L(GF A)(L,λ) it suffices to show that λ◇ is ̃LF-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' However, the latter assertion follows immediately from the fact that λ◇ extends to N ̃ G(L)F λ according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 and [Spä10, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As an immediate consequence of the lemma above, we deduce that every character of NG(L)F lying above λ extends to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This can be considered as a local analogue of [DM91, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Every irreducible character of NG(L)F lying above λ extends to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 whose hypothesis is satisfied by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 14 We point out that, under our assumptions, every irreducible character of NG(L)F lying above λ extends to its stabiliser in N ̃ G(L)F because the quotient N ̃ G(L)F /NG(L)F is cyclic according to [GM20, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' However, in the lemma above we are also showing, using independent methods, that each such character is N ̃ G(L)F -invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6, we can now define bijections Ω ∶= ΩG (L,λ) and ̃Ω ̃ G (̃L,̃λ) as described in Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In what follows, we consider the sets of characters G ∶= E(GF ,(L,λ)), L ∶= Irr(NG(L)F ∣ λ), ̃G ∶= E(̃GF ,(̃L,̃λ)) and ̃L ∶= Irr(N ̃ G(L)F ∣ ̃λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Our next aim is to show that the parametrisation Ω is compatible with GF -block isomorphisms of charac- ter triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We start by checking the group theoretic properties required for the existence of such isomorphisms (see [Spä17, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7 (i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every χ ∈ G and ψ ∶= Ω(χ) ∈ L we have (̃GF A)L,χ = (̃GF A)L,ψ and ̃GFAχ = GF (̃GF A)L,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We argue as in the proof of [Ros22c, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To start, we observe that (̃GF A)(L,λ),χ = (̃GF A)(L,λ),ψ since the map Ω is (̃GF A)(L,λ)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Set U(χ) ∶= (̃GF A)L,χ and U(ψ) ∶= (̃GF A)L,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, consider x ∈ U(χ) and observe that according to [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)] there exists y ∈ NG(L)F such that (L,λ)xy = (L,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, xy ∈ (̃GF A)(L,λ),χ = (̃GF A)(L,λ),ψ and hence x ∈ U(ψ) since ψy = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This shows that U(χ) ≤ U(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, suppose that x ∈ U(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Clifford’s theorem there exists y ∈ NG(L)F such that λxy = λ and so xy ∈ (̃GF A)L,ψ = (̃GF A)L,χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since χy = χ we deduce that x ∈ U(χ) and hence U(χ) = U(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To conclude, it is enough to show that ̃GFAχ = GF U(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, notice that GF U(χ) ≤ ̃GF Aχ since χ is ̃GF -invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, for x ∈ ̃GF Aχ we know that (L,λ)x is GF -conjugate to (L,λ) thanks to [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, we obtain x ∈ GF U(χ) and as explained above this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8 to show that the map ̃Ω satisfies some useful equivariance properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Before doing so, we need to introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For this purpose, consider a pair (G∗,F ∗) dual to (G,F) and a pair (̃G∗,F ∗) dual to (̃G,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let i∗ ∶ ̃G∗ → G∗ be the surjection induced by duality from the inclusion G ≤ ̃G and observe that Ker(i∗) = Z(̃G∗) since G is simply connected (see [CE04, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As shown in [CE04, (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2)], there exists an isomorphism Ker(i∗)F → Irr(̃GF /GF) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) z ↦ ̂z̃ G Furthermore, if L is an F-stable Levi subgroup of G and z ∈ Ker(i∗), then we define ̂z̃L to be the restriction of ̂z̃ G to ̃LF and ̂zN ̃ G(L) to be the restriction of ̂z̃ G to N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We set K ∶= Ker(i∗) and obtain an action of the group K on the characters of ̃GF , ̃LF and N ̃ G(L)F as defined in [Ros22d, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, we consider the external semidirect product (̃GF A)⋉K given by defining zx as the unique element of K corresponding to the character (̂z̃ G)x of the quotient ̃GF /GF via the isomorphism specified in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2), whenever x ∈ ̃GF A and z ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, for every F-stable Levi subgroup L of G, we obtain an action of (̃GF A)L ⋉ K on the irreducible characters of ̃LF and N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by ((̃GF A)L ⋉ K)̃λ the stabiliser of ̃λ ∈ Irr(̃LF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, it follows that ((̃GF A)L ⋉ K)̃λ acts on the sets of characters ̃G and ̃L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we show that the bijection ̃Ω is compatible with this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The bijection ̃Ω is (N ̃ G(L)F (̃GF A)(L,λ) ⋊ K)̃λ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ̃χ ∈ ̃G and ̃ψ ∈ ̃L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By the definition of ̃Ω, we have ̃Ω(̃χ) = ̃ψ if and only if Ω(χ) = ψ where χ ∶= ̃χGF and ψ ∶= ̃ψNG(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, if we consider g ∈ N ̃ G(L)F , x ∈ (̃GF A)(L,λ) and z ∈ K such that (gx,z) stabilises ̃λ, then we obtain ̃Ω(̃χ(gx,z)) = ̃ψ(gx,z) if and only if Ω((̃χ(gx,z)) GF ) = (̃ψ(gx,z)) NG(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) However, since the restriction of ̃χ(gx,z) to GF coincides with χx and the restriction of ̃ψ(gx,z) to NG(L)F coincides with ψx, we deduce that the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) holds by the equivariant properties of Ω as described in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' One of the main ingredients for the construction of the projective representations needed to obtain GF -block isomorphisms of character triples is given by the following two lemmas on maximal extendibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Maximal extendibility holds for G with respect to the inclusion GF ⊴ GF A, that is, every character χ ∈ G extends to GF Aχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If G is of type B or C then the result follows from [Isa76, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='22] since A is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, we can assume that G is of type A in which case the result follows from [CS17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] (see also [Mal08, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The local version of the lemma above is a consequence of the results obtained in [BS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Maximal extendibility holds for L with respect to the inclusion NG(L)F ⊴ (GF A)L, that is, every character ψ ∈ L extends to (GF A)L,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10, it is enough to prove the result in the case where G is of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In fact, if G is of type B or C, then the quotient (GF A)(L,ψ)/NG(L)F is cyclic because it it a subquotient of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, if G is of type A the result follows from [BS20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, we can start constructing isomorphisms of character triples for the bijection Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As a first step, we obtain a weaker isomorphism, know as GF -central isomorphism of character triples and denoted by ∼c GF , whose requirements are given by [Spä17, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7 (i)-(iii)] and replacing the condition on defect groups by imposing that CG(N) ≤ H1 ∩ H2 with the notations used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We refer the reader to [Ros22b, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4] for a precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have (̃GFAχ,GF ,χ) ∼c GF ((̃GF A)L,ψ,NG(L)F ,ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We start by constructing projective representations associated with χ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' According to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 we can find a unipotent extension ̃χ ∈ ̃G of χ to ̃GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10 there exists an extension χ′ of χ to GF Aχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ̃Dglo be a representation of ̃GF affording ̃χ and D′ glo a representation of GF Aχ affording χ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, [Spä12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11] implies that Pglo ∶ (̃GF A)χ → GLχ(1)(C) defined by Pglo(x1x2) ∶= ̃ Dglo(x1)D′ glo(x2) for every x1 ∈ ̃GF and x2 ∈ GF Aχ is a projective representation associated with χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, observe that ̃ψ ∶= ̃Ω(̃χ) ∈ ̃L is an extension of ψ to N ̃ G(L)F and consider an extension ψ′ of ψ to (GF A)L,ψ given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ̃Dloc be a representation of N ̃ G(L)F affording ̃ψ and D′ loc a representation of (GF A)L,ψ affording ψ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Once again, [Spä12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11] shows that the map Ploc ∶ (̃GF A)L,ψ → GLψ(1)(C) given by Ploc(x1x2) ∶= ̃Dloc(x1)D′ loc(x2) for every x1 ∈ N ̃ G(L)F and x2 ∈ (GF A)L,ψ is a pro- jective representation associated with ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by αglo and αloc the factor set of Pglo and Ploc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As explained in the proof of [Ros22d, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3], in order to prove that αglo coincides with αloc via the isomorphism ̃GF Aχ/GF ≃ (̃GF A)L,ψ/NG(L)F , it suffices to show that (µglo x )N ̃ G(L)F = µloc x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4) for every x ∈ (GF A)L,χ and where µglo x ∈ Irr(̃GF /GF ) and µloc x ∈ Irr(N ̃ G(L)F /NG(L)F ) are determined by Gallagher’s theorem (see [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17]) via the equalities ̃χ = µglo x ̃χx and ̃ψ = µloc x ̃ψx respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Because (GF A)L,χ = NG(L)F (GF A)(L,λ),χ, we may assume that x stabilises λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let z ∈ K such that µglo x = ˆz̃ G and observe that (x,z) is an element of (GF A)(L,λ),χ ⋊ K that stabilises ̃χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, applying [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)], we deduce that ̃λ and ̃λ(x,z) are N ̃ G(L)F - conjugate and we may choose g ∈ N ̃ G(L)F such that ̃λ = (̃λ(x,z))g = ̃λ(xg,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In other words (xg,z) ∈ (N ̃ G(L)F (̃GF A)(L,λ) ⋊ K)̃λ and thus Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 implies that the equality ̃χ = ̃χ(xg,z) holds if and only if ̃ψ = ̃ψ(xg,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' From this, we immediately deduce the equality required in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, denote by ζglo and ζloc the scalar functions associated to Pglo and Ploc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To con- clude the proof, it remains to show that ζglo and ζloc coincide on C(̃ GF A)χ(GF ) = Z(̃GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' As in the proof of [Ros22d, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3], it is enough to show that the restrictions of ̃χ and ̃ψ to Z(̃GF ) are multiples of a common irreducible constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This follows from the fact that unipotent characters contain the center in their kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In fact, on one hand, 1Z(̃ GF ) is the unique irreducible constituent of ̃χZ(̃ GF ) because ̃χ is unipotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, ̃ψ lies above ̃λ and, since Z(̃GF ) ≤ Z(̃LF ) and ̃λ is unipotent, we deduce that 1Z(̃ GF ) is the unique irreducible constituent of ̃ψZ(̃ GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We conclude this section by verifying the remaining condition [Spä17, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7 (iv)] and obtain the required GF -block isomorphisms of character triples for the map Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have (̃GFAχ,GF ,χ) ∼GF ((̃GF A)L,ψ,NG(L)F ,ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='12 it is enough to check the block theoretic requirement given by [Spä17, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7 (ii) and (iv)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, observe that under our assumption [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (ii)] shows that LF = CGF (E) where E ∶= Z(L)F ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, NJ(L) = NJ(E) for every GF ≤ J ≤ ̃GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, for every block C0 of NJ(L) and every defect group D of C0 we have E ≤ Oℓ(NJ(L)) ≤ D and hence C ̃ GF (D) ≤ N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, [KS15, Theorem B] implies that for every block C of N ̃ G(L)F covering C0, the induced blocks B ∶= C ̃GF and B0 ∶= CJ 0 are well-defined and B covers B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let ̃χ ∈ ̃G be an extension of χ and set ̃ψ ∶= ̃Ω(̃χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 the block of ̃C of ̃ψ coincides with the induced block bl(̃λ)N ̃ G(L)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, by [CE94, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] we know that the block ̃B of ̃χ coincides with b ̃ GF (̃L,̃λ) = bl(̃λ)̃ GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, by the transitivity of block induction we get ̃ B = ̃C ̃ GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider now GF ≤ J ≤ ̃GF as in the previous paragraph and notice that bl(̃χJ) is the unique block of J covered by ̃B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, since bl(̃ψNJ (L)) is covered by ̃C, we deduce that bl(̃ψNJ (L))J is covered by ̃B and therefore bl(̃χJ) = bl( ̃ψNJ (L)) J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) As explained in the proof of [Ros22d, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8] we can now use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) together with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='12 to conclude the proof via an application of [Spä17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 (i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 Proof of Theorem C Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The hypothesis of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 is satisfied under our restrictions on G accord- ing to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 and therefore we obtain an AutF(GF )(L,λ)-equivariant bijection ΩG (L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ) that preserves the ℓ-defect of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, observe that the groups ̃GF A and X ∶= GF ⋊ AutF(GF ) induce the same automorphisms on GF according to the description given in [GLS98, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, by applying [Spä17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3] and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13, we conclude that (Xχ,GF ,χ) ∼GF (NX(L)ψ,NG(L)F ,ψ) for every χ ∈ E(GF ,(L,λ)) and where ψ ∶= ΩG (L,λ)(χ) and the proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 4 Consequences of Theorem C In this section, we collect some consequences of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, we extend the parametrisation obtained in Theorem C from unipotent e-Harish-Chandra series of the simple group G to pseudo- unipotent (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) e-Harish-Chandra series of the Levi subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, for every F-stable Levi subgroup K of G, we construct a parametrisation of the e-Harish-Chandra series associated to e-cuspidal pairs of the form (L,λ) for some (K,F)-pseudo-unipotent character λ ∈ psK(LF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In a second step, we construct character bijections above this parametrisation by ex- ploiting results on isomorphisms of character triples (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This will allow us to control the characters of e-chain stabilisers lying above pseudo-unipotent characters (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Parametrisation of pseudo-unipotent characters of Levi subgroups Let K be an F-stable Levi subgroup of G and set K0 ∶= [K,K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that since the group G is simply connected, the subgroup K0 is also simply connected according to [MT11, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In addition, under our assumption on the type of G, we deduce that the simple components of K0 can only be of some of the types A, B or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every unipotent e-cuspidal pair (L0,λ0) of (K0,F) there exists a defect pre- serving AutF(KF 0 )(L0,λ0)-equivariant bijection ΩK0 (L0,λ0) ∶ E (KF 0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0) such that (Yϑ,KF 0 ,ϑ) ∼KF 0 (NYϑ(L0),NK0(L0)F ,ΩK0 (L0,λ0)(ϑ)) for every ϑ ∈ E(KF 0 ,(L0,λ0)) and where Y ∶= KF 0 ⋊ AutF(KF 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Notice that K0 is the direct product of simple algebraic groups K1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' ,Kn and that the action of F permutes the simple components Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Denote the direct product of the simple components in each F-orbit by Hj for j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The (Hj,F) are the irreducible rational components of (K,F) and we have KF 0 = HF 1 × ⋯ × HF t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Similarly, if we define the intersections Mj ∶= L0 ∩ Hj, then we have a decomposition LF 0 = MF 1 × ⋯ × MF t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, we can write λ0 = µ1 × ⋯ × µt with µj ∈ Irr(MF j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, notice that (Mj,µj) is a unipotent e-cuspidal pair of (Hj,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, suppose that Hj = Hj,1 × ⋅⋅⋅ × Hj,mj and observe that HF j ≃ HF mj j,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By the discussion at the beginning of this section we know that Hj,1 is a simple, simply connected group of type A, B or C and hence it satisfies the assumptions of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, via the isomorphism HF j ≃ HF mj j,1 , we obtain an AutF(HF j )(Mj,µj)-equivarint bijection ΩHj (Mj,µj) ∶ E (HF j ,(Mj,µj)) → Irr(NHj(Mj)F ∣µj) that preserves the defect of characters and such that (Yj,ϑ,HF j ,ϑ) ∼HF j (NYj,ϑ(Mj),NHj(Mj)F ,ΩHj (Mj,µj)(ϑ)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) for every ϑ ∈ E(HF j ,(Mj,µj)) and where Yj ∶= HF j ⋊ AutF(HF j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since the characters in the sets E(KF 0 ,(L0,λ0)) and Irr(NK0(L0)F ∣ λ0) are direct products of characters belonging to the sets E(HF j ,(Mj,µj)) and Irr(NHj(Mj)F ∣ µj) respectively, we obtain a bijection ΩK0 (L0,λ0) ∶ E (KF 0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0) by setting ΩK0 (L0,λ0) (ϑ1 × ⋅⋅⋅ × ϑt) ∶= ΩH1 (M1,µ1)(ϑ1) × ⋅⋅⋅ × ΩHt (Mt,µt)(ϑt) for every ϑj ∈ E(HF j ,(Mj,µj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, arguing as in the proof of [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5], we de- duce that the bijection ΩK0 (L0,λ0) preserves the defect of characters, is AutF(KF 0 )(L0,λ0)-equivariant, and, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1), it induces the KF 0 -block isomorphisms of character triples required in the state- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 19 In our next result, we replace the automorphism group Y ∶= KF 0 ⋊ AutF(KF 0 ) with the group of automorphisms of GF stabilising K, that is, X ∶= (GF ⋊ AutF(GF ))K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To do so, we apply the so-called Butterfly theorem [Spä17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3] which basically states that, for any finite group G, the notion of G-block isomorphism of character triples only depends on the automorphisms induced on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L0,λ0) be a unipotent e-cuspidal pair of (K0,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The map ΩK0 (L0,λ0) given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 is AutF(GF )K,(L0,λ0)-equivariant and satisfies (Xϑ,KF 0 ,ϑ) ∼KF 0 (NXϑ(L0),NK0(L0)F ,ΩK0 (L0,λ0)(ϑ)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) for every ϑ ∈ E(KF 0 ,(L0,λ0)) and where X ∶= (GF ⋊ AutF(GF ))K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, observe that AutF(GF )K is contained in AutF(KF 0 ) because K0 is an F-stable char- acteristic subgroup of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, we deduce that the map ΩK0 (L0,λ0) is AutF(GF )K,(L0,λ0)- equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, to obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2), we apply [Spä17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3] to the isomorphism of character triples given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 as explained in the proof of [Ros22c, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Isomorphisms of character triples play a fundamental role in representation theory of finite groups and in the study of the local-global conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' One of the most important consequences of the existence of isomorphisms of character triples is the possibility to lift character bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For in- stance, the main result of [NS14], shows how to apply this technique to construct bijections above characters of height zero in the context of the Alperin–McKay Conjecture [NS14, Theorem B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The main consequence of this result, which follows from an argument introduced by Murai [Mur12], is a reduction theorem for the celebrated Brauer’s Height Zero Conjecture [NS14, Theorem A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This strategy ultimately lead to the solution of Brauer’s conjecture [Ruh22a] and [MNSFT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For other applications of isomorphisms of character triples see [Tur17], [NSV20], [Ros22a], [Ruh22b] [Ros23] and [MR22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In our next result, we exploit this idea in order to lift the bijections given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 to the Levi subgroup K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consequently, we extend the parametrisation of unipotent e-Harish-Chandra series given by Theorem C for the simple group G to a parametrisation of e-Harish-Chandra series associated to (K,F)-pseudo-unipotent characters for every F-stable Levi subgroup K of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, we need a preliminary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let (L,λ) be a unipotent e-cuspidal pair of (K,F) and define X ∶= (GF ⋊AutF(GF ))K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If KF ≤ H ≤ NG(L)F and Q is an ℓ-radical subgroup of NH(L), then CX(Q) ≤ NX(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let E ∶= Z(L)F ℓ and observe that L = C○ G(E) according to [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, since Oℓ(NH(L)) is the smallest ℓ-radical subgroup of NH(L) [Dad92, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4], we deduce that E ≤ Oℓ(NH(L)) ≤ Q and it follows that CX(Q) ≤ CX(E) ≤ NX(L) as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 20 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every unipotent e-cuspidal pair (L,λ) of (K,F) there exists a defect preserving AutF(GF )K,(L,λ)-equivariant bijection ΩK (L,λ) ∶ E (KF ,(L,psK(λ))) → Irr(NK(L)F ∣ psK(λ)) such that (Xχ,KF ,χ) ∼KF (NXχ(L),NK(L)F ,ΩK (L,λ)(χ)) for every χ ∈ E(KF ,(L,psK(λ))) and where X ∶= (GF ⋊ AutF(GF ))K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Recall that K0 = [K,K] and define L0 ∶= L ∩ K0 and λ0 the restriction of λ to LF 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that (L0,λ0) is a unipotent e-cuspidal pair of (K0,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let z ∈ Z(K∗)F ∗ and consider a character χ belonging to E(KF ,(L,λˆzL)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since the restriction of λˆzL to LF 0 coincides with λ0, [GM20, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='25] implies that χ lies above some character in E(KF 0 (L0,λ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, suppose that χ ∈ Irr(KF ) lies above some χ0 ∈ E(KF 0 ,(L0,λ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] the character χ0 has an extension χ′ ∈ E(KF ,(L,λ)) and hence, using Gallagher’s theorem [Isa76, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='17] and [CE04, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='19)], we can find z ∈ Z(K∗)F ∗ such that χ = χ′ˆzK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since χ′ˆzK is a character of E(KF ,(L,λˆzL)) according to [CE04, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='20)], we conclude that E (KF,(L,psK(λ))) = Irr(KF ∣ E (KF 0 ,(L0,λ0))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) Next, suppose that ψ ∈ Irr(NK(L)F ∣ λˆzL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, ψ lies above the restriction of λˆzL to LF 0 which coincides with λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, there exists some ϕ ∈ Irr(NK0(L0)F ∣ λ0) such that ψ lies above ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other, if χ lies above such a character ϕ ∈ Irr(NK0(L0)F ∣ λ0), then it lies above λ0 and therefore we can find z ∈ Z(K∗)F ∗ such that ψ ∈ Irr(NK(L)F ∣ λˆzL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This shows that Irr(NK(L)F ∣ psK(λ)) = Irr(NK(L)F ∣ Irr(NK0(L0)F ∣ λ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4) Finally, consider the map ΩK0 (L0,λ0) given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, the result follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4) by applying [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] as explained in the proof of [Ros22c, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10] and using the KF -block isomorphisms of character triples obtained in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Here, we consider A ∶= GF ⋊ AutF(GF ), A0 ∶= NA(L), K ∶= KF 0 , K0 = NK0(L)F = NK0(L0)F , G ∶= GF , X ∶= (GF ⋊ AutF(GF ))K, S ∶= E(KF 0 ,(L0,λ0)), S0 ∶= Irr(NK0(L0)F ∣ λ0), V ∶= (GF ⋊ AutF(GF ))K,S and U ∶= (GF ⋊ AutF(GF ))K,L,Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that the condition on defect groups required by [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] is satisfied by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 Above e-Harish-Chandra series We now further extend Theorem C by lifting the character bijections from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 with respect to normal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider the setup of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 and let KF ≤ H ≤ NG(K)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, there exists a defect preserving AutF(GF )H,K,(L,λ)-equivariant bijection ΩK,H (L,λ) ∶ Irr(H ∣E (KF,(L,psK(λ)))) → Irr(NH(L)∣psK(λ)) such that (NX(H)χ,H,χ) ∼H (NX(H,L)χ,NH(L),ψ) for every χ ∈ Irr(H ∣ E(KF ,(L,psK(λ)))) and where X ∶= (GF ⋊ AutF(GF ))K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We apply [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] to the bijection given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We consider A ∶= GF ⋊ AutF(GF ), G ∶= GF, K ∶= KF , A0 ∶= NA(L), X ∶= NA(K), S ∶= E(KF ,(L,psK(λ))), S0 ∶= Irr(NK(L)F ∣ psK(λ)), U ∶= X0,λ, V ∶= XS and J ∶= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Notice that the conditions (i)-(iii) of [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1] are satisfied by [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, the requirements about defect groups are satisfied by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, as explained in [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11], we obtain the claimed result by applying [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Before proceeding further, we point out an interesting analogy with another important character correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The Glauberman correspondence plays a fundamental role in the study of the local- global counting conjectures and lies at the heart of most reduction theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In its most basic form, it states that for every finite ℓ-group L acting on a finite ℓ′-group K, there exists a bijection fL ∶ IrrL(K) → Irr(NK(L)) between the set of L-invariant characters of K and the characters of the normaliser NK(L) (see, for instance, [Nav18, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' A very deep result due to Dade [Dad80] and recently reproved by Turull [Tur08], shows that, if K and L are subgroups of a finite group G and KP ≤ H ≤ KNG(L), then the Glauberman correspondence fL can be lifted to a character correspondence for H, that is, there exists a bijection f H L ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ fL(χ)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) for every χ ∈ IrrL(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, the parametrisation of unipotent e-Harish-Chandra series obtained by Broué, Malle and Michel [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] lies at the centre of the proofs of the local-global counting conjectures for finite reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It is interesting to note that our methods yield a character bijection above e-Harish-Chandra series which is analogous to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) in the context of the Glauberman correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This is an immediate consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider the setup of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 and let KF ≤ H ≤ NG(K)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, there exists a bijection ΨH χ ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ ΩK (L,λ)(χ)) for every χ ∈ E(KF ,(L,psK(λ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This follows immediately from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 by following the construction made in [Ros22c, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 5 Towards Theorem A and Theorem B Finally, we apply the results obtained in the previous sections to prove Theorem A which is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, we obtain Theorem B as a corollary by applying the e-Harish-Chandra theory for unipotent characters developed by Broué, Malle and Michel [BMM93] and by Cabanes and En- guehard [CE94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Before doing so, we introduce the relevant notation and prove some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 Preliminaries on e-chains Our first aim is to define e-local structures for finite reductive groups that play a role analogue to that of ℓ-chains in the context of Dade’s Conjecture and the Character Triple Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The connection between the set of e-chains and that of ℓ-chains has already been studied in [Ros22c, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' These results provide a way to obtain Dade’s Conjecture and the Character Triple Conjecture as a consequence of [Ros22c, Conjecture C and Conjecture D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The possibility to use different types of chains is crucial in the study of Dade’s Conjecture and has been introduced by Knörr and Robinson [KR89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Their results were insipred by previous studies conducted by many authors including Brown [Bro75] and Quillen [Qui78] who analised the homotopy theory of associated simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by Le(G,F) the set of e-chains of the finite reductive group (G,F), that is, chains of the form σ = {G = L0 > L1 > ⋅⋅⋅ > Ln} where n is a non-negative integer and each Li is an e-split Levi subgroup of (G,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by ∣σ∣ ∶= n the length of the e-chain σ and by L(σ) its last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, we define Le(G,F)>0 to be the set of e-chains having length strictly larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that the notion of length defined above, induces a partition of the set Le(G,F) into e- chains of even and odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, we denote by Le(G,F)± the subset of those e-chains σ ∈ Le(G,F) that satisfy (−1)∣σ∣ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In what follows, given an e-chain σ and an e-split Levi subgroup M of (L(σ),F), we denote by σ+M the e-chain obtainedby adding M at the end of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We also allow the possibility that M = L(σ), in which case we have σ + L(σ) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Vice versa, we denote by σ − L(σ) the e-chain obtained by removing the last term L(σ) from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this way we obtain (σ + M) − L(σ + M) = σ where as usual L(σ + M) denotes the final term of the e-chain σ + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Here, we use the convention that σ0 − L(σ0) = σ0 = σ0 + G where σ0 = {G} is the trivial e-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, consider the action of GF on the set of e-chains Le(G,F) induced by conjugation: for every g ∈ GF and σ = {Li}i, we define σg ∶= {G = L0 > Lg 1 > ⋅⋅⋅ > Lg n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It follows from this definition that the stabiliser GF σ coincides with the intersection of the normalis- ers NG(Li)F for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Similarly, we can define an action of AutF(GF ) on Le(G,F) and give an analogous description of the chains stabilisers AutF(GF )σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In particular, notice that the last term of the chain satisfies L(σ)F ⊴ GF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using this observation, we can use the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 to control the characters of GF σ that lie above pseudo-unipotent series of L(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every e-chain σ ∈ Le(G,F) we denote by CPu(σ) the set of unipotent e- cuspidal pairs (M,µ) ∈ CPu(L(σ),F) that satisfy M < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, for any such pair (M,µ) ∈ CPu(σ), we define the character set Uch(GF σ ,(M,µ)) ∶= ⎧⎪⎪⎨⎪⎪⎩ Irr(GF σ ∣ E (L(σ)F ,(M,psL(σ)(µ)))) L(σ) > M (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) Irr(GF σ ∣ E (L(σ)F ,(M,psL(σ−L(σ))(µ)))) L(σ) = M (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) 23 The need to distinguish the cases (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) will become apparent in the proofs of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that in the definition above, we are excluding the degenerate case where G = L(σ) = M and therefore the chain σ − L(σ) in the case (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) is always defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To understand the reason why we are excluding this case, we can consider an analogy with Dade’s Con- jecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every finite group G, recall that k(G) denotes the number of its irreducible characters and that, for any non-negative integer d, the symbol kd(G) denotes the number of those irreducible characters of ℓ-defect d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The local-global counting conjectures provide a way to determine the global invariants kd(G) in terms of ℓ-local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This idea was made precise by Isaacs and Navarro [IN20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' According to their definitions, the block-free version of Dade’s Conjecture can be stated by saying that the functions kd are chain local for every d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consequently, and because a sum of chain local functions is chain local, we deduce that the difference k − k0 = ∑d>0 kd is a chain local function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' On the other hand, using the fact that groups admitting a character of ℓ-defect zero have trivial ℓ-core, it is easy to see that k0 is not chain local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The exclusion of the case G = L(σ) = M can be explained by interpreting these observations in the context of unipotent characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Recall that ku(GF ) and kc,u(GF ) denote the number of unipotent characters of GF and unipotent e-cuspidal characters of GF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If ℓ does not divide the order of Z(GF ), then [CE94] implies that the unipotent e-cuspidal characters of GF have defect zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, as in the case of Dade’s Con- jecture, the global invariant we want to determine e-locally is the difference ku(GF ) − kc,u(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, notice that kc,u(GF ) is exactly the number of unipotent e-cuspidal pairs (M,µ) of L(σ) satisfying G = L(σ) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In the following lemma, we show that if the set Uch(GF σ ,(M,µ)) is non-empty then (M,µ) is uniquely defined up to GF σ -conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let σ ∈ Le(G,F) and consider two unipotent e-cuspidal pairs (M,µ) and (K,κ) in CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If the sets Uch(GF σ ,(M,µ)) and Uch(GF σ ,(K,κ)) have non-trivial intersection, then (M,µ) and (K,κ) are GF σ -conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Suppose that ϑ is a character belonging to Uch(GF σ ,(M,µ)) and Uch(GF σ ,(K,κ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If we set L ∶= L(σ), then we can find elements s,t ∈ Z(L∗)F ∗ and characters ϕ ∈ E(LF ,(M,µ)) and ψ ∈ E(LF ,(K,κ)) such that ϑ lies above ϕˆsL and ψˆtL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Clifford’s theorem, we deduce that ϕˆsL = (ψˆtL)g for some g ∈ GF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, since ˆs is a linear character, we obtain that ϕ = ψg(ˆtL)g(ˆsL)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since both ϕ and ψg are unipotent characters of LF , using [CE04, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='26] we deduce that (ˆtL)g(ˆsL)−1 = 1L and therefore ϕ = ψg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' But then, [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2(1)] shows that (M,µ) and (K,κ)g are LF-conjugate and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we describe the block theory associated to characters in the sets introduced in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let σ ∈ Le(G,F) and consider a unipotent e-cuspidal pair (M,µ) ∈ CPu(σ) and a character ϑ ∈ Uch(GF σ ,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then: (i) the block bl(ϑ) is L(σ)F -regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (ii) if the character ϑ lies above a given ϕˆzL(σ) ∈ E(L(σ)F ,(M,µˆzM)) for some z ∈ Z(L(σ)∗)F ∗, then we have bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F and bl(ϑ) = bl(ϕˆzL(σ))GF σ = bl(µˆzM)GF σ 24 (iii) the induced block bl(ϑ)GF is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The first point follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 by choosing L = L(σ) and H = GF σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, in the case of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) observe that L(σ) ≤ L(σ − L(σ)) and hence Z(L(σ − L(σ))∗) ≤ Z(L(σ)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, we can always find ϕ and z as in the statement of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Since ϕ is an irreducible constituent of the virtual character RL(σ) M (µ), it follows from [CE94, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2] (whose assumptions are satisfied by [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 (ii)]) that bl(ϕ) = bL(σ)F (M,µ) = bl(µ)L(σ)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, since ˆzM is the restriction of the linear character ˆzL(σ) to MF , we deduce from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1 that bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, [Nav98, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='19] implies that bl(ϑ) = bl(ϕˆzL(σ)) GF σ and the second point follows by the transitivity of block induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, set Q ∶= Z(M)F ℓ and observe that QCGF (Q) = MF ≤ NGF (Q) by [CE94, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3(ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, [Nav98, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='14] implies that bl(µˆzM)GF is well defined and so is bl(ϑ)GF by (ii) and transitivity of block induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using the lemma above, we can now define the following character set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This yields the e-local object through which we can determine the number of unipotent characters in a given block of B of GF and with a given defect d ≥ 0 (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let B be a block of GF and d a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every e-chain σ ∈ Le(G,F) and unipotent e-cuspidal pair (L,λ) ∈ CPu(σ) we define the character set Uchd (Bσ,(M,µ)) ∶= {ϑ ∈ Uch (GF σ ,(M,µ)) ∣ d(ϑ) = d,bl(ϑ)GF = B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' where bl(ϑ)GF is defined according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, we denote the cardinality of this set by kd u (Bσ,(M,µ)) ∶= ∣Uchd (Bσ,(M,µ))∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To conclude this section, we show that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 can be used to parametrise the character sets from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let B be a block of G and d a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' If σ ∈ Le(G,F) and (M,µ) is a unipotent e-cuspidal pair in CPu(σ) then there exists an AutF(GF )B,σ,(M,µ)-equivariant bijection ΩB,d σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bσ+M,(M,µ)) such that (Xσ,ϑ,GF σ ,ϑ) ∼GFσ (Xσ+M,ϑ,GF σ+M,ΩB,d σ,(M,µ)(ϑ)) for every ϑ ∈ Uchd(Bσ,(M,µ)) and where X ∶= GF ⋊ AutF(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, observe that if M coincides with the last term L(σ) of the chain σ, then we have σ + M = σ which implies Uchd(Bσ,(M,µ)) = Uchd(Bσ+M,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case the result holds by defining ΩB,d σ,(M,µ) as the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, we can assume that M < L(σ) and define ρ ∶= σ+M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='1) we have Uch (GF σ ,(M,µ)) = Irr(GF σ ∣ E (L(σ)F ,(M,psL(σ)(µ)))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) On the other hand, noticing that M coincides with the last term L(ρ) of the chain ρ and that ρ−L(ρ) = σ, we obtain the equality E(L(ρ)F ,(M,psL(ρ−L(ρ))(µ))) = psL(σ)(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, observing that GF ρ = NGFσ (M), we can apply (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2) to obtain the equality Uch (GF ρ ,(M,µ)) = Irr(NGFσ (M) ∣ psL(σ)(µ))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4) Next, we apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 by choosing the groups in that statement to be H = GF σ , K = L(σ) and (L,λ) = (M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='4), we deduce that there exists an AutF(GF )σ,(M,µ)- equivariant bijection ΩL(σ),GF σ (M,µ) ∶ Uch(GF σ ,(M,µ)) → Uch(GF ρ ,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) Moreover, using the H-block isomorphisms given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5 together with [Spä17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8 (b)], we deduce that (Xσ,ϑ,GF σ ,ϑ) ∼GFσ (Xρ,ϑ,GF ρ ,ΩL(σ),GF σ (M,µ) (ϑ)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6) for every ϑ ∈ Uchd(GF σ ,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To conclude, observe first that ΩL(σ),GF σ (M,µ) sends characters of defect d to characters of defect d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, by the transitivity of block induction and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6), we deduce that bl(ϑ)GF = bl(ΩL(σ),GF σ (M,µ) (ϑ)) GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This shows that the bijection from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5) sends characters in the set Uchd(Bσ,(M,µ)) to charac- ters in the set Uchd(Bσ+M,(M,µ)) and therefore it restricts to a bijection, denoted by ΩB,d σ,(M,µ), satisfying the properties required in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We conclude this section with a remark on the isomorphisms of character triples obtained in Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Suppose that ℓ does not divide q ± 1 if G is of type A(±q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, every e-split Levi subgroup L of G satisfies L = C○ G(Z(L)F ℓ ) according to [CE04, Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' This fact can be used to show that the GF σ -block isomorphisms of character triples given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 can be extended to GF -block isomorphisms of character triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' First, we claim that CGF Xσ,ϑ(D) ≤ Xσ,ϑ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7) for every irreducible character ϑ of GF σ and every ℓ-radical subgroup D of GF σ+M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Define Qi ∶= Z○(Li)F ℓ for every e-split Levi subgroup Li appearing in the chain σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, using the fact that D is 26 ℓ-radical, we obtain the inclusions Qi ≤ Oℓ(GF σ ) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, every element x ∈ GF Xσ,ϑ that centralises D centralises also each Qi and hence normalises each Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' It follows that CGF Xσ,ϑ(D) ≤ (GF Xσ,ϑ)σ = Xσ,ϑ as required by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We can now apply [Ros22a, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11] to the GF σ -block isomorphisms given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 to show that (Xσ,ϑ,GF σ ,ϑ) ∼GF (Xσ+M,ϑ,GF σ+M,ΩB,d σ,(M,µ)(ϑ)) for every ϑ ∈ Uchd(Bσ,(M,µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 Proof of Theorem A We are finally ready to prove our main theorem which provides a bijection for unipotent characters in the spirit of the Character Triple Conjecture [Spä17, Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this section, we prove a slightly stronger result that provides further information on the type of e-chains and isomorphisms of character triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In the following definition we introduce the analogue of the set Cd(B)± con- sidered in the Character Triple Conjecture as defined in [Spä17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 1097].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Let B be a block of GF and consider a non-negative integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We define the set Ld u(B)± = {(σ,M,µ,ϑ) ∣ σ ∈ Le(G,F)±,(M,µ) ∈ CPu(σ),ϑ ∈ Uchd (Bσ,(M,µ))} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The conjugacy action of GF induces an action of GF on Ld u(B)± defined by (σ,M,µ,ϑ)g ∶= (σg,Mg,µg,ϑg) for every element g ∈ GF and (σ,M,µ,ϑ) ∈ Ld u(B)±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We denote by Ld u(B)±/GF the corresponding set of GF -orbits of tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every such orbit ω, we denote by ω● the corresponding GF -orbit of pairs (σ,ϑ) such that (σ,M,µ,ϑ) ∈ ω for some (M,µ) ∈ CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In other words, if we indicate by (σ,M,µ,ϑ) the GF-orbit of (σ,M,µ,ϑ), then (σ,M,µ,ϑ) is the GF -orbit of the pairs (σg,ϑg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In a similar way, if AutF(GF )B denotes the set of those automorphisms α ∈ AutF(GF ) that sta- bilise B, then we can define (σ,M,µ,ϑ)α ∶= (σα,Mα,µα,ϑα) for every α ∈ AutF(GF )B and (σ,M,µ,ϑ) ∈ Ld u(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this way, we obtain an action of the group AutF(GF )B on the set Ld u(B)± and on the corresponding set of orbits Ld u(B)±/GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every block B of GF and every non-negative integer d, there exists an AutF(GF )B- equivariant bijection Λ ∶ Ld u(B)+/GF → Ld u(B)−/GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, for every ω ∈ Ld u(B)+/GF, any (σ,ϑ) ∈ ω● and any (ρ,χ) ∈ Λ(ω)● we have ∣σ∣ = ∣ρ∣ ± 1 and (Xσ,ϑ,GF σ ,ϑ) ∼J (Xρ,χ,GF ρ ,χ) with J = GF σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF ρ , if ∣σ∣ = ∣ρ∣ + 1, and where X ∶= GF ⋊ AutF(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Define A ∶= AutF(GF ) and observe that X = GF ⋊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In a first step, we construct an equivariant bijection between triples of the form (σ,M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' More precisely, let S denote the set of such triples (σ,M,µ) with σ ∈ Le(G,F) and (M,µ) ∈ CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' We define a map ∆ ∶ S → S by setting ∆((σ,M,µ)) ∶= ⎧⎪⎪⎨⎪⎪⎩ (σ + M,M,µ) , L(σ) > M (σ − M,M,µ) , L(σ) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Observe that the chain σ−M is always defined since M < G by the definition of CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, it is clear from the definition above that the map ∆ is A-equivariant and satisfies ∆2 = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Therefore, observing that ∣σ ± M∣ = ∣σ∣ ± 1, we conclude that ∆ restricts to an A-equivariant bijection ∆ ∶ S+ → S− where S± denotes the set of those triples (σ,M,µ) of S that satisfy σ ∈ Le(G,F)±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Furthermore, notice once again that if ∆((σ,M,µ)) = (ρ,K,κ), then ∣σ∣ = ∣ρ∣ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8) Now, fix an AB-transversal T+ in S+ and observe that the image of T+ under the map ∆, denoted by T−, is an AB-transversal in S because of the equivariance property of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consider (σ,M,µ) ∈ T+ and write ∆((σ,M,µ)) = (ρ,M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In what follows, we may assume without loss of generality that L(σ) > M and that ρ = σ + M, otherwise we repeat the arguments verbatim by replacing (σ,M,µ) with (ρ,M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='6 we obtain an AB,σ,(M,µ)-equivariant bijection ΩB,d σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bρ,(M,µ)) such that (Xσ,ϑ,GF σ ,ϑ) ∼GFσ (Xρ,χ,GF ρ ,χ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9) for every ϑ ∈ Uchd(Bσ,(M,µ)) and where χ is the image of ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consequently, if U(σ,M,µ) + is an AB,(σ,M,µ)-transversal in the character set Uchd(Bσ,(M,µ)), then its image, denoted by U(ρ,M,µ) − , under the bijection above is an AB,(ρ,M,µ)-transversal in the character set Uchd(Bρ,(M,µ)) be- cause AB,(σ,M,µ) = AB,(ρ,M,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, by the discussion in the previous paragraph and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3, we conclude that the sets of GF -orbits L+ ∶= {(σ,M,µ,ϑ) ∣ (σ,M,µ) ∈ T+,ϑ ∈ U(σ,M,µ) + } and L− ∶= {(ρ,M,µ,χ) ∣ (ρ,M,µ) ∈ T−,χ ∈ U(ρ,M,µ) − } are AB-transversals in the sets Ld u(B)+/GF and Ld u(B)−/GF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Finally, we can define the bijection Λ by setting Λ((σ,M,µ,ϑ) x) ∶= (ρ,M,µ,χ)x 28 for every x ∈ AB and every (σ,M,µ,ϑ) ∈ L+ and (ρ,M,µ,χ) ∈ L− satisfying ∆(σ,M,µ) = (ρ,M,µ) and such that χ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ΩB,d σ,(M,µ)(ϑ), ρ = σ + M (ΩB,d ρ,(M,µ)) −1 (ϑ), ρ = σ − M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9) together with the definition of Λ, we conclude that the properties required in the statement are satisfied and the proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Now, as a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7, we can finally prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Assume that ℓ does not divide q ± 1 whenever (G,F) is of type A(±q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Con- sider the bijection Λ from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 and chose ω ∈ Ld u(B)+/GF, (σ,ϑ) ∈ ω● and (ρ,χ) ∈ Λ(ω)●.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Then, we have (Xσ,ϑ,GF σ ,ϑ) ∼J (Xρ,χ,GF ρ ,χ) with J = GF σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF ρ , if ∣σ∣ = ∣ρ∣ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In either cases, applying Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='7, we deduce that (Xσ,ϑ,GF σ ,ϑ) ∼GF (Xρ,χ,GF ρ ,χ) as required by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3 Proof of Theorem B Our final goal is to obtain a counting argument for unipotent characters as a consequence of The- orem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Recall that Dade’s Conjecture provides a way to determine the number of characters in a given ℓ-block B and with a given defect d in terms of ℓ-local structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Theorem B provides an adaptation of this idea to the unipotent characters of finite reductive groups by means of e-local structures compatible with e-Harish-Chandra theory (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' For every σ ∈ Le(G,F) we define kd u(Bσ) ∶= ∑ (M,µ) kd u(Bσ,(M,µ)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='10) where (M,µ) runs over a set of representatives for the action of GF σ on CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Moreover, recall that kd u(B) and kd c,u(B) denote the number of irreducible characters belonging to the block B and with defect d that are unipotent and unipotent e-cuspidal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' To start, we determine the cardinality of the sets of GF-orbits Ld u(B)±/GF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' By applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='3, we obtain ∣Ld u(B)±/GF ∣ = ∑ σ,(M,µ) kd u(Bσ,(M,µ)) = ∑ σ kd u(Bσ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11) where σ runs over a set of representatives, say L±, for the action of GF on Le(G,F)± and (M,µ) runs over a set of representatives for the action of GF σ on CPu(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, we isolate the contribution given by the trivial chain σ0 ∶= {G} ∈ Le(G,F)+ to the sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In this case, we have L(σ0) = G and hence psL(σ)(µ) = {µ} for every (M,µ) ∈ CPu(σ0) because the center Z(G∗)F ∗ 29 is trivial under our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Consequently, using Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 and Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='5, we deduce that kd u(Bσ0) = ∑ (M,µ) kd u(Bσ0,(M,µ)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='12) = ∑ (M,µ) ∣Irrd(B) ∩ E(GF ,(M,µ))∣ = kd u(B) − kd c,u(B) where the last equality holds by [BMM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='2 (1)] and recalling that every pair (M,µ) ∈ CPu(σ0) satisfies M < G = L(σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Next, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='9 implies that the sets Ld u(B)+/GF and Ld u(B)−/GF have the same cardinality and therefore we conclude from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='12) that kd u(B) − kd c,u(B) + ∑ σ∈L+ σ≠σ0 kd u(Bσ) = ∑ σ∈L+ kd u(Bσ) = ∑ σ∈L− kd u(Bσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13) Finally, noticing that (−1)∣σ∣+1 = ∓1 for every σ ∈ L±, we can rewrite (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='13) as kd u(B) − kd c,u(B) = ∑ σ∈L−∪L+ (−1)∣σ∣+1kd u(Bσ) which is exactly the equality in the statement of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' References [Alp76] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Alperin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' The main problem of block theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In Proceedings of the Conference on Finite Groups (Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Utah, Park City, Utah, 1975), pages 341–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Academic Press, New York, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [Alp87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Alperin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Weights for finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In The Arcata Conference on Representations of Finite Groups (Arcata, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=', 1986), volume 47 of Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=', pages 369–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=', Providence, RI, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [BM11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Bonnafé and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Michel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Computational proof of the Mackey formula for q > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Algebra, 327:506–526, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [Bra56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Brauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Number theoretical investigations on groups of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' In Proceedings of the international symposium on algebraic number theory, Tokyo and Nikko, 1955, pages 55–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Science Council of Japan, Tokyo, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [Bro22a] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Broué.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Gunter is sixty something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Presented at the workshop Counting conjectures and beyond of the Isaac Newton Institute, Cambridge, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [BM92] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Broué and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Malle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=', 292(2):241–262, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' [BMM93] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Broué, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' Algebra, 474:424–465, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' DIPARTIMENTO DI MATEMATICA E INFORMATICA U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content=' DINI, VIALE MORGAGNI 67/A, FIRENZE, ITALY Email address: damiano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='rossi00@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} +page_content='com 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtE4T4oBgHgl3EQfkg1D/content/2301.05151v1.pdf'} diff --git a/RdE3T4oBgHgl3EQfDQlE/content/2301.04284v1.pdf 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intent +detection and slot filling. To improve the per- +formance of these two sub-tasks, we propose +to use consistency regularization based on a +hybrid data augmentation strategy. The consis- +tency regularization enforces the predicted dis- +tributions for an example and its semantically +equivalent augmentation to be consistent. We +conduct experiments on the MASSIVE dataset +under both full-dataset and zero-shot settings. +Experimental results demonstrate that our pro- +posed method improves the performance on +both intent detection and slot filling tasks. Our +system1 ranked 1st in the MMNLU-22 compe- +tition under the full-dataset setting. +1 +Introduction +The MMNLU-22 evaluation focuses on the prob- +lem of multilingual natural language understanding. +It is based on the MASSIVE dataset (FitzGerald +et al., 2022), a multilingual spoken language under- +standing (SLU) dataset with two sub-tasks, includ- +ing intent detection and slot filling. Specifically, +given a virtual assistant utterance in an arbitrary +language, the model is designed to predict the cor- +responding intent label and extract the slot results. +An English example is illustrated in Figure 1. +Fine-tuning pre-trained cross-lingual language +models allows task-specific supervision to be +shared and transferred across languages (Conneau +and Lample, 2019; Conneau et al., 2020; Xue et al., +2021). +This motivates the two setting for the +MMNLU-22 evaluation, namely the full-dataset +setting and the zero-shot setting. Participants are +allowed to use training data in all languages under +the full-dataset setting, while they can only access +the English training data under the zero-shot setting. +∗Email corresponding. +1The code will be available at https://github.com/ +bozheng-hit/MMNLU-22-HIT-SCIR. +Utterance +Slot +Intent +Wake me +up +at +five +am +Friday +this week +O +O +O +O +time time +date +date +date +set alarm +Figure 1: An English example from the MASSIVE +dataset. The slot label ‘O’ stands for the ‘Other’ label. +The latter is also called zero-shot cross-lingual SLU +in previous work (Qin et al., 2020, 2022). +Cross-lingual data augmentation methods have +been proven effective to improve cross-lingual +transferability, e.g., code-switch substitution (Qin +et al., 2020) and machine translation (Conneau and +Lample, 2019; Singh et al., 2019). Most previ- +ous work directly utilizes the data augmentations +as additional training data for fine-tuning. How- +ever, they ignore the inherent correlation between +the original example and its semantically equiva- +lent augmentation, which can be fully exploited +with the consistency regularization (Zheng et al., +2021b). The consistency regularization enforces +the model predictions to be more consistent for +semantic-preserving augmentations. +Motivated by this, we propose to apply consis- +tency regularization based on a hybrid data aug- +mentation strategy, including data augmentation of +machine translation and subword sampling (Kudo, +2018). We use machine translation augmentation +to align the model predictions of the intent detec- +tion task. Meanwhile, subword sampling augmen- +tation is used to align the model predictions of +both intent detection and slot filling tasks. The +proposed method consistently improves the SLU +performance on the MASSIVE dataset under both +full-dataset and zero-shot settings. It is worth men- +tioning that our system ranked 1st in the MMNLU- +22 competition under the full-dataset setting. We +achieved an exact match accuracy of 49.65 points, +outperforming the 2nd system by 1.02 points. +arXiv:2301.02010v1 [cs.CL] 5 Jan 2023 + +2 +Background +2.1 +Task Description +The task of SLU is that given an utterance with +a word sequence x = (x1, ..., xn) with length n. +The model is required to solve two sub-tasks. The +intent detection task can be seen as an utterance +classification task to decide the intent label oI, and +the slot filling task is a sequence labeling task that +generates a slot label for each word in the utterance +to obtain the slot sequence oS = (oS +1 , ..., oS +n). +2.2 +Dataset Description +The MASSIVE dataset is composed of realistic, +human-created virtual assistant utterance text span- +ning 51 languages, 60 intents, 55 slot types, and +18 domains (FitzGerald et al., 2022). There are +11,514 training utterances for each language. For +the full-dataset setting, all training data can be used. +For the zero-shot setting, only English training data +can be used, yet we can translate them into other +languages using commercial translators. There are +2,033, 2,974, and 3,000 utterances for each lan- +guage in the development, test, and evaluation set, +respectively. The average performance in all lan- +guages should be reported under the full-dataset +setting. Meanwhile, the average performance in all +languages except English should be reported under +the zero-shot setting. +2.3 +Related Work +Pre-trained cross-lingual language models (Con- +neau and Lample, 2019; Conneau et al., 2020; Chi +et al., 2021a,b, 2022; Xue et al., 2021) encode dif- +ferent languages into universal representations and +significantly improve cross-lingual transferability. +These models usually consist of a multilingual vo- +cabulary (Conneau and Lample, 2019; Conneau +et al., 2020; Xue et al., 2021; Zheng et al., 2021a) +and a Transformer model (Vaswani et al., 2017). +A simple yet effective way to improve cross- +lingual fine-tuning is to populate the training data +with cross-lingual data augmentation (Conneau +et al., 2020). Singh et al. (2019) replace a segment +of source language input text with its translation +in another language as data augmentation. Qin +et al. (2020) randomly replace words in the source- +language training example with target-language +words using the bilingual dictionaries. Then the +model is fine-tuned on the generated code-switched +data. Instead of directly treating cross-lingual data +augmentation as extra training data, Zheng et al. +(2021b) proposed to better use data augmentations +based on consistency regularization. +3 +Method +Given the input utterance x = (x1, ..., xn) with +length n and the corresponding intent label oI and +slot labels oS = (oS +1 , ..., ...oS +n) from training cor- +pus D, we define the loss for the two sub-tasks of +SLU in our fine-tuning process as: +LI = +� +(x,oI)∈D +CE(fI(x), oI), +LS = +� +(x,oS)∈D +CE(fS(x), oS), +where LI and LS stand for the intent detection task +and the slot filling task, fI(·) and fS(·) denote the +model which predicts task-specific probability dis- +tributions for the input example x, CE(·, ·) denotes +cross-entropy loss. +3.1 +Consistency Regularization +In order to make better use of data augmentations, +we introduce the consistency regularization used +in Zheng et al. (2021b), which encourages consis- +tent predictions for an example and its semantically +equivalent augmentation. We apply consistency +regularization on intent detection and slot filling +tasks, which is defined as follows: +RI = +� +x∈D +KLS(fI(x)∥fI(A(x, z))), +RS = +� +x∈D +KLS(fS(x)∥fS(A(x, z))), +KLS(P∥Q) = KL(stopgrad(P)∥Q)+ +KL(stopgrad(Q)∥P) +where KLS(·∥·) is the symmertrical Kullback- +Leibler divergence, A(x, z) denotes the augmented +version of input utterance x with data augmenta- +tion strategy z. The regularizer encourages the +predicted distributions of the original training ex- +ample and its augmented version to agree with +each other. The stopgrad(·) operation2 is used to +stop back-propagating gradients, which is also em- +ployed in (Jiang et al., 2020; Liu et al., 2020; Zheng +et al., 2021b). +3.2 +Data Augmentations +We consider two types of data augmentation strate- +gies for our consistency regularization method, in- +cluding subword sampling and machine translation. +2Implemented by .detach() in PyTorch. + +Subword +Sampling +Machine +Translation +𝑥 +Pretrained Language Model +Slot Classifier +Intent Classifier +Intent +Task +Loss +Intent +Consistency +Regularization +Slot +Consistency +Regularization +Slot +Task +Loss +Hybrid Data Augmentation +Wake me up at five am +_Wa/ke/_me/_up/_at/_five/_am +_Wake/_me/_up/_at/_f/ive/_am +_Wa/ke/_me/_up/_at/_fiv/e/_am +早上五点叫醒我 +朝5時に起こして +Maak me om vijf uur wakker +Subword +Sampling +Machine +Translation +Input Utterance +Figure 2: Illustration of our fine-tuning framework. ‘MT’ denotes machine translation augmentation and ‘SS’ +denotes subword sampling augmentation. +3.2.1 +Subword Sampling +Subword sampling is to generate multiple subword +sequences from the original text as data augmen- +tation. We apply the on-the-fly subword sampling +algorithm from the unigram language model (Kudo, +2018) in SentencePiece (Kudo and Richardson, +2018). The output distributions of slot labels are +generated on the first subword of each word in the +input utterance. Therefore, the subword sampling +augmentation can be used to align the output dis- +tribution of both intent detection and slot filling +tasks. +3.2.2 +Machine Translation +Machine translation is a common and effective +data augmentation strategy in the cross-lingual sce- +nario (Conneau and Lample, 2019; Singh et al., +2019). Due to the difficulty of accessing ground- +truth labels in translation examples, machine trans- +lation can not be an available data augmentation +strategy in the slot filling task. To improve the +quality of our translations, we employ a variety +of approaches (See Section 4.2). Unlike subword +sampling, the output distributions of slot labels be- +tween the translation pairs can not be aligned. Thus, +we only use machine translation to align the output +distributions of the intent detection task. +3.3 +Consistency Regularization based on +Hybrid Data Augmentations +We illustrate our fine-tuning framework in Figure 2. +We propose to use consistency regularization based +on a hybrid data augmentation strategy, which in- +cludes data augmentation of machine translation +and subword sampling. During the training pro- +cess, we perform task fine-tuning and consistency +regularization for an input example simultaneously. +Then the final training loss is defined as follows: +L = LI + λ1LS + λ2RI + λ3RS +where λ1 is the slot loss coefficient, λ2 and λ3 +are the corresponding weights of the consistency +regularization for two tasks. We sample different +data augmentation for the input example with the +pre-defined distribution. +4 +Experiments +4.1 +Experimental Setup +We consider two types of pre-trained cross-lingual +language models, which are encoder-only models +and Text-to-Text models. +We use XLM-Align Base (Chi et al., 2021b) for +the encoder-only model setting. We use a two-layer +feed-forward network with a 3,072 hidden size. We +use the first representation of sentences “” for +the intent detection task and the first subword of +each word for the slot filling task. +We use mT5 Base (Xue et al., 2021) for the Text- +to-Text model setting. We follow FitzGerald et al. +(2022) to concatenate “Annotate: ” and the unla- +beled input utterance as the input of the encoder, +and generate the text concatenation of the intent +label and the slot labels as the decoder output. The +labels are separated with white spaces and then +tokenized into subwords. +We select the model that performs the best on +the development dataset to run prediction on the +test and evaluation dataset. We mainly select the +batch size in [32, 64, 128, 256], dropout rate in + +Text Type +Text Content +Slot Translation +Text Translation +Aligned or Not +Plain Text +Wake me up at five am Friday this week +five am: 凌晨五点 +Friday this week: 本周周五 +本周周五凌晨五点叫我起床 +Yes +Text with Slots in Brackets +Wake me up at [five am] [Friday this week] +在[凌晨五点][本周星期五]叫醒我 +No +Plain Text +set an alarm for two hours from now +two hours from now: +从现在起两小时后 +从现在开始设置两个小时的闹钟 +No +Text with Slots in Brackets +set an alarm for [two hours from now] +设置[从现在起两小时后]的闹钟 +Yes +Table 1: Examples of aligning slots into machine translations. +Model +Test Set +Evaluation Set +Intent Acc +Slot F1 +EMA +Intent Acc +Slot F1 +EMA +XLM-R Base +85.10 +73.60 +63.69 +- +- +- +XLM-Align Base +86.16 +76.36 +66.42 +- +- +- +mT5 Base Text-to-Text +85.33 +76.77 +66.64 +- +- +- +XLM-Align Base + Ours +87.12 +77.99 +68.76 +85.00 +68.45 +48.64 +mT5 Base Text-to-Text + Ours +87.60 +78.22 +69.60 +85.10 +69.08 +49.65 +Table 2: Test and evaluation results on the MASSIVE dataset under the full-dataset setting. Results of XLM-R +Base and mT5 Base Text-to-Text are taken from FitzGerald et al. (2022). +[0.05, 0.1, 0.15], and the hyper-parameters used in +our proposed method, including slot loss coeffi- +cient λ1 in [1, 2, 4], weights of consistency regu- +larization λ2 and λ3 in [2, 3, 5, 10]. We select the +learning rate in [5e−5, 8e−5, 1e−4] for Text-to-Text +models. As for encoder-only models, we select the +learning rate in [4e−6, 6e−6, 8e−6]. +4.2 +Data Processing +For the full-dataset setting, we use examples with +the same id in different languages as machine trans- +lation augmentation in our fine-tuning framework. +For the zero-shot setting, we translated the entire +English training set into 50 languages using com- +mercial translation APIs, such as DeepL translator +and Google translator. These translations refer to +plain text translations and can be used for intent +detection training and consistency regularization. +We used two methods to obtain a translated ex- +ample that aligned at the slot level. One is based +on the plain text translation. Each slot value in an +English training example is translated into a target +language. If the translation results of each slot can +be found in the plain text translation, a slot-aligned +translation is obtained. The other is based on the +annotated English training examples. We translated +the annotated English training example with brack- +ets for slot values (without slot type in brackets). +Using brackets explicitly allows the translator to +align slots to consecutive spans. And we also trans- +lated each slot value into the target language. If +the translation result of each slot can be found in +the annotated utterance translation, we obtain a slot +alignment example after removing the brackets. +In practice, slot-aligned examples based on plain +text translations are preferred as the final result of +the slot alignment. If no such example is avail- +able, we use the slot-aligned results from annotated +translations. Examples of slot alignment are shown +in Table 1. For those plain text translations where +we can not align the slot labels, we only use them +for the training of the intent detection task. +4.3 +Evaluation Metrics +The evaluation in competition is mainly conducted +using three metrics: +• Exact Match Accuracy (EMA): The percent- +age of utterance-level predictions where the +intent and all slots are exactly correct. +• Intent Accuracy (Intent Acc): The percentage +of predictions in which the intent is correct. +• Slot Micro F1 (Slot F1): The micro-averaged +F1 score is calculated over all slots. +4.4 +Results +Table 2 shows our results on the MASSIVE dataset +under the full-set setting. We tried different cross- +lingual pre-trained language models under the base- +line setting. Among them, XLM-Align Base per- +forms the best on the intent detection task, while +the mT5 Base Text-to-Text model performs the +best on the slot filling task and exact match ac- +curacy. When applying our consistency regular- +ization method, the mT5 Base Text-to-Text model +outperforms the XLM-Align Base model by 0.84 +points and 0.99 points on exact match accuracy on +the test dataset and the evaluation set, respectively. +Meanwhile, compared to the baseline model, us- +ing consistency regularization achieves an absolute + +Model +Test Set +Evaluation Set +Intent Acc +Slot F1 +EMA +Intent Acc +Slot F1 +EMA +XLM-R Base +70.62 +50.27 +38.70 +- +- +- +XLM-Align Base +68.49 +54.69 +40.91 +- +- +- +mT5 Base Text-to-Text +62.92 +44.77 +34.72 +- +- +- +XLM-Align Base + Ours +85.12 +71.27 +62.18 +83.18 +62.84 +43.05 +XLM-Align Base + Ours + KD +85.76 +73.55 +64.44 +83.89 +64.60 +44.84 +mT5 Base Text-to-Text + Ours +84.58 +69.24 +60.59 +82.56 +60.00 +40.93 +Table 3: Test and evaluation results on the MASSIVE dataset under the zero-shot setting. Results of XLM-R Base +and mT5 Base Text-to-Text are taken from FitzGerald et al. (2022). +Model +Intent Acc Slot F1 EMA +XLM-Align Base + Ours +87.12 +77.99 +68.76 +- Subword Sampling +87.50 +76.08 +67.40 +- Consistency Regularization +86.16 +76.32 +66.57 +Table 4: Ablation studies on the MASSIVE test dataset +under the full-dataset setting. +2.96-point improvement on exact match accuracy +with the mT5 Base Text-to-Text model. +Table 3 shows our results on the MASSIVE +dataset under the zero-shot setting. For the base- +line models, XLM-Align Base performs the best on +all three metrics. Difference from the full-dataset +setting, mT5 Base Text-to-Text models perform +poorly under the zero-shot setting. We attribute +it to the fact that Text-to-Text models strongly +rely on the training data quality since most of the +training data under the zero-shot setting are ob- +tained with machine translation systems. When +applying our consistency regularization method, +the XLM-Align Base model outperforms the base- +line model by 21.27 points. Distilled from the In- +foXLM Large (Chi et al., 2021a) model will further +improve the performance by an absolute 2.26-point. +4.5 +Ablation Studies +We conduct ablation studies on the test dataset of +MASSIVE under the two settings. Table 4 shows +the results under the full-dataset setting. Ablating +subword sampling will degrade the performance +by 1.36 points on the exact match accuracy, where +the performance drop comes mainly from the slot +filling task, indicating the subword sampling aug- +mentation mainly works on slot filling. Ablating +consistency regularization will degrade the perfor- +mance by 2.19 points on the exact match accuracy. +The performances on both intent detection and slot +filling tasks are decreased. +The zero-shot setting results are presented in Ta- +Model +Intent Acc Slot F1 EMA +XLM-Align Base + Ours +85.12 +71.27 +62.18 +- Subword Sampling +85.14 +69.52 +60.94 +- Machine Translation +72.27 +58.37 +45.50 +- Consistency Regularization +83.90 +69.37 +59.95 +Table 5: Ablation studies on the MASSIVE test dataset +under the zero-shot setting. +ble 5. It can be observed that when machine trans- +lation augmentation is removed, the exact match +accuracy drops by 16.68 points, while the perfor- +mance on intent detection and slot filling are also +significantly worse. We also removed the subword +sampling augmentation, and the performance is +found to have the same trend as in the full-dataset +setting. An absolute 1.24-point drop on the exact +match accuracy and an absolute 1.75-point drop +on slot micro F1 demonstrate that subword sam- +pling is more beneficial for the slot filling task. By +removing the consistency regularization, the per- +formance of exact match accuracy will degrade by +2.23 points. The performance shows a significant +performance drop on both intent detection and slot +filling tasks. +5 +Conclusion +We propose to use consistency regularization based +on a hybrid data augmentation strategy to improve +the performance of multilingual SLU. The pro- +posed method is flexible and can be easily plugged +into the fine-tuning process of both the encoder- +only model and the Text-to-Text model. The ex- +perimental results demonstrate the importance of +consistency regularization and the hybrid data aug- +mentation strategy, respectively. +Acknowledgments +This work was supported by the National Key R&D +Program of China via grant 2020AAA0106501 and + +the National Natural Science Foundation of China +(NSFC) via grant 62236004 and 61976072. +References +Zewen Chi, Li Dong, Furu Wei, Nan Yang, Sak- +sham Singhal, Wenhui Wang, Xia Song, Xian-Ling +Mao, Heyan Huang, and Ming Zhou. 2021a. +In- +foXLM: An information-theoretic framework for +cross-lingual language model pre-training. In Pro- +ceedings of the 2021 Conference of the North Amer- +ican Chapter of the Association for Computational +Linguistics: Human Language Technologies, pages +3576–3588, Online. 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Association for Computational Lin- +guistics. + diff --git a/SNA0T4oBgHgl3EQfD_8a/content/tmp_files/load_file.txt b/SNA0T4oBgHgl3EQfD_8a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2541563d2004f4aebdf64b94bcf0431d0266d69a --- /dev/null +++ b/SNA0T4oBgHgl3EQfD_8a/content/tmp_files/load_file.txt @@ -0,0 +1,442 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf,len=441 +page_content='HIT-SCIR at MMNLU-22: Consistency Regularization for Multilingual Spoken Language Understanding Bo Zheng, Zhouyang Li, Fuxuan Wei, Qiguang Chen, Libo Qin, Wanxiang Che∗ Harbin Institute of Technology {bzheng,zhouyangli,fxwei,qgchen,lbqin,car}@ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='cn Abstract Multilingual spoken language understanding (SLU) consists of two sub-tasks, namely intent detection and slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' To improve the per- formance of these two sub-tasks, we propose to use consistency regularization based on a hybrid data augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The consis- tency regularization enforces the predicted dis- tributions for an example and its semantically equivalent augmentation to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We conduct experiments on the MASSIVE dataset under both full-dataset and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Experimental results demonstrate that our pro- posed method improves the performance on both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Our system1 ranked 1st in the MMNLU-22 compe- tition under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 1 Introduction The MMNLU-22 evaluation focuses on the prob- lem of multilingual natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' It is based on the MASSIVE dataset (FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2022), a multilingual spoken language under- standing (SLU) dataset with two sub-tasks, includ- ing intent detection and slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Specifically, given a virtual assistant utterance in an arbitrary language, the model is designed to predict the cor- responding intent label and extract the slot results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' An English example is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Fine-tuning pre-trained cross-lingual language models allows task-specific supervision to be shared and transferred across languages (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' This motivates the two setting for the MMNLU-22 evaluation, namely the full-dataset setting and the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Participants are allowed to use training data in all languages under the full-dataset setting, while they can only access the English training data under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' ∗Email corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 1The code will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='com/ bozheng-hit/MMNLU-22-HIT-SCIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Utterance Slot Intent Wake me up at five am Friday this week O O O O time time date date date set alarm Figure 1: An English example from the MASSIVE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The slot label ‘O’ stands for the ‘Other’ label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The latter is also called zero-shot cross-lingual SLU in previous work (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Cross-lingual data augmentation methods have been proven effective to improve cross-lingual transferability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', code-switch substitution (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020) and machine translation (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Most previ- ous work directly utilizes the data augmentations as additional training data for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' How- ever, they ignore the inherent correlation between the original example and its semantically equiva- lent augmentation, which can be fully exploited with the consistency regularization (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The consistency regularization enforces the model predictions to be more consistent for semantic-preserving augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Motivated by this, we propose to apply consis- tency regularization based on a hybrid data aug- mentation strategy, including data augmentation of machine translation and subword sampling (Kudo, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We use machine translation augmentation to align the model predictions of the intent detec- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Meanwhile, subword sampling augmen- tation is used to align the model predictions of both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The proposed method consistently improves the SLU performance on the MASSIVE dataset under both full-dataset and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' It is worth men- tioning that our system ranked 1st in the MMNLU- 22 competition under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We achieved an exact match accuracy of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='65 points, outperforming the 2nd system by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='02 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='02010v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='CL] 5 Jan 2023 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='1 Task Description The task of SLU is that given an utterance with a word sequence x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', xn) with length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The model is required to solve two sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The intent detection task can be seen as an utterance classification task to decide the intent label oI, and the slot filling task is a sequence labeling task that generates a slot label for each word in the utterance to obtain the slot sequence oS = (oS 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', oS n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2 Dataset Description The MASSIVE dataset is composed of realistic, human-created virtual assistant utterance text span- ning 51 languages, 60 intents, 55 slot types, and 18 domains (FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' There are 11,514 training utterances for each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' For the full-dataset setting, all training data can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' For the zero-shot setting, only English training data can be used, yet we can translate them into other languages using commercial translators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' There are 2,033, 2,974, and 3,000 utterances for each lan- guage in the development, test, and evaluation set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The average performance in all lan- guages should be reported under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Meanwhile, the average performance in all languages except English should be reported under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='3 Related Work Pre-trained cross-lingual language models (Con- neau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021a,b, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021) encode dif- ferent languages into universal representations and significantly improve cross-lingual transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' These models usually consist of a multilingual vo- cabulary (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021a) and a Transformer model (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' A simple yet effective way to improve cross- lingual fine-tuning is to populate the training data with cross-lingual data augmentation (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2019) replace a segment of source language input text with its translation in another language as data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2020) randomly replace words in the source- language training example with target-language words using the bilingual dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Then the model is fine-tuned on the generated code-switched data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Instead of directly treating cross-lingual data augmentation as extra training data, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2021b) proposed to better use data augmentations based on consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3 Method Given the input utterance x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', xn) with length n and the corresponding intent label oI and slot labels oS = (oS 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='oS n) from training cor- pus D, we define the loss for the two sub-tasks of SLU in our fine-tuning process as: LI = � (x,oI)∈D CE(fI(x), oI), LS = � (x,oS)∈D CE(fS(x), oS), where LI and LS stand for the intent detection task and the slot filling task, fI(·) and fS(·) denote the model which predicts task-specific probability dis- tributions for the input example x, CE(·, ·) denotes cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='1 Consistency Regularization In order to make better use of data augmentations, we introduce the consistency regularization used in Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2021b), which encourages consis- tent predictions for an example and its semantically equivalent augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We apply consistency regularization on intent detection and slot filling tasks, which is defined as follows: RI = � x∈D KLS(fI(x)∥fI(A(x, z))), RS = � x∈D KLS(fS(x)∥fS(A(x, z))), KLS(P∥Q) = KL(stopgrad(P)∥Q)+ KL(stopgrad(Q)∥P) where KLS(·∥·) is the symmertrical Kullback- Leibler divergence, A(x, z) denotes the augmented version of input utterance x with data augmenta- tion strategy z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The regularizer encourages the predicted distributions of the original training ex- ample and its augmented version to agree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The stopgrad(·) operation2 is used to stop back-propagating gradients, which is also em- ployed in (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2 Data Augmentations We consider two types of data augmentation strate- gies for our consistency regularization method, in- cluding subword sampling and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 2Implemented by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='detach() in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Subword ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Pretrained Language Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Slot Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Intent Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Intent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Intent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Consistency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Regularization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Slot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Consistency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Regularization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Slot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Hybrid Data Augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Wake me up at five am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='_Wa/ke/_me/_up/_at/_five/_am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='_Wake/_me/_up/_at/_f/ive/_am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='_Wa/ke/_me/_up/_at/_fiv/e/_am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='早上五点叫醒我 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='朝5時に起こして ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Maak me om vijf uur wakker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Subword ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Input Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Figure 2: Illustration of our fine-tuning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' ‘MT’ denotes machine translation augmentation and ‘SS’ denotes subword sampling augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='1 Subword Sampling Subword sampling is to generate multiple subword sequences from the original text as data augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We apply the on-the-fly subword sampling algorithm from the unigram language model (Kudo, 2018) in SentencePiece (Kudo and Richardson, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The output distributions of slot labels are generated on the first subword of each word in the input utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Therefore, the subword sampling augmentation can be used to align the output dis- tribution of both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2 Machine Translation Machine translation is a common and effective data augmentation strategy in the cross-lingual sce- nario (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Due to the difficulty of accessing ground- truth labels in translation examples, machine trans- lation can not be an available data augmentation strategy in the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' To improve the quality of our translations, we employ a variety of approaches (See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Unlike subword sampling, the output distributions of slot labels be- tween the translation pairs can not be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Thus, we only use machine translation to align the output distributions of the intent detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='3 Consistency Regularization based on Hybrid Data Augmentations We illustrate our fine-tuning framework in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We propose to use consistency regularization based on a hybrid data augmentation strategy, which in- cludes data augmentation of machine translation and subword sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' During the training pro- cess, we perform task fine-tuning and consistency regularization for an input example simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Then the final training loss is defined as follows: L = LI + λ1LS + λ2RI + λ3RS where λ1 is the slot loss coefficient, λ2 and λ3 are the corresponding weights of the consistency regularization for two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We sample different data augmentation for the input example with the pre-defined distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='1 Experimental Setup We consider two types of pre-trained cross-lingual language models, which are encoder-only models and Text-to-Text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We use XLM-Align Base (Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021b) for the encoder-only model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We use a two-layer feed-forward network with a 3,072 hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We use the first representation of sentences “” for the intent detection task and the first subword of each word for the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We use mT5 Base (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021) for the Text- to-Text model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We follow FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2022) to concatenate “Annotate: ” and the unla- beled input utterance as the input of the encoder, and generate the text concatenation of the intent label and the slot labels as the decoder output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The labels are separated with white spaces and then tokenized into subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We select the model that performs the best on the development dataset to run prediction on the test and evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We mainly select the batch size in [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 256],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' dropout rate in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Text Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Text Content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Slot Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Text Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Aligned or Not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Plain Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Wake me up at five am Friday this week ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='five am: 凌晨五点 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Friday this week: 本周周五 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='本周周五凌晨五点叫我起床 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Text with Slots in Brackets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Wake me up at [five am] [Friday this week] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='在[凌晨五点][本周星期五]叫醒我 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Plain Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='set an alarm for two hours from now ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='two hours from now: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='从现在起两小时后 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='从现在开始设置两个小时的闹钟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Text with Slots in Brackets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='set an alarm for [two hours from now] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='设置[从现在起两小时后]的闹钟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='Table 1: Examples of aligning slots into machine translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Model Test Set Evaluation Set Intent Acc Slot F1 EMA Intent Acc Slot F1 EMA XLM-R Base 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='10 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='60 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='69 XLM-Align Base 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='36 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='42 mT5 Base Text-to-Text 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='33 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='77 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='64 XLM-Align Base + Ours 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='12 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='99 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='76 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='00 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='45 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='64 mT5 Base Text-to-Text + Ours 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='60 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='22 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='60 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='10 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='08 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='65 Table 2: Test and evaluation results on the MASSIVE dataset under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Results of XLM-R Base and mT5 Base Text-to-Text are taken from FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='15], and the hyper-parameters used in our proposed method, including slot loss coeffi- cient λ1 in [1, 2, 4], weights of consistency regu- larization λ2 and λ3 in [2, 3, 5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We select the learning rate in [5e−5, 8e−5, 1e−4] for Text-to-Text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' As for encoder-only models, we select the learning rate in [4e−6, 6e−6, 8e−6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='2 Data Processing For the full-dataset setting, we use examples with the same id in different languages as machine trans- lation augmentation in our fine-tuning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' For the zero-shot setting, we translated the entire English training set into 50 languages using com- mercial translation APIs, such as DeepL translator and Google translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' These translations refer to plain text translations and can be used for intent detection training and consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We used two methods to obtain a translated ex- ample that aligned at the slot level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' One is based on the plain text translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Each slot value in an English training example is translated into a target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' If the translation results of each slot can be found in the plain text translation, a slot-aligned translation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The other is based on the annotated English training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We translated the annotated English training example with brack- ets for slot values (without slot type in brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Using brackets explicitly allows the translator to align slots to consecutive spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' And we also trans- lated each slot value into the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' If the translation result of each slot can be found in the annotated utterance translation, we obtain a slot alignment example after removing the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' In practice, slot-aligned examples based on plain text translations are preferred as the final result of the slot alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' If no such example is avail- able, we use the slot-aligned results from annotated translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Examples of slot alignment are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' For those plain text translations where we can not align the slot labels, we only use them for the training of the intent detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='3 Evaluation Metrics The evaluation in competition is mainly conducted using three metrics: Exact Match Accuracy (EMA): The percent- age of utterance-level predictions where the intent and all slots are exactly correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Intent Accuracy (Intent Acc): The percentage of predictions in which the intent is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Slot Micro F1 (Slot F1): The micro-averaged F1 score is calculated over all slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='4 Results Table 2 shows our results on the MASSIVE dataset under the full-set setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We tried different cross- lingual pre-trained language models under the base- line setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Among them, XLM-Align Base per- forms the best on the intent detection task, while the mT5 Base Text-to-Text model performs the best on the slot filling task and exact match ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' When applying our consistency regular- ization method, the mT5 Base Text-to-Text model outperforms the XLM-Align Base model by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='84 points and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='99 points on exact match accuracy on the test dataset and the evaluation set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Meanwhile, compared to the baseline model, us- ing consistency regularization achieves an absolute Model Test Set Evaluation Set Intent Acc Slot F1 EMA Intent Acc Slot F1 EMA XLM-R Base 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='62 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='27 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='70 XLM-Align Base 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='49 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='69 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='91 mT5 Base Text-to-Text 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='92 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='77 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='72 XLM-Align Base + Ours 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='27 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='18 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='18 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='84 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='05 XLM-Align Base + Ours + KD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='76 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='55 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='44 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='89 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='60 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='84 mT5 Base Text-to-Text + Ours 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='58 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='24 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='59 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='56 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='00 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='93 Table 3: Test and evaluation results on the MASSIVE dataset under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Results of XLM-R Base and mT5 Base Text-to-Text are taken from FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Model Intent Acc Slot F1 EMA XLM-Align Base + Ours 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='12 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='99 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='76 Subword Sampling 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='50 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='08 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='40 Consistency Regularization 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='32 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='57 Table 4: Ablation studies on the MASSIVE test dataset under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='96-point improvement on exact match accuracy with the mT5 Base Text-to-Text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Table 3 shows our results on the MASSIVE dataset under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' For the base- line models, XLM-Align Base performs the best on all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Difference from the full-dataset setting, mT5 Base Text-to-Text models perform poorly under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We attribute it to the fact that Text-to-Text models strongly rely on the training data quality since most of the training data under the zero-shot setting are ob- tained with machine translation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' When applying our consistency regularization method, the XLM-Align Base model outperforms the base- line model by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='27 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Distilled from the In- foXLM Large (Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=', 2021a) model will further improve the performance by an absolute 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='26-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='5 Ablation Studies We conduct ablation studies on the test dataset of MASSIVE under the two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Table 4 shows the results under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Ablating subword sampling will degrade the performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='36 points on the exact match accuracy, where the performance drop comes mainly from the slot filling task, indicating the subword sampling aug- mentation mainly works on slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Ablating consistency regularization will degrade the perfor- mance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='19 points on the exact match accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The performances on both intent detection and slot filling tasks are decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The zero-shot setting results are presented in Ta- Model Intent Acc Slot F1 EMA XLM-Align Base + Ours 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='27 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='18 Subword Sampling 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='14 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='52 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='94 Machine Translation 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='27 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='37 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='50 Consistency Regularization 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='90 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='37 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='95 Table 5: Ablation studies on the MASSIVE test dataset under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' It can be observed that when machine trans- lation augmentation is removed, the exact match accuracy drops by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='68 points, while the perfor- mance on intent detection and slot filling are also significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' We also removed the subword sampling augmentation, and the performance is found to have the same trend as in the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' An absolute 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='24-point drop on the exact match accuracy and an absolute 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='75-point drop on slot micro F1 demonstrate that subword sam- pling is more beneficial for the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' By removing the consistency regularization, the per- formance of exact match accuracy will degrade by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content='23 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The performance shows a significant performance drop on both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 5 Conclusion We propose to use consistency regularization based on a hybrid data augmentation strategy to improve the performance of multilingual SLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The pro- posed method is flexible and can be easily plugged into the fine-tuning process of both the encoder- only model and the Text-to-Text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' The ex- perimental results demonstrate the importance of consistency regularization and the hybrid data aug- mentation strategy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Key R&D Program of China via grant 2020AAA0106501 and the National Natural Science Foundation of China (NSFC) via grant 62236004 and 61976072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' References Zewen Chi, Li Dong, Furu Wei, Nan Yang, Sak- sham Singhal, Wenhui Wang, Xia Song, Xian-Ling Mao, Heyan Huang, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' 2021a.' 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of the 59th Annual Meeting of the Associa- tion for Computational Linguistics and the 11th In- ternational Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 3403– 3417, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'} diff --git a/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf b/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..73f2f77ee590931b8adf5898a16c68ebea38a6ec Binary files /dev/null and b/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf differ diff --git a/SdE3T4oBgHgl3EQfzAun/content/tmp_files/2301.04725v1.pdf.txt b/SdE3T4oBgHgl3EQfzAun/content/tmp_files/2301.04725v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..089062374b3ef221472f0e099df04c2517f70a66 --- /dev/null +++ b/SdE3T4oBgHgl3EQfzAun/content/tmp_files/2301.04725v1.pdf.txt @@ -0,0 +1,321 @@ +arXiv:2301.04725v1 [cs.CR] 11 Jan 2023 +GeNeDis manuscript No. +(will be inserted by the editor) +Blockchain For Mobile Health Applications +Acceleration With GPU Computing +Georgios Drakopoulos · Michail Marountas · +Xenophon Liapakis · Giannis Tzimas · Phivos +Mylonas · Spyros Sioutas +Received: date / Accepted: date +Abstract Blockchain is a linearly linked, distributed, and very robust data structure. +Originally proposed as part of the Bitcoin distributed stack, it found a number of ap- +plications in a number of fields, most notably in smart contracts, social media, secure +IoT, and cryptocurrency mining. It ensures data integrity by distributing strongly en- +crypted data in widely redundant segments. Each new insertion requires verification +and approval by the majority of the users of the blockchain. Both encryption and ver- +ification are computationally intensive tasks which cannot be solved with ordinary +off-the-shelf CPUs. This has resulted in a renewed scientific interest in secure dis- +tributed communication and coordination protocols. Mobile health applications are +growing progressively popular and have the enormous advantage of timely diagnosis +of certain conditions. However, privacy concerns have been raised as mobile health +application by default have access to highly sensitive personal data. This chapter +presents concisely how blockchain can be applied to mobile health applications in +order to enhance privacy. +Keywords Blockchains · Digital health · Edge computing · Mobile computing · +Mobile applications · Majority protocols · GPU computing +Mathematics Subject Classification (2010) 65Y05 · 68Q05 · 68Q10 · 68W10 +G. Drakopoulos and P. Mylonas +Department of Informatics, Ionian University, Greece +E-mail: {c16drak, fmylonas}@ionio.gr +X. Liapakis +Interamerican SA, Greece +E-mail: liapakisx@interamerican.gr +G. Tzimas +Technological and Educational Institute of Western Greece, Antirrio Campus, Greece +E-mail: tzimas@teimes.gr +M. Marountas and S. Sioutas +Computer Engineering and Informatics Department, University of Patras, Greece +E-mail: {marounta, sioutas}@ceid.upatras.gr + +2 +Drakopoulos et al. +1 Introduction +Perhaps the most well studied recent advent in the domain of distributed comouting +and data structures is that of blockchain. The latter acts as a public or private ledger +and from a structural perspective is a linear, distributed, and robust data structure in +the sense that not only the insertion of new data requires special permission from its +stakeholders, mostly but not necessarily ordinary netizens with a legitimate vested +interest in a given blockchain, which is obtained from specially designed consen- +sus protocols, but also the true netizen identities participating to a given block chain +as well as data contained therein are strongly encrypted, typically with a public key +scheme such as SHA-256. Additionally, the exact location of data insertion is decided +on the basis of a secure hash function. Finally, when the number of netizens partici- +pating to a blockchain is large, typically in the thousands, it becomes difficult to hack +or game it as any malicious changes become visible almost immediately. +Having the computational properties just described, a blockchain is an excellent data +structure for securely storing large volumes of information for a very broad spec- +trum of purposes including but not limited to smart contracts, digital health informa- +tion, smart city and smart infrastructure status, financial macrotransactions as well +as gaming and social media microtransactions, and insurance information. In fact, +the blockchain as a data structure was initially part of the Bitcoin stack as described +in Nakamoto (2008) or as later explored in cryptocurrency surveys as for instance +Antonopoulos (2014) or Antonopoulos (2017). Since then, however it took a life of +its own with numerous parties developing some version of the original blockchain for +their own purposes. +Blockchain is not the only recent computationally intensive development. Fields +like numerical and distributed deep learning such as the training of multilyayer con- +volutional and recurrent neural networks, complex systems simulation such as brain +networks and protein-to-protein interaction networks, as large scale social network +analysis are notorious for their quick scaling. One response to the need for additional +computational power was the development of hardware aiming at massive parallelism +through special purpose GPUs along with the associated software which can take ad- +vantage of such specialized hardware and can orchestrate the appopriate sequence +of computations to derive the desired result. Google TensorFlow, a low level frame- +work whose primary unit is a tensor as explained among others in Abadi et al. (2016), +namely a multidimensional array, belongs to this category. +The primary objective of this chapter is to concentrate and succintly present the +ways TensorFlow and GPU computation in conjunction with blockchain can em- +power applications in the domains of digital health and insurance market. As a sec- +ondary objective, the computational capabilities and the dataflow model of Tensor- +Flow are analysed. + +Blockchain For Mobile Health Applications +3 +Table 1 Notation of this chapter. +symbol +meaning +△= +Definition or equality by definition +⟨sk⟩ +Sequence with elements sk +|⟨sk⟩| +Sequence cardinality +The remaining of this work is structured as follows. In section 2 the relevant scien- +tific literature regarding blockchain, GPU computing, and their applications is briefly +reviewed. The properties of the blockchain as well as these of TensorFlow are de- +scribed in section 3, whereas the blockchain applications in the domains of digital +health and insurance are explored in section 4. The main findings of this chapter as +well as possible future research directions are stated in section 5. Finally, table 1 +summarises the notation of this work. +2 Previous Work +Blockchains were formally introduced in the seminal Bitcoin work of Nakamoto +(2008). Their technological innovation and the potential to become a disruptive tech- +nology was explored among others in Barber et al. (2012) and in Cachin (2016). The +combination of blockchains with the IoT and their applications to the mainstream in- +dustrial sector in conjunction with the upcoming digital transformations of Industry +4.0 are the focus of Miller (2018). Practical ways and the associated challenges to +implement a blockchain over IoT and edge computing are shown in Zyskind et al. +(2015). The financial prospects of Bitcoin in terms of wealth accumulation as well as +the properties of Bitcoin versus the traditional fiat currency are the focus of a num- +ber of works, for instance Antonopoulos (2014), Antonopoulos (2017), Kosba et al. +(2016), Swan (2015), and B¨ohme et al. (2015). The distributed implementation of +blockchains is discussed in Abbas et al. (2018) and in Pass et al. (2017), whereas se- +curity aspects of the blockchains are treated in Puthal et al. (2018). A large number +of blockchains besides the Bitcoin stack can be found in Underwood (2016). +Since the original public description of TensorFlow in Abadi et al. (2016) and in +Abadi (2016) it was widely adopted from the deep learning community.In Matthews et al. +(2017) a Gaussian process generator implemented with rudimentary TensorFlow op- +erations is described in detail. For a new graph resilience metric based on paths +see Drakopoulos et al. (2018b) along the lines of the regularization methodology of +Kanavos et al. (2017). The advantages of and the ways for visualising the TensorFlow +computations are Wongsuphasawat et al. (2018). For tensor applications in social net- +work analysis such as multiway digital influence estimation see Drakopoulos et al. +(2017), community structure discovery Drakopoulos et al. (2018a), and graph based +k-means initialization Drakopoulos et al. (2016). Finally, for a genetic algorithm for +clustering tensors containing linguistic and spatial data see Drakopoulos et al. (2019). + +4 +Drakopoulos et al. +3 Parallelism and Blockchain +3.1 Blockchains +As their collective name suggests, from a structural point of view blockchains are, +typically very long, linearly linked nodes. Each such node contains part of the post- +marked information stored in the data structure along with some administrative in- +formation. The data stored in a blockchain can never be erased, although it can be +updated provided all interested parties agree on that. Thus, both the original and the +updated data are stored, making audits efficient. +Perhaps the most important advantages of selecting a blockchain scheme besides +the increased security are the following: +– Blockchains support a very large volume of transactions which can take place +almost simultaneously because of their very inherent distributed nature. There- +fore, their stakeholders can perform any desired number of transactions within +a very reasonable amount of time without worrying about the exact transaction +execution time, which in certain cases may influence the transaction cost. +– The stakeholders of a given blockchain can stay informed of the global status of +the blockchain in almost real time. Thus, not only can they perform transactions +but they can also know their results almost immediately or at least at the moment +the latter are actually executed. +– Blockchains, either public or private, offer full transparency since every partic- +ipant to a given blockchain is free to validate any tranaction which took place +within that blockchain. Additionally, the verification protocols are deliberately +built so that verification be easy even for netizens with low computational re- +sources, for instance a smartphone or a tablet. This reinforces the trust toward +properly implemented and managed blockchains. +– In the case of a catastrophic loss, a properly implemented blockchain can at least +partially rebuild itself from the segments stored at the computers of its stakehold- +ers. This is feasible given the increased redundancy integrated into a blockchain. +– From a software engineering viewpoint, each blockchain node is a relatively sim- +ple construct and, therefore, it can be managed with little or no human interven- +tion. Thus, a blockchain administrator is only required to control certain a few key +parts of the data structure, making blockchains easy and inexpensive to maintain. +– Last but not least, any third parties and intermediaries are no longer necessary. +The interested parties can directly communicate and get current quotes or any +other vital pieces of information from each other. +Notice that blockchains are not immune to various sophisticated attacks, although +the latter typically require considerable resources which are nowadays well within +the capabilities of a dedicated hacker group or of a government agency. Although +directly attacking the encryption protocols may not be a wise course of action, unless +some knowledge of the private key is available, using a zero day exploit is. + +Blockchain For Mobile Health Applications +5 +As with any new technology, blockchain management software is by no means error +free. However, most known attacks so far take on a completely different approach +akin to a brute force attack by relying on big botnet networks in order to take charge +of a small or medium sized blockchain. +Yet another method, holistic in nature, for attacking a blockchain is through the use +of control theory concepts. The current state, in any way that is estimated by the +attacker, of a large blockchain is represented as a control vector x[n]. Then a usually +linear state space model is formulated as follows: +x[n + 1] +△=Ax[n]+ bu[n] +y[n + 1] +△=cTx[n + 1]+ du[n] +(1) +If the attacker can insert an appropriate input sequence ⟨uk⟩, then, depending on the +modelling correctness, he may bring the entire system to an undersirable state. Of +course, such a sequence may not exist or its cardinality |⟨uk⟩| might approach infinity. +3.2 TensorFlow +Google TensorFlow is a low level programming framework based on the dataflow +programming paradigm and using tensors, namely multidimensional arrays, as its +primary data structure. Originally developed for simulating brain networks, it is a +powerful tool for deep learning. It has official APIs for Python and C++, whereas +unofficial APIs are being developed for a number of established programming lan- +guages. Moreover, it has computational kernels for CPUs, GPUs, and TPUs. +Besides the methods for elementary operations such as Kronecker and Hadamard +tensor multiplication, minimum location, tensor reshaping, and tensor factorizations +such as Kruskal and Tucker decompositions, TensorFlow has a number of numerical +optimizers which are common in deep learning such as AdaGrad. Also, TensorFlow +supports checkpoints, allowing the early termination of a training process. +Within a blockchain context, TensorFlow can accelerate numerical computations +for hashing or encryption. Additionally, it can be used to train a neural network, +recurrent, convolutional, autoencoding, or otherwise, which can predict the volume +in the immediate future, so that a bursty load of transactions can be better balanced +throughout the blockchain nodes. Moreover, similar networks can be built in order +to predict which blockchain user will be the next to generate a chunk of data or will +ask for a transaction verification, again for load balancing purposes. Finally, large +deep learning networks can in theory be deployed in order to mount an attack on the +encryption protocol used by a given blockchain, but to the best of the knowledge of +the authors, no such use has been recorded. + +6 +Drakopoulos et al. +4 Applications +The blockchain as a ledger structure, either public or private, because of its secure +and distributed design is a place for storing sensitive data such as health condition +and financial transactions. Additionally, as stated earlier any intermediaries are elim- +inated, at least in a higher level. Thus, any fees and premiums such as taxes or bank +processing fees are also, in theory at least, automatically gone. +Regarding the digital health world, blockchain-based applications have an enormous +potential. The following list contains some of the most prominent ones. +– The medical records of a netizen can be stored with safety in a blockchain and +can be recovered only by the certified health professional who cure the netizen +regardless of their location or whether they have cured her before. +– Blockchain can facilitate automated monitoring of selected biomarkers by smart- +phones and the measurements can be compared against personalized baselines. +– Netizens have much improved control over their personal records and their con- +sent can be obtained under more transparent and clear conditions. +– Netizens can use micropayments or mobile payments in order to procure medicines, +further protecting their privacy. +Concerning the growing insurance market, there is also a significant room for blo- +ckchain-based applications. Some indicative are the following: +– Netizens can search easier for attractive offers and can contact insurance agents +directly in order to negotiate for even better offers. This can also be done through +software agents configured to look specific offers or terms. +– Netizens and insurance agents can hold smart contracts such as property and ve- +hicle electronic contract purchses in blockchains. At a later point, should the need +arise, they can directly renegotiate contract terms which will be also recorded in +the blockchain, provided the interested parties reach an agreement. +– Once smart contracts are recorded, ordinary shallow or deep learning algorithms +can be run atop the blockchain in order to identify possible fraud cases. +– Blockchains simplify considerably payments and can even be combined with mo- +bile payments. Payment records remain immutable and constitute proof that a +payment indeed took place at the time indicated. +– Claims can be automatically verified by smartphones and other personal devices +which are connected to the blockchain, reducing thus the administrative burden +and the overhead. +At this point it should also be reminded that the general advantages of section 3 also +hold in addition to those listed above. +5 Conclusions +The twofold epicenter of this chapter was the blockchain applications in the domains +of digital health and insurance market and the ways Google TensorFlow, a low level + +Blockchain For Mobile Health Applications +7 +computational framework for computationally intensive applications, can be used to +accelerate the associated computations. The blockchain has numerous applications in +the domains of medical healthcare and insurance. Moreover, it reinforces the privacy +and transparency conditions and, thus, help establish a viable and scalable market. +Further research directions include the developmentof extensively tested blockchain +management systems so that most, if not all, zero day exploits are eliminated. More- +over, given the recent advances in quantum computing which make large scale brute +force attacks feasible, stronger cryptographic schemes should be sought in order to +protect the sensitive personal data stored in blockchains. +Acknowledgements We gratefully acknowledge the support of NVIDIA Corporation with the donation +of the Titan Xp GPU used for this research. +References +Abadi M (2016) TensorFlow: Learning functions at scale. ACM SIGPLAN Notices +51(9):1–1 +Abadi M, et al. (2016) TensorFlow: A system for large-scale machine learning. 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O’Reilly Media, Inc. +Underwood S (2016) Blockchain beyond Bitcoin. Communications of the ACM +59(11):15–17 +Wongsuphasawat K, et al. (2018) Visualizing dataflow graphs of deep learning mod- +els in TensorFlow. Transactions on visualization and computer graphics 24(1):1–12 +Zyskind G, Nathan O, et al. (2015) Decentralizing privacy: Using blockchain to pro- +tect personal data. In: SPW, IEEE, pp 180–184 + diff --git a/SdE3T4oBgHgl3EQfzAun/content/tmp_files/load_file.txt b/SdE3T4oBgHgl3EQfzAun/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef60df6503172ae5195e5a2349c02e299226a646 --- /dev/null +++ b/SdE3T4oBgHgl3EQfzAun/content/tmp_files/load_file.txt @@ -0,0 +1,179 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf,len=178 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='04725v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='CR] 11 Jan 2023 GeNeDis manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (will be inserted by the editor) Blockchain For Mobile Health Applications Acceleration With GPU Computing Georgios Drakopoulos · Michail Marountas · Xenophon Liapakis · Giannis Tzimas · Phivos Mylonas · Spyros Sioutas Received: date / Accepted: date Abstract Blockchain is a linearly linked, distributed, and very robust data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Originally proposed as part of the Bitcoin distributed stack, it found a number of ap- plications in a number of fields, most notably in smart contracts, social media, secure IoT, and cryptocurrency mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' It ensures data integrity by distributing strongly en- crypted data in widely redundant segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Each new insertion requires verification and approval by the majority of the users of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Both encryption and ver- ification are computationally intensive tasks which cannot be solved with ordinary off-the-shelf CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' This has resulted in a renewed scientific interest in secure dis- tributed communication and coordination protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Mobile health applications are growing progressively popular and have the enormous advantage of timely diagnosis of certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' However, privacy concerns have been raised as mobile health application by default have access to highly sensitive personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' This chapter presents concisely how blockchain can be applied to mobile health applications in order to enhance privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Keywords Blockchains · Digital health · Edge computing · Mobile computing · Mobile applications · Majority protocols · GPU computing Mathematics Subject Classification (2010) 65Y05 · 68Q05 · 68Q10 · 68W10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Drakopoulos and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Mylonas Department of Informatics, Ionian University, Greece E-mail: {c16drak, fmylonas}@ionio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='gr X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Liapakis Interamerican SA, Greece E-mail: liapakisx@interamerican.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='gr G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Tzimas Technological and Educational Institute of Western Greece, Antirrio Campus, Greece E-mail: tzimas@teimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='gr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Marountas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Sioutas Computer Engineering and Informatics Department, University of Patras, Greece E-mail: {marounta, sioutas}@ceid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='upatras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='gr 2 Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 1 Introduction Perhaps the most well studied recent advent in the domain of distributed comouting and data structures is that of blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The latter acts as a public or private ledger and from a structural perspective is a linear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' distributed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' and robust data structure in the sense that not only the insertion of new data requires special permission from its stakeholders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' mostly but not necessarily ordinary netizens with a legitimate vested interest in a given blockchain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' which is obtained from specially designed consen- sus protocols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' but also the true netizen identities participating to a given block chain as well as data contained therein are strongly encrypted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' typically with a public key scheme such as SHA-256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Additionally, the exact location of data insertion is decided on the basis of a secure hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Finally, when the number of netizens partici- pating to a blockchain is large, typically in the thousands, it becomes difficult to hack or game it as any malicious changes become visible almost immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Having the computational properties just described, a blockchain is an excellent data structure for securely storing large volumes of information for a very broad spec- trum of purposes including but not limited to smart contracts, digital health informa- tion, smart city and smart infrastructure status, financial macrotransactions as well as gaming and social media microtransactions, and insurance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' In fact, the blockchain as a data structure was initially part of the Bitcoin stack as described in Nakamoto (2008) or as later explored in cryptocurrency surveys as for instance Antonopoulos (2014) or Antonopoulos (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Since then, however it took a life of its own with numerous parties developing some version of the original blockchain for their own purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Blockchain is not the only recent computationally intensive development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Fields like numerical and distributed deep learning such as the training of multilyayer con- volutional and recurrent neural networks, complex systems simulation such as brain networks and protein-to-protein interaction networks, as large scale social network analysis are notorious for their quick scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' One response to the need for additional computational power was the development of hardware aiming at massive parallelism through special purpose GPUs along with the associated software which can take ad- vantage of such specialized hardware and can orchestrate the appopriate sequence of computations to derive the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Google TensorFlow, a low level frame- work whose primary unit is a tensor as explained among others in Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2016), namely a multidimensional array, belongs to this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The primary objective of this chapter is to concentrate and succintly present the ways TensorFlow and GPU computation in conjunction with blockchain can em- power applications in the domains of digital health and insurance market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' As a sec- ondary objective, the computational capabilities and the dataflow model of Tensor- Flow are analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Blockchain For Mobile Health Applications 3 Table 1 Notation of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' symbol meaning △= Definition or equality by definition ⟨sk⟩ Sequence with elements sk |⟨sk⟩| Sequence cardinality The remaining of this work is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' In section 2 the relevant scien- tific literature regarding blockchain, GPU computing, and their applications is briefly reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The properties of the blockchain as well as these of TensorFlow are de- scribed in section 3, whereas the blockchain applications in the domains of digital health and insurance are explored in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The main findings of this chapter as well as possible future research directions are stated in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Finally, table 1 summarises the notation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 2 Previous Work Blockchains were formally introduced in the seminal Bitcoin work of Nakamoto (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Their technological innovation and the potential to become a disruptive tech- nology was explored among others in Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2012) and in Cachin (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The combination of blockchains with the IoT and their applications to the mainstream in- dustrial sector in conjunction with the upcoming digital transformations of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='0 are the focus of Miller (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Practical ways and the associated challenges to implement a blockchain over IoT and edge computing are shown in Zyskind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The financial prospects of Bitcoin in terms of wealth accumulation as well as the properties of Bitcoin versus the traditional fiat currency are the focus of a num- ber of works, for instance Antonopoulos (2014), Antonopoulos (2017), Kosba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2016), Swan (2015), and B¨ohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The distributed implementation of blockchains is discussed in Abbas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2018) and in Pass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2017), whereas se- curity aspects of the blockchains are treated in Puthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' A large number of blockchains besides the Bitcoin stack can be found in Underwood (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Since the original public description of TensorFlow in Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2016) and in Abadi (2016) it was widely adopted from the deep learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='In Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2017) a Gaussian process generator implemented with rudimentary TensorFlow op- erations is described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' For a new graph resilience metric based on paths see Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2018b) along the lines of the regularization methodology of Kanavos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The advantages of and the ways for visualising the TensorFlow computations are Wongsuphasawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' For tensor applications in social net- work analysis such as multiway digital influence estimation see Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2017), community structure discovery Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2018a), and graph based k-means initialization Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Finally, for a genetic algorithm for clustering tensors containing linguistic and spatial data see Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 4 Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 3 Parallelism and Blockchain 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='1 Blockchains As their collective name suggests, from a structural point of view blockchains are, typically very long, linearly linked nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Each such node contains part of the post- marked information stored in the data structure along with some administrative in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The data stored in a blockchain can never be erased, although it can be updated provided all interested parties agree on that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Thus, both the original and the updated data are stored, making audits efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Perhaps the most important advantages of selecting a blockchain scheme besides the increased security are the following: – Blockchains support a very large volume of transactions which can take place almost simultaneously because of their very inherent distributed nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' There- fore, their stakeholders can perform any desired number of transactions within a very reasonable amount of time without worrying about the exact transaction execution time, which in certain cases may influence the transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – The stakeholders of a given blockchain can stay informed of the global status of the blockchain in almost real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Thus, not only can they perform transactions but they can also know their results almost immediately or at least at the moment the latter are actually executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Blockchains, either public or private, offer full transparency since every partic- ipant to a given blockchain is free to validate any tranaction which took place within that blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Additionally, the verification protocols are deliberately built so that verification be easy even for netizens with low computational re- sources, for instance a smartphone or a tablet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' This reinforces the trust toward properly implemented and managed blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – In the case of a catastrophic loss, a properly implemented blockchain can at least partially rebuild itself from the segments stored at the computers of its stakehold- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' This is feasible given the increased redundancy integrated into a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – From a software engineering viewpoint, each blockchain node is a relatively sim- ple construct and, therefore, it can be managed with little or no human interven- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Thus, a blockchain administrator is only required to control certain a few key parts of the data structure, making blockchains easy and inexpensive to maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Last but not least, any third parties and intermediaries are no longer necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The interested parties can directly communicate and get current quotes or any other vital pieces of information from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Notice that blockchains are not immune to various sophisticated attacks, although the latter typically require considerable resources which are nowadays well within the capabilities of a dedicated hacker group or of a government agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Although directly attacking the encryption protocols may not be a wise course of action, unless some knowledge of the private key is available, using a zero day exploit is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Blockchain For Mobile Health Applications 5 As with any new technology, blockchain management software is by no means error free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' However, most known attacks so far take on a completely different approach akin to a brute force attack by relying on big botnet networks in order to take charge of a small or medium sized blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Yet another method, holistic in nature, for attacking a blockchain is through the use of control theory concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The current state, in any way that is estimated by the attacker, of a large blockchain is represented as a control vector x[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Then a usually linear state space model is formulated as follows: x[n + 1] △=Ax[n]+ bu[n] y[n + 1] △=cTx[n + 1]+ du[n] (1) If the attacker can insert an appropriate input sequence ⟨uk⟩, then, depending on the modelling correctness, he may bring the entire system to an undersirable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Of course, such a sequence may not exist or its cardinality |⟨uk⟩| might approach infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content='2 TensorFlow Google TensorFlow is a low level programming framework based on the dataflow programming paradigm and using tensors, namely multidimensional arrays, as its primary data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Originally developed for simulating brain networks, it is a powerful tool for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' It has official APIs for Python and C++, whereas unofficial APIs are being developed for a number of established programming lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Moreover, it has computational kernels for CPUs, GPUs, and TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Besides the methods for elementary operations such as Kronecker and Hadamard tensor multiplication, minimum location, tensor reshaping, and tensor factorizations such as Kruskal and Tucker decompositions, TensorFlow has a number of numerical optimizers which are common in deep learning such as AdaGrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Also, TensorFlow supports checkpoints, allowing the early termination of a training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Within a blockchain context, TensorFlow can accelerate numerical computations for hashing or encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Additionally, it can be used to train a neural network, recurrent, convolutional, autoencoding, or otherwise, which can predict the volume in the immediate future, so that a bursty load of transactions can be better balanced throughout the blockchain nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Moreover, similar networks can be built in order to predict which blockchain user will be the next to generate a chunk of data or will ask for a transaction verification, again for load balancing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Finally, large deep learning networks can in theory be deployed in order to mount an attack on the encryption protocol used by a given blockchain, but to the best of the knowledge of the authors, no such use has been recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 6 Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 4 Applications The blockchain as a ledger structure, either public or private, because of its secure and distributed design is a place for storing sensitive data such as health condition and financial transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Additionally, as stated earlier any intermediaries are elim- inated, at least in a higher level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Thus, any fees and premiums such as taxes or bank processing fees are also, in theory at least, automatically gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Regarding the digital health world, blockchain-based applications have an enormous potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The following list contains some of the most prominent ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – The medical records of a netizen can be stored with safety in a blockchain and can be recovered only by the certified health professional who cure the netizen regardless of their location or whether they have cured her before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Blockchain can facilitate automated monitoring of selected biomarkers by smart- phones and the measurements can be compared against personalized baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Netizens have much improved control over their personal records and their con- sent can be obtained under more transparent and clear conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Netizens can use micropayments or mobile payments in order to procure medicines, further protecting their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Concerning the growing insurance market, there is also a significant room for blo- ckchain-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Some indicative are the following: – Netizens can search easier for attractive offers and can contact insurance agents directly in order to negotiate for even better offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' This can also be done through software agents configured to look specific offers or terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Netizens and insurance agents can hold smart contracts such as property and ve- hicle electronic contract purchses in blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' At a later point, should the need arise, they can directly renegotiate contract terms which will be also recorded in the blockchain, provided the interested parties reach an agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Once smart contracts are recorded, ordinary shallow or deep learning algorithms can be run atop the blockchain in order to identify possible fraud cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Blockchains simplify considerably payments and can even be combined with mo- bile payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Payment records remain immutable and constitute proof that a payment indeed took place at the time indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' – Claims can be automatically verified by smartphones and other personal devices which are connected to the blockchain, reducing thus the administrative burden and the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' At this point it should also be reminded that the general advantages of section 3 also hold in addition to those listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' 5 Conclusions The twofold epicenter of this chapter was the blockchain applications in the domains of digital health and insurance market and the ways Google TensorFlow, a low level Blockchain For Mobile Health Applications 7 computational framework for computationally intensive applications, can be used to accelerate the associated computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' The blockchain has numerous applications in the domains of medical healthcare and insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Moreover, it reinforces the privacy and transparency conditions and, thus, help establish a viable and scalable market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Further research directions include the developmentof extensively tested blockchain management systems so that most, if not all, zero day exploits are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' More- over, given the recent advances in quantum computing which make large scale brute force attacks feasible, stronger cryptographic schemes should be sought in order to protect the sensitive personal data stored in blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' Acknowledgements We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' References Abadi M (2016) TensorFlow: Learning functions at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' ACM SIGPLAN Notices 51(9):1–1 Abadi M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' (2016) TensorFlow: A system for large-scale machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' In: OSDI, vol 16, pp 265–283 Abbas N, Zhang Y, Taherkordi A, Skeie T (2018) Mobile edge computing: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} +page_content=' IEEE Internet of Things Journal 5(1):450–465 Antonopoulos AM (2014) 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfzAun/content/2301.04725v1.pdf'} diff --git a/TdAyT4oBgHgl3EQf8foB/content/tmp_files/2301.00855v1.pdf.txt b/TdAyT4oBgHgl3EQf8foB/content/tmp_files/2301.00855v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..004a19788ece677a8f0b7f406d93265f8dd82dd5 --- /dev/null +++ b/TdAyT4oBgHgl3EQf8foB/content/tmp_files/2301.00855v1.pdf.txt @@ -0,0 +1,702 @@ +Experimental Status of Jets in Heavy-Ion Collisions +Jaime Norman1,∗ +1University of Liverpool, Oliver Lodge Laboratory, Oxford St, Liverpool L69 7ZE +Abstract. Jet quenching has been one of the most important indicators that +ultra-relativistic heavy-ion collisions produce a deconfined state of quarks and +gluons, known as the Quark-Gluon Plasma. While the quenching of jets tra- +ditionally refers to the energy loss of high-momentum partons, the study of +jet quenching has grown into a multi-pronged field where the measurement of +jets and their modification in heavy-ion collisions is used as an important tool +to study many aspects of QCD deconfinement. This contribution reviews the +current experimental status of jets at the LHC and RHIC, and reports recent +experimental highlights. +1 Introduction +The collision of ultra-relativistic heavy-ion collisions at the LHC and RHIC generates tem- +peratures hot enough to create a deconfined state of quarks and gluons, the Quark-Gluon +Plasma (QGP). Measurements up to now have enabled a detailed study of the QGP, which +has been determined to exist as a low-viscocity, collectively-expanding, strongly-interacting +fluid. The nature of the fundamental degrees of freedom within the QGP, and how a strongly- +interacting fluid emerges from the asymptotically free gauge theory of QCD, is however an +open question. +One of the most powerful probes of the QGP at a range of length scales are QCD jets +- high-energy partons (quarks or gluons) which are observed as a high-energy ‘spray’ of +hadrons. Jets are produced in hard-scattering processes at the start of particle collisions, and +their production in vacuum (in e+e−/ep collisions) is well understood in QCD. This makes +them a useful, ‘calibrated’ probe with which the QGP can be studied, where the in-medium +evolution and parton shower of the jet is modified at all stages of the QGP lifetime. The +study of jets in heavy-ion collisions aims to address some of the most pressing questions in +the study of the different phases of nuclear matter: what are the emergent ‘bulk’ properties +of QCD matter at high temperatures? By what physical mechanisms does a partonic ‘probe’ +interact with this matter? Can the individual degrees of freedom of deconfined QCD matter +be resolved with jets, and what is their nature? +Theoretical description of the propogation of jets through the QGP requires a consistent +description of the jet production, the parton shower and its thermalisation in the medium, +parton-medium interactions, plus relativistic hydrodynamic evolution of the medium and its +response to the propagating jet. An overview of jet quenching theory and recent developments +can be found in [1]. +∗e-mail: jknorman@liverpool.ac.uk +arXiv:2301.00855v1 [nucl-ex] 2 Jan 2023 + +2 Experimental considerations +The most significant challenge in measuring jets in heavy-ion collisions is the huge ‘under- +lying event’ created in these collisions, i.e. particles coming from sources uncorrelated to the +hard scattering in which the jet is produced. This leads to a given fraction of the jet trans- +verse momentum (pT) originating from these uncorrelated sources which must be corrected +for. In addition, at small jet energies it is expected that a non-negligible fraction of jets con- +tain constituents which do not originate at all from any hard scattering, but instead are made +up of completely uncorrelated sources (also referred to as ‘fake jets’ or ‘combinatorial jets’). +Techniques have been developed to correct for both cases. +The underlying event can be subtracted from the jet using an event-by-event procedure. +ALICE generally estimates the underlying event of a jet as the median underlying event +density of the full event [2]. Other approaches (see e.g. [3]) take into account the event- +plane dependence of the underlying event due to flow effects. It has also recently been shown +that the underlying event density resolution can be improved by calculating it on a jet-by-jet +basis, using a regression Neural Network trained on simulated jets embedded into a heavy-ion +background [4]; A preliminary measurement from this method is shown in section 3.1. +Combinatorial jets can be suppressed in different ways. The most simple is to impose a jet +pT or leading hadron pT cut to suppress the combinatorial background. One drawback here +is that this does restrict measurements to higher pT jets, or bias the fragmentation pattern +of the jet, respectively. To push to lower pT, statistical techniques have been developed +to subtract the uncorrelated background. One is a ‘mixed event’ technique, where tracks +are randomly selected from many events to generate a mixed event background which is +subtracted from the measured distribution(s) (see e.g. [5, 6]). Another technique has been +developed where the difference is taken between two ‘triggered’ distributions containing the +same uncorrelated background component (in particular, a trigger(hadron/γ/π0)-normalised +recoil jet distributions in a lower trigger pT interval is subtracted from the same distribution +in a higher trigger pT interval, see e.g. [7], also the hadron-jet measurement in section 3.2). +In order to compare to theory, one has to factor into account that reconstructed jet ob- +servables are smeared by detector inefficiencies and resolution effects (in heavy-ion and pp +collisions). In heavy-ion collisions, jets are additionally smeared due to underlying event +fluctuations. Nowadays, most measurements correct for these effects by unfolding the detec- +tor level quantities to particle level, using statistical unfolding techniques. +3 Results +Presented in the following are experimental highlights from the LHC and RHIC. The +measurements are grouped into three broad catagories: inclusive jet measurements, semi- +inclusive jet measurements and jet substructure. Searches for jet quenching effects in small +systems are then briefly discussed. +3.1 Modification of inclusive jets +The nuclear modification of jets has been measured by CMS [3], ATLAS [8] and ALICE [9] +at the LHC, and STAR [10] at RHIC. Measurements have shown jets to be suppressed in AA +collisions with respect to pp collisions, indicating the jet loses significant energy outside the +jet cone. A suppression persists up to a pjet +T of ∼ 1 TeV/c in central Pb–Pb collisions [3, 8]. +How this energy is redistributed can be studied further by measuring the R (jet radius param- +eter) dependence of jet production. Figure 1 (left) shows the ratios of the jet nuclear mod- + +20 +40 +60 +80 +100 +120 +140 +) +c +(GeV/ +T, ch jet +p +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 += 0.2) +R +( +AA +R += 0.6)/ +R +( +AA +R +R +| < 0.9- +jet +η +, | +T +k +Ch-particle jets, anti- += 5.02 TeV +NN +s +ALICE Preliminary, 0-10% Pb-Pb +ALICE Data +LIDO +ALI-PREL-511674 +Figure 1. Left: Ratios of the nuclear modification factor RAA for different jet radii R, in pjet +T regions +300-400 GeV/c, 400-500 GeV/c and 500-1000 GeV/c, measured by CMS in central 0-10% Pb–Pb +collisions [3]. Right: The ratio of the nuclear modification factor RAA(R = 0.6)/RAA(R = 0.2), measured +by ALICE in central 0-10% Pb–Pb collisions. +ification factor RAA 1 for different jet R measured by CMS as a function of the jet R in the +numerator of the ratio [3]. The increased statistics in this dataset allowed for measurements +at higher pT and also larger jet R (up to R = 1.0) than previous measurements. This measure- +ment indicates that the amount of jet suppression has minimal dependence on R. ALICE has +measured large-R jet production (up to R = 0.6) at lower pT than ATLAS and CMS - figure 1 +(right) shows the ratio of the nuclear modification factors RAA(R = 0.6)/RAA(R = 0.2), where +a hint of a stronger suppression of larger R jets is seen. It is noted however that these mea- +surements display tension with an ATLAS measurement [11] which shows less suppression +for larger R jets, and further measurements are needed to resolve this tension. Jets tagged +for their beauty flavour content have been measured by ATLAS [12], which suggest that the +RAA for b-jets is larger than that for inclusive jets, indicating that b-jets lose less energy than +inclusive jets. A recent measurement by ALICE of charm-tagged jets also shows that there is +a hint that charm-jets lose less energy than inclusive jets. +Measurements can be performed aiming to study the path-length and event-shape depen- +dence of jet production. The centrality dependence of jet azimuthal anisotropy vn has been +measured by ATLAS for the 2nd, 3rd and 4th order cumulants [13], where the v2,3,4 values +follow a similar trend with centrality for measurements of vn that are driven by hydrodynam- +ics, indicating that event geometry plays a significant role in jet quenching. CMS measured +1The nuclear modification factor RAA is defined as the ratio of jet yields in AA collisions with respect to pp +collision, RAA = +dNAA/dpT +⟨Ncoll⟩dNpp/dpT , where the pp yields are scaled by the average number of binary nucleon-nucleon +collisions in the collision ⟨Ncoll⟩. An RAA < 1 indicates a suppression in AA collisions with respect to pp collisions. + +CMS +0-10% +PbPb404μb,pp27.4pb1 +300

5 GeV/ +lead track +T +p +V0C +2 +q +30% large +V0C +2 +q +30% small +Out-of-Plane/In-Plane +ALI-PREL-503397 +Figure 2. Ratio of R = 0.2 jet yields measured in-plane and out-of-plane, for high and low q2 events, +measured in semi-central 30 − 50% Pb–Pb collisions by ALICE. +a positive dijet v2 which follows a similar trend to the inclusive jet v2 with centrality, and a v3 +and v4 consistent with 0 [14]. ALICE performed a new measurement where events are classi- +fied by centrality and their anisotropy. This is done by calculating the ’second-order harmonic +reduced flow vector’ q2 = |(�M +i=1 cos(2φi), �M +i=1 sin(2φi))|/ +√ +M, where M is the event multi- +plicity, the sum is over all tracks in the event and φi is the azimuthal angle of track i. In this +case events with large q2 are less anisotropic, and events with small q2 are more anisotropic. +Figure 2 shows the ratio of jet yields measured in-plane and out-of-plane, for high and low +q2 events in semi-central (30 − 50%) collisions. While the ratio is consistent with unity for +small q2 (isotropic) events, the ratio is less than unity for large q2 (anisotropic events) indicat- +ing that jets are more suppressed out-of-plane with respect to in-plane in highly anisotropic +events. +3.2 Modification of semi-inclusive jets +While inclusive jet measurements are the most straightforward, the drawback is that the initial +energy and/or flavour of the jet can remain unconstrained. More stringent constraints on the +initial energy of the jet and flavour of the jet can be obtained using semi-inclusive coincidence +measurements - the measurement of jets recoiling from another object determined to originate +from the same hard scattering. These measurements also allow to define an axis to study the +deflection of jets due to in-medium scattering. These measurements can include: +• photon-jet or electroweak boson (W/Z)-jet coincidence: Since photons and electroweak +bosons do not interact strongly, they do not lose energy when traversing the QGP, and can +thus be used to tag/constrain the momentum transfer of the hard scattering. +• hadron - jet coincidence: Here a high-pT hadron is used as a proxy for a jet, and the +recoiling jets are measured. +Figure 3 (top left) shows the momentum imbalance xjγ = pγ +T/pjet +T for γ-tagged jets mea- +sured by ATLAS [15]. CMS measured the momentum imbalance pγ +T/pjet +T , azimuthal angle +difference ∆φγ, jet and pjet +T distribution [16], where a significant shift in the momentum im- +balance in Pb–Pb with respect to pp is seen, also indicating significant jet energy loss in +Pb–Pb collisions. ALICE recently measured xjγ for lower-pT jets, shown in figure 3 (top +right)(not unfolded due to statistical limitations). In this measurement no significant modi- +fication is seen in central collisions with respect to peripheral collisions. The production of + +ALI-PREL-511826 +Figure 3. Top left: the momentum imbalance xjγ = pγ +T/pjet +T for jets in central 0-10% Pb–Pb collisions +measured by ATLAS [15]. Top right: the truncated mean of the momentum imbalance x jγ = pγ +T/pjet +T as +measured by ALICE as a function of centrality. Bottom: The γ-tagged jet RAA as measured by ATLAS, +compared with the inclusive jet RAA. +jets in coincidence with photons or electroweak bosons are also significantly more likely to +be initiated by a quark (through e.g. Compton scattering gq → qγ) than inclusive jets, which +are dominated by gluon-initiated jets, and thus can be used to constrain quark/gluon energy +loss. ATLAS recently measured the RAA of γ-tagged jets [17], which is shown in figure 3 +(bottom). γ-tagged jets are measured to be less suppressed than inclusive jets, indicating +quarks lose less energy than gluons. +ALICE recently measured jets recoiling from a trigger hadron in Pb–Pb and pp collisions, +building on previous measurements by ALICE [7] and STAR [5]. This measurement utilised +statistical techniques [7] to remove the combinatorial jet background, allowing to measure jets +down to very low (∼ 10 GeV/c) pjet +T . Figure 4 (left) shows the distribution of the azimuthal +angle between the trigger hadron and jet for R = 0.4 jets in selected pjet +T intervals. It is shown +that for 10 < pjet +T < 20 GeV/c jets the ∆ϕ distribution is significantly broadened in Pb–Pb +collisions with respect to pp collisions. This broadening is accompanied with an overall +yield enhancement in the back-to-back region (|∆ϕ − π| < 0.6). The same observation of +azimuthal broadening was made by STAR in γ-jet and π0-jet correlations, which is shown +in figure 4 (right). It is noted that CMS measured the azimuthal angular distribution of γ-jet +correlations [16] for larger jet pT and found consistency in pp and Pb–Pb collisions, further +suggesting that the broadening effect occurs just for low-pT jets. + +("Txp/Np)(*N/ L) +1.6 +ATLAS +1.4 +5.02 TeV, 0.49 nb-1 +1.2 +p=63.1-79.6GeV +Pb+Pb0-10% +JEWEL+PYTHIA +0.8 +Hybrid +0.6 +BDMPS-Z +(9=2-8 GeV2/fm) +0.4 +0.2 +(g = 2.0-2.2) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +5Xvfilsingo +0-3000 +20-000 +0.0 +er > O IC' >8 +. > i S.0= -is +Cugiaeq-bgificje lef2 +O'S +T( +S250 GeV, Iml< 2.37 +10 - 30% +Imell < 2.8, △Φ(,jet) > 元/2 +30 - 80% +0 +60 +80 +100 +120 +140 +160 +y-tagged jet p. [GeV]4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +1 +1 +− + rad) +× + +c + (GeV/ +recoil +∆ +ALICE Preliminary + = 5.02 TeV +NN +s +T +k +Ch-particle jets, anti- +| < 0.5 +jet +η + = 0.4, | +R +TT(20,50) - TT(5,7) +c + < 20 GeV/ +ch +T,jet +p +10 < +1.6 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 + (rad) +ϕ +∆ +1 +10 +Pb / pp +− +Pb +1 +1 +c + < 30 GeV/ +ch +T,jet +p +20 < +10 % +− +Pb 0 +− +Pb +pp +Sys. uncertainty +1.6 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 + (rad) +ϕ +∆ +1 +1 +1 +c + < 50 GeV/ +ch +T,jet +p +30 < +1.6 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +3 + (rad) +ϕ +∆ +1 +10 +ALI−PREL−507953 +Figure 4. Left: The background-subtracted hadron+jet trigger normalised recoil jet distributions in cen- +tral 0-10% Pb–Pb collisions and pp collisions, as a function of the azimuthal angle between the trigger +and the jet ∆ϕ, for different pjet +T intervals, measured by ALICE. Right: The γ-jet and π0-jet trigger- +normalised recoil jet distribution as a function of ∆ϕ in central 0-15% Au–Au collisions, measured by +STAR. +3.3 Modification of jet substructure +The time evolution of jets can be visualised as an initial parton-parton scattering, followed +by hard parton splittings, followed by successive, softer splittings, finally followed by hadro- +nisation. The study of the substructure of the jet can give important information on how the +core of the jet, and how the splittings, are modified by the QGP. Recent developments in +jet substructure have developed techniques where a reconstructed jet is reclustered and the +clustering procedure is rewound to isolate the ‘subjets’ within the jet. This technique aims +to study the jet splittings by filling the ‘Lund Plane’ [18, 19] which is a representation of the +relative angular and transverse momentum of a radiative emission with respect to its emitter +(here, jet splittings). Grooming techniques can be used to separate out the hard jet core and +hard parton splittings from the softened constituents and medium response. These techniques +have been proposed [20] to study different aspects of jet quenching such as whether the subjet +structure is resolved by the medium, how the medium response is redistributed, and also how +the space-time evolution of the jet is modified. +ATLAS and ALICE have measured the groomed jet radius corresponding to the distance +between the leading and sub-leading sub-jets within a jet, rg = +� +∆η2 +1,2 + ∆ϕ2 +1,2 [21, 22]. +Shown in figure 5 (left) is the measurement of the RAA as a function of rg for groomed jets +in four pjet +T intervals by ATLAS. This result shows that jets are narrowed in Pb–Pb collisions, +or wider jets are found to be more suppressed than narrower jets, a result that is also seen by +ALICE at lower jet pT. ALICE also measured the groomed jet momentum splitting fraction +defined as zg = +pT,subleading +pT,leading+pT,subleading , shown in figure 5 (right) [22]. This result indicates that there +is minimal modification to the relative pT scale of leading and subleading subjets. +3.4 Search for jet quenching in small systems +One of the most significant surprises from the LHC physics program is that collective effects +traditionally seen as indicative of a deconfined state of quarks and gluons have been measured +in smaller collision systems (p–Pb and pp). It is therefore an important question whether the +quenching of jets also extends to these collision systems. Measurements of inclusive jet +production [23] and semi-inclusive h+jet production [24] in p–Pb collisions as a function of +the centrality of the collision, and b-jet production in p–Pb collisions [25], do not provide +any evidence for jet quenching. ATLAS recently measured the near-side and away-side yield + +[GeV/c]-1 [rad]-1 +STAR Preliminary +Au+Au V snn = 200 GeV, 0-15% ++jet +10-1 + 元°+jet +PYTHIA-8 +up (Φ V)p +10-2 +10-3 +T,jet +anti-kt,R=0.5 +10-4 +L +10< +ch +<15GeV/c +10-5 +jet +A-8 +10 +Data +PYTHI +10- +2 +2.5 +3 +AΦ (= Φ +.,- Φ.)[rad] +trig +letALI-PUB-521472 +Figure 5. Left: The RAA as a function of the groomed jet radius rg in central 0-10% Pb–Pb collisions +measured by ATLAS [21]. Right: The groomed jet momentum splitting fraction zg for R = 0.2 jets in +central 0-10% Pb–Pb collisions and pp collisions measured by ALICE [22]. +of charged hadrons correlated with reconstructed jets as a function of the p–Pb collision +centrality [26]. The ratios of these yields in pp and p–Pb collisions IAA is consistent with MC +generator AGANTYR which does not include any final state effects producing collectivity or +jet quenching, thus also providing no evidence for jet quenching in p–Pb collisions. +4 Summary and Outlook +Many new insights into the QGP have been obtained from recent measurements of jets at +the LHC and RHIC. Significant modification to jet kinematics and jet substructure have been +measured. To date no evidence for jet quenching in small systems (pp and p–Pb) has been +seen. The measurement highlights shown in this contribution offer constraining power to +theoretical calculations, which use different approaches to calculate jet production and modi- +fication in heavy-ion collisions. Recent approaches to estimate transport properties, connect- +ing theory and experiment, have been performed (see e.g. Bayesian parameter estimation +of single particle spectra to estimate the jet transport coefficient ˆq from JETSCAPE [27]). +Such methods offer a promising way to combine multiple jet measurements and rigorously +compare with theory to extract quantitative information about the QGP. +Run 3 at the LHC has begun after major upgrades to the LHC experiments and a heavy- +ion run is scheduled for 2023. At RHIC, sPHENIX is a significant experiment upgrade which +will begin collecting data in 2023, complementary to the LHC program. The next few years +will thus open up a more precise era in the measurement of jets, allowing to study the QGP +with unprecedented accuracy. +References +[1] L. Apolinário (2022), these proceedings +[2] B. Abelev et al. (ALICE), JHEP 03, 053 (2012), 1201.2423 +[3] A.M. Sirunyan et al. (CMS), JHEP 05, 284 (2021), 2102.13080 + +a +S.0 +0'3 +0'4 +2.0 +8.0 +J'S +162 +2ldsq +8 +bp-bp +bgpjo2 +-1G2 +14 +bspjo2 +CUIGU +Cgncg] +TEMEL、LGCO!2 OU +E2CVbE +EMEF' LGcoli2 ot +S +4 +rgaaeg +rgaaeg +88.0 = +18.0 +AA +bb +Q +20tf Dlob cnt=0's' =0 +' cμ lef +eoIo.m/ ,S.0 = +!Uc +bp-bp 0-↓0oo +or +bb +V9T SO. = +VTICE +pA +1.2 +ATLAS Preliminary +R +0 - 10 % +anti-kt R = 0.4 jets, lyl < 2.1 +pp 5.02 TeV, 260 pb-1 +Pb+Pb 5.02 TeV, 1.72 nb-1 +Zcut = 0.2, β = 0 +0.8 +0.6 +et + 158 GeV +0.4 +jet +158 < +< 200 GeV +200

0 islands’, [21, Theorem 2]. Theorem 1.2 in [12] has also been applied +to obtain a positive mass theorem for asymptotically hyperbolic manifolds with +boundary; see [9]. This theorem will be a useful tool in the present work as well. +In this paper, we present some further initial data rigidity results for compact +initial data sets, in both the boundary and no boundary cases. In [15], the authors +considered 3-dimensional initial data sets containing spherical MOTS. It was shown, +roughly speaking, that in a matter-filled spacetime, perhaps with positive cosmo- +logical constant, a stable marginally outer trapped 2-sphere must satisfy a certain +area inequality; namely, its area must be bounded above by 4π/c, where c > 0 is +a lower bound on a natural energy-momentum term. We then established rigidity +results for stable, or weakly outermost, marginally outer trapped 2-spheres when +this bound is achieved. In particular, we prove a local splitting result, [15, Theo- +rem 3.2], that extends to the spacetime setting a result of H. Bray, S. Brendle, and +1 +arXiv:2301.00639v1 [gr-qc] 2 Jan 2023 + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +2 +A. Neves [8] concerning area minimizing 2-spheres in Riemannian 3-manifolds with +positive scalar curvature. These spacetime results have interesting connections to +the Vaidya and Nariai spacetimes [15]. +One of the main aims of the present work is to obtain a global version of [15, +Theorem 3.2]; see Theorem 3.1 in Section 3 for a statement. +The proof makes +use of certain techniques introduced in [12]. In this work, we have also been led +to consider certain variations of [12, Theorem 5.2]; see Theorems 3.2 and 3.3 in +Section 3. Here, it becomes useful to consider the so-called ‘brane action’, as well +as the area functional. +These results are then used to examine the question of +the existence of MOTS in closed (compact without boundary) initial data sets in +Section 4. The relationship to known spacetimes is also discussed. +The paper is organized as follows: in Section 2, we review some background +material on MOTS; in Section 3, we state and prove several global rigidity results +for compact-with-boundary initial data sets; and, in Section 4, we apply the results +obtained in Section 3 to prove some global rigidity statements for closed initial data +sets. In Section 4, we also give various examples in order to illustrate the results +presented in this paper. +Acknowledgements. The work of GJG was partially supported by the Simons +Foundation, under Award No. 850541. The work of AM was partially supported +by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq, +Brazil (Grant 305710/2020-6), the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de +N´ıvel Superior - CAPES, Brazil (CAPES-COFECUB 88887.143161/2017-0), and the +Funda¸c˜ao de Amparo `a Pesquisa do Estado de Alagoas - FAPEAL, Brazil (Process +E:60030.0000002254/2022). The authors would like to thank Ken Baker and Da +Rong Cheng for helpful comments. +2. Preliminaries +All manifolds in this paper are assumed to be connected and orientable except +otherwise stated. +An initial data set (M, g, K) consists of a Riemannian manifold (M, g) with +boundary ∂M (possibly ∂M = ∅) and a symmetric (0, 2)-tensor K on M. +Let (M, g, K) be an initial data set. +The local energy density µ and the local current density J of (M, g, K) are given +by +µ = 1 +2(S − |K|2 + (tr K)2) +and +J = div(K − (tr K)g), +where S is the scalar curvature of (M, g). We say that (M, g, K) satisfies the domi- +nant energy condition (DEC for short) if +µ ≥ |J| +on +M. +Consider a closed embedded hypersurface Σ ⊂ M. Since, by assumption, Σ and +M are orientable, we can choose a unit normal field ν on Σ. If Σ separates M, by +convention, we say that ν points to the outside of Σ. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +3 +The null second fundamental forms χ+, χ− of Σ in (M, g, K) with respect to ν +are given by +χ+ = K|Σ + A +and +χ− = K|Σ − A, +where A is the second fundamental form of Σ in (M, g) with respect to ν. More +precisely, +A(X, Y ) = g(∇Xν, Y ) +for +X, Y ∈ X(Σ), +where ∇ is the Levi-Civita connection of (M, g). +The null expansion scalars θ+, θ− of Σ in (M, g, K) with respect to ν are given by +θ+ = trΣ(K) + H +and +θ− = trΣ(K) − H, +(2.1) +where H = tr A is the mean curvature of Σ in (M, g) with respect to ν. Observe +that θ± = tr χ±. +R. Penrose introduced the now famous notion of a trapped surface, when both +θ+ and θ− are negative. Restricting to one side, we say that Σ is outer trapped if +θ+ < 0, weakly outer trapped if θ+ ≤ 0, and marginally outer trapped if θ+ = 0. +In the latter case, we refer to Σ as a marginally outer trapped surface (MOTS for +short). +Assume now that Σ is a MOTS in (M, g, K), with respect to a unit normal ν, +that is a boundary in M. +More precisely, assume that ν points towards a top- +dimensional submanifold M + ⊂ M such that ∂M + = Σ ⊔ S, where S (possibly +S = ∅) is a union of components of ∂M (in particular, if Σ separates M). We think +of M + as the region outside of Σ. Then we say that Σ is outermost (resp. weakly +outermost) if there is no closed embedded hypersurface in M + with θ+ ≤ 0 (resp. +θ+ < 0) that is homologous to and different from Σ. The notions of locally weakly +outermost and locally outermost MOTS can be given in an analogous way. +Remark 2.1. It is important to mention that initial data sets arise naturally in +general relativity. In fact, let M be a spacelike hypersurface in a spacetime, i.e. a +time-oriented Lorentzian manifold, ( ¯N, ¯h). Let g be the Riemannian metric on M +induced from ¯h and K be the second fundamental form of M in ( ¯N, ¯h) with respect +to the future-pointing timelike unit normal u on M. Then (M, g, K) is an initial +data set. As before, let Σ be a closed embedded hypersurface in M. In this setting, +χ+ and χ− are the null second fundamental forms of Σ in ( ¯N, ¯h) with respect to the +null normal fields +ℓ+ = u|Σ + ν +and +ℓ− = u|Σ − ν, +respectively. Observe that θ± = divΣ ℓ±. Physically, θ+ (resp. θ−) measures the +divergence of the outward pointing (resp. inward pointing) light rays emanating +from Σ. +An initial data set (M, g, K) is said to be time-symmetric or Riemannian if K = 0. +In this case, a MOTS in (M, g, K) is nothing but a minimal hypersurface in (M, g). +Moreover, the energy condition µ − |J| ≥ c, for some constant c, reduces to the +requirement on the scalar curvature S ≥ 2c. +Quite generally, marginally outer + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +4 +trapped surfaces share many properties with minimal hypersurfaces, which they +generalize; see e.g. the survey article [1]. +As in the minimal hypersurfaces case, an important notion for the theory of MOTS +is the notion of stability introduced, in the context of MOTS, by L. Andersson, +M. Mars, and W. Simon [2, 3], which we now recall. +Let Σ be a MOTS in (M, g, K) with respect to ν. Consider a normal variation of +Σ in M, i.e. a variation t → Σt of Σ = Σ0 with variation vector field +∂ +∂t|t=0 = φ ν, +φ ∈ C∞(Σ). Let θ±(t) denote the null expansion scalars of Σt with respect to νt, +ν = νt|t=0. Computations as in [2, p. 2] or [3, p. 861] give, +∂θ+ +∂t +���� +t=0 = Lφ, +(2.2) +where +Lφ = −∆φ + 2⟨X, ∇φ⟩ + (Q + div X − |X|2)φ +and +Q = 1 +2SΣ − (µ + J(ν)) − 1 +2|χ+|2. +Here, ∆ is the negative semi-definite Laplace-Beltrami operator, ∇ the gradient, div +the divergence, and SΣ the scalar curvature of Σ with respect to the induced metric +⟨ · , · ⟩. Moreover, X is the tangent vector field on Σ that is dual to the 1-form +K(ν, · )|Σ. +It is possible to prove (see [3, Lemma 4.1]) that L has a real eigenvalue λ1 = λ1(L), +called the principal eigenvalue of L, such that Re λ ≥ λ1 for any other complex +eigenvalue λ. +Furthermore, the corresponding eigenfunction φ1, Lφ1 = λ1φ1, is +unique up to a multiplicative constant and can be chosen to be real and everywhere +positive. +Then a MOTS Σ is said to be stable if λ1(L) ≥ 0. This is equivalent to the +existence of a positive function φ ∈ C∞(Σ) such that Lφ ≥ 0. It follows directly +from (2.2) with φ = φ1 that every locally weakly outermost (in particular, locally +outermost) MOTS is stable. +Observe that in the Riemannian case, L reduces to the classical stability operator, +also known as the Jacobi operator, for minimal hypersurfaces. +As such, in the +literature, L is known as the MOTS stability operator or the stability operator for +MOTS. +The study of rigidity results for minimal surfaces in Riemannian manifolds with +a lower scalar curvature bound has been, and continues to be, an active area of re- +search. From the point of view of initial data sets, these are time-symmetric results, +as noted above. It has been of interest to extend some of these results to general +initial data sets. In the context of general relativity, black hole horizons within +initial data sets are often modeled by MOTS, and, in particular, minimal surfaces +in the time-symmetric case. These rigidity results often shed light on properties of +spacetimes with black holes, as noted in the introduction. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +5 +The next proposition and theorem extend to the general non-time-symmetric set- +ting some results of Bray, Brendle, and Neves [8]. +Proposition 2.2 (Infinitesimal rigidity, [15]). Let Σ be a stable MOTS in a 3- +dimensional initial data set (M, g, K) with respect to a unit normal field ν. Suppose +there exists a constant c > 0 such that µ + J(ν) ≥ c on Σ. Then the area of Σ +satisfies, +A(Σ) ≤ 4π +c . +Moreover, if A(Σ) = 4π/c, then the following hold: +(a) Σ is a round 2-sphere with Gaussian curvature κΣ = c, +(b) the second fundamental form χ+ of Σ with respect to ν vanishes, and +(c) µ + J(ν) = c on Σ. +The proposition above is used in the proof of the following local splitting theorem. +But, before stating the next result, which is also used in the proof of Theorem 3.1, +let us remember the notion of an area minimizing surface. +With respect to a fixed Riemannian metric g on a 3-dimensional manifold M, a +closed embedded surface Σ ⊂ M is said to be area minimizing if Σ is of least area +in its homology class in M, that is, A(Σ) ≤ A(Σ′) for any closed embedded surface +Σ′ that is homologous to Σ in M. In this case, we also say that Σ minimizes area. +Similarly, Σ is said to be locally area minimizing if A(Σ) ≤ A(Σ′) for any such Σ′ +in a neighborhood of Σ in M. +Theorem 2.3 (Local splitting, [15]). Let (M, g, K) be a 3-dimensional initial data +set with boundary. Suppose that (M, g, K) satisfies the energy condition µ − |J| ≥ c +for some constant c > 0. Let Σ0 be a closed connected component of ∂M such that +the following conditions hold: +(1) Σ0 is a MOTS with respect to the normal that points into M and +(2) Σ0 is locally weakly outermost and locally area minimizing. +Then Σ0 is topologically S2 and its area satisfies, +A(Σ0) ≤ 4π +c . +Furthermore, if A(Σ0) = 4π/c, then a collar neighborhood U of Σ in M is such that: +(a) (U, g) is isometric to ([0, δ) × Σ0, dt2 + g0) for some δ > 0, where g0 - the +induced metric on Σ0 - has constant Gaussian curvature κΣ0 = c, +(b) K = a dt2 on U, where a ∈ C∞(U) depends only on t ∈ [0, δ), and +(c) µ = c and J = 0 on U. +This theorem extends to the general non-time-symmetric setting the local rigidity +statements in [8]. The local rigidity obtained in [8] is then used to obtain a global +rigidity result; see [8, Proposition 11]. In Theorem 3.1 in the next section, we obtain +a global version of Theorem 2.3. A key improvement in this global rigidity result +is that it does not require the ‘weakly outermost’ assumption, and hence parallels +somewhat more closely the global result in [8]. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +6 +Now, we recall two topological concepts that are important for our purposes; see +also [12]. +We say that M satisfies the homotopy condition with respect to Σ ⊂ M provided +there exists a continuous map ρ : M → Σ such that ρ ◦ i : Σ → Σ is homotopic +to idΣ, where i : Σ �→ M is the inclusion map (for example, if Σ is a retract of M). +On the other hand, a closed not necessarily connected manifold N of dimension m +is said to satisfy the cohomology condition if there are m classes ω1, . . . , ωm in the +first cohomology group H1(N), with integer coefficients, whose cup product +ω1 ⌣ · · · ⌣ ωm ∈ Hm(N) +is nontrivial. For example, the m-torus T m = S1 × · · · × S1 satisfies the cohomol- +ogy condition. More generally, the connected sums T m ♯ Q satisfy the cohomology +condition for any closed m-manifolds Q. A version of this condition is considered in +[23, Theorem 5.2]. Here, we are using the form of the condition as it appears in [19, +Theorem 2.28]. A manifold N satisfying this cohomology condition has a component +that does not carry a metric of positive scalar curvature; see the discussion in [19]. +We will make use of the following theorem (mentioned in the introduction) in +several situations. +Theorem 2.4 ([12, Theorem 1.2]). Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, +compact-with-boundary initial data set. Suppose that (M, g, K) satisfies the domi- +nant energy condition, µ ≥ |J|. Suppose also that the boundary can be expressed +as a disjoint union ∂M = Σ0 ∪ S of nonempty unions of components such that the +following conditions hold: +(1) θ+ ≤ 0 on Σ0 with respect to the normal that points into M, +(2) θ+ ≥ 0 on S with respect to the normal that points out of M, +(3) M satisfies the homotopy condition with respect to Σ0, and +(4) Σ0 satisfies the cohomology condition. +Then the following hold: +(a) M ∼= [0, ℓ] × Σ0 for some ℓ > 0. +Let Σt ∼= {t} × Σ0 with unit normal νt in direction of the foliation. +(b) χ+ = 0 on Σt for every t ∈ [0, ℓ]. +(c) Σt is a flat (n−1)-torus with respect to the induced metric for every t ∈ [0, ℓ]. +(d) µ + J(νt) = 0 on Σt for every t ∈ [0, ℓ]. In particular, µ = |J| on M. +The following is the basic existence result for MOTS due to L. Andersson and +J. Metzger in 3-dimensions, and M. Eichmair in dimensions 3 ≤ n ≤ 7. It is used in +the proof of Theorem 3.1, and is the source of the dimension restriction appearing +in various results discussed herein. +Theorem 2.5 (Existence of MOTS, [4, 10, 11]). Let (M, g, K) be an n-dimensional, +3 ≤ n ≤ 7, compact-with-boundary initial data set. Suppose that the boundary can +be expressed as a disjoint union ∂M = Σin ∪ Σout, where Σin and Σout are nonempty +unions of components of ∂M with θ+ ≤ 0 on Σin with respect to the normal pointing + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +7 +into M and θ+ > 0 on Σout with respect to the normal pointing out of M. Then +there is an outermost MOTS in (M, g, K) that is homologous to Σout. +3. The compact-with-boundary cases +In this section, we obtain several global initial data results. These results, in +turn, will be applied in the next section to the case that the initial data manifold is +closed. The first is a global version of Theorem 2.3; see the comments above, after +the statement of Theorem 2.3. +Theorem 3.1. Let (M, g, K) be a 3-dimensional compact-with-boundary initial data +set. +Suppose that (M, g, K) satisfies the energy condition µ − |J| ≥ c for some +constant c > 0. Suppose also that the boundary can be expressed as a disjoint union +∂M = Σ0 ∪ S of nonempty unions of components such that the following conditions +hold: +(1) θ+ ≤ 0 on Σ0 with respect to the normal that points into M, +(2) θ+ ≥ 0 on S with respect to the normal that points out of M, +(3) M satisfies the homotopy condition with respect to Σ0, +(4) the relative homology group H2(M, Σ0) vanishes, and +(5) Σ0 minimizes area. +Then Σ0 is topologically S2 and its area satisfies, +A(Σ0) ≤ 4π +c . +Moreover, if A(Σ0) = 4π/c, then the following hold: +(a) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + g0) for some ℓ > 0, where g0 - the +induced metric on Σ0 - has constant Gaussian curvature κΣ0 = c, +(b) K = a dt2 on M, where a ∈ C∞(M) depends only on t ∈ [0, ℓ], and +(c) µ = c and J = 0 on M. +Proof. First, observe that Σ0 is connected, since M is connected and satisfies the +homotopy condition with respect to Σ0. +If Σ0 is not homeomorphic to S2, then Σ0 is homeomorphic to T 2 ♯ Q for some +closed orientable surface Q. In particular, Σ0 satisfies the cohomology condition and +so Theorem 2.4 applies to (M, g, K). Therefore, 0 = µ − |J| ≥ c on M, which is a +contradiction. Then Σ0 is topologically S2. +Claim: Σ0 is a weakly outermost MOTS in (M, g, K) of area A(Σ0) = 4π/c +unless A(Σ0) < 4π/c. +Assume that A(Σ0) ≥ 4π/c. +If θ+ +K ≤ 0 is not identically zero on Σ0, it follows from [4, Lemma 5.2] that there is +a surface Σ ⊂ M - obtained by a small perturbation of Σ0 into M - such that θ+ +K < 0 +on Σ with respect to the normal pointing away from Σ0. Let W be the connected +compact region bounded by Σ and S in M. Observe that θ+ +−K ≤ 0 on S with respect +to the normal that points into W and θ+ +−K > 0 on Σ with respect to the normal that +points out of W. Applying the MOTS existence theorem (Theorem 2.5), we obtain + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +8 +an outermost MOTS Σ′ in (W, g, −K) that is homologous to and disjoint from Σ. +Clearly, Σ′ is homologous to Σ0 in M. +Without loss of generality, we may assume that each connected component of Σ′ +is homologically nontrivial in M. Also, because H2(M, Σ0) = 0, Σ′ is connected. +Since we are assuming that Σ0 minimizes area in its homology class, we have +4π +c ≤ A(Σ0) ≤ A(Σ′). +On the other hand, because Σ′ is an outermost MOTS in (W, g, −K), in particular +stable, the infinitesimal rigidity (Proposition 2.2) gives that A(Σ′) = 4π/c. There- +fore, Σ′ is an area minimizing outermost MOTS in (W, g, −K) of area A(Σ′) = 4π/c +and then the local splitting theorem (Theorem 2.3) applies so that an outer neigh- +borhood of Σ′ in W is foliated by MOTS, which is a contradiction. +This proves that Σ0 is a MOTS in (M, g, K). +Now, we claim that Σ0 is weakly outermost in (M, g, K). If not, there is a surface +Σ that is homologous to Σ0 in M and such that θ+ +K < 0 on it. Perturbing Σ a bit, +we may assume that Σ ∩ Σ0 = ∅. Also, by the strong maximum principle as in e.g. +[4, Proposition 2.4] or [5, Proposition 3.1], Σ ∩ S = ∅. +As before, without loss of generality, we may assume that each connected compo- +nent of Σ is homologically nontrivial in M and, in particular, Σ is connected. Let +W be the region in M bounded by Σ and S. Arguing with (W, g, −K) as above, we +have a contradiction. Thus Σ0 is weakly outermost. +We have then proved that, if A(Σ0) ≥ 4π/c, then Σ0 is a weakly outermost MOTS +in (M, g, K). In this case, by the infinitesimal rigidity, A(Σ0) = 4π/c. +This finishes the proof of the Claim. +We have then obtained that Σ0 is homeomorphic to S2 and its area satisfies +A(Σ0) ≤ 4π/c. +Furthermore, if A(Σ0) = 4π/c, then Σ0 is an area minimizing +weakly outermost MOTS in (M, g, K). In this case, by the local splitting theorem, +there is a collar neighborhood U ∼= [0, δ) × Σ0 of Σ0 in M such that conclusions +(a), (b), and (c) of the theorem hold on U. Clearly, Σt ∼= {t} × Σ0 converges to a +closed embedded MOTS Σδ of area 4π/c as t ↗ δ. If Σδ ∩ S ̸= ∅, by the strong +maximum principle, Σδ = S. If Σδ ∩ S = ∅, we can replace Σ0 by Σδ and M by the +complement of U and run the process again. The result then follows by a continuity +argument. +□ +The next two theorems make use of the notion of (n−1)-convexity of a symmetric +(0, 2)-tensor. Imposing such convexity leads to stronger rigidity. +We say that a symmetric (0, 2)-tensor P on (M, g) is (n − 1)-convex if, at every +point p ∈ M, the sum of the smallest (n − 1) eigenvalues of P with respect to g is +nonnegative (in particular, if P is positive semi-definite). This is equivalent to the +trace of P with respect to any (n − 1)-dimensional linear subspace of TpM being +nonnegative, for every p ∈ M. In particular, if P is (n − 1)-convex, then trΣ P ≥ 0 +for every hypersurface Σ ⊂ M. +This convexity condition has been used by the +second-named author in [22] and by the authors, together with M. Eichmair, in [12] +in related contexts. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +9 +Let (M, g, K) be as in Theorem 2.4, and let Σ be a closed embedded hypersurface +homologous to Σ0. The next theorem makes use of the functional, +Bϵ(Σ) = A(Σ) − (n − 1) ϵ V(Σ), +ϵ = 0, 1, +where A(Σ) is the area of Σ and V(Σ) is the volume of the region bounded by Σ +and Σ0. In the case ϵ = 0, we are just talking about the area functional. In the case +ϵ = 1, we are talking about the functional associated with hypersurfaces of constant +mean curvature n − 1, sometimes referred to as the brane action and denoted by B. +The following theorem extends in a couple of directions Theorem 5.2 in [12]. +Theorem 3.2. Let (M, g, K) be as in Theorem 2.4. Assume that +(i) K + ϵ g is (n − 1)-convex, where ϵ = 0 or ϵ = 1, and +(ii) Σ0 and S are such that Bϵ(Σ0) ≤ Bϵ(S). +Then the following hold: +(a) Σ0 is a flat (n − 1)-torus with respect to the induced metric g0, +(b) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + e2 ϵ tg0) for some ℓ > 0, +(c) K = (1 − ϵ)a − ϵ g on M, where a ∈ C∞(M) depends only on t ∈ [0, ℓ], and +(d) µ = 0 and J = 0 on M. +The convexity assumption holds if, in particular, K satisfies, K ≥ −ϵ g. In the +case ϵ = 0, this would apply to cosmological models that are expanding to the future +(in all directions). +Proof. By Theorem 2.4, +- M ∼= [0, ℓ] × Σ0 for some ℓ > 0, and +- each leaf Σt ∼= {t} × Σ0 is a MOTS with respect to the unit normal νt in +direction of the foliation. +On the other hand, since K + ϵ g is (n − 1)-convex, we have +H(t) − (n − 1) ϵ ≤ H(t) + trΣt K = 0, +(3.1) +where H(t) = divΣt νt is the mean curvature of Σt. +Now, the first variation of Bϵ gives that +d +dtBϵ(Σt) = +� +Σt +φ (H(t) − (n − 1) ϵ) dΣt ≤ 0, +(3.2) +where φ = ⟨νt, ∂t⟩ is the lapse function of the foliation. Therefore, Bϵ(t) = Bϵ(Σt) is +a nonincreasing function on [0, ℓ] satisfying Bϵ(0) ≤ Bϵ(ℓ). Thus Bϵ(t) = Bϵ(Σt) is +constant. Inequalities (3.1) and (3.2) give that H(t) = (n − 1) ϵ = − trΣt K for all +t ∈ [0, ℓ]. In particular, θ− = −2 (n − 1) ϵ on Σℓ = S. +The result then follows directly from [12, Theorem 1.3] (observe that our sign +convention in the definition of θ− in this work is the opposite of that one in [12]). +□ +In the next theorem we consider B = B1 under a modified convexity condition. +Theorem 3.3. Let (M, g, K) be as in Theorem 2.4. +Assume that −(K + g) is +(n − 1)-convex. +Then B(Σ0) ≤ B(S). +Moreover, if equality holds, we have the +following: + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +10 +(a) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + gt) for some ℓ > 0, where gt is the +induced metric on Σt ∼= {t} × Σ0. +(b) Each Σt is a flat (n − 1)-torus with respect to gt and has constant mean +curvature H(t) = n − 1. +(c) The scalar curvature of (M, g) satisfies S ≤ −n(n − 1). If equality holds, +(M, g) is isometric to ([0, ℓ] × Σ0, dt2 + e2 tg0). +(d) For each t ∈ [0, ℓ], µ + J(νt) = 0 on Σt. In particular, µ = |J| on M. +(e) tr K ≤ −n on M. If equality holds, K = −g, S = −n(n − 1), µ = 0, and +J = 0 on M. +The convexity assumption holds if, in particular, K satisfies, K ≤ −g. If one +views K as being defined with respect to the past directed unit normal, this would +apply to cosmological models that are strongly contracting to the past, e.g. that +begin with a ‘big bang’. +Proof. By Theorem 2.4, +- M ∼= [0, ℓ] × Σ0 with +g = φ2dt2 + gt, +(3.3) +where gt is the induced metric on Σt ∼= {t} × Σ0. +- Each (Σt, gt) is a flat (n − 1)-torus. +- Every leaf Σt is a MOTS in (M, g, K). In fact, +0 = χ+(t) = A(t) + K|Σt, +where A(t) is the second fundamental form of Σt computed with respect to +the unit normal νt in direction of the foliation. +- For each t ∈ [0, ℓ], µ + J(νt) = 0 on Σt. In particular, µ = |J| on M. +Now, since −(K + g) is (n − 1)-convex, we have +H(t) − (n − 1) ≥ H(t) + trΣt K = 0, +(3.4) +where H(t) = tr A(t) is the mean curvature of Σt. Then the first variation of B gives +that +d +dtB(Σt) = +� +Σt +φ (H(t) − (n − 1)) dΣt ≥ 0. +(3.5) +Therefore, B(t) = B(Σt) is a nondecreasing function defined on [0, ℓ]. In particular, +B(0) ≤ B(ℓ), that is, B(Σ0) ≤ B(S). +If B(Σ0) = B(S), then B(t) = B(Σt) is constant. Therefore, inequalities (3.4) +and (3.5) imply that H(t) = n − 1 = − trΣt K for all t ∈ [0, ℓ]. +Now, fix t ∈ [0, ℓ], p ∈ Σt, and let {e1, . . . , en−1} be an orthonormal basis for TpΣt. +Define +η(s) = cos s · en−1 + sin s · νt, +s ∈ R, +and let π(s) be the (n − 1)-dimensional linear subspace of TpM generated by +{e1, . . . , en−2, η(s)}. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +11 +Since −(K + g) is (n − 1)-convex and trΣt K = −(n − 1), we have +f(s) := trπ(s) K ≤ −(n − 1) +and +f(0) = trΣt K = −(n − 1). +Therefore, s = 0 is a critical point of f(s). Observing that +f(s) = +n−2 +� +i=1 +K(ei, ei) + K(η(s), η(s)), +we obtain, +0 = f ′(0) = 2K(η′(0), η(0)) = 2K(νt, en−1). +Analogously, K(νt, ei) = 0 for i = 1, . . . , n−2. This gives that X♭ = K(νt, · )|Σt = 0 +for all t ∈ [0, ℓ]. +On the other hand, the first variation of θ+(t) = 0 reads as +∂θ+ +∂t = −∆φ + 2⟨X, ∇φ⟩ + (Q + div X − |X|2)φ = −∆φ + Qφ, +where +Q = 1 +2SΣt − (µ + J(νt)) − 1 +2|χ+(t)|2 = 0. +Thus ∆φ = 0 on Σt and then φ = φ(t) is constant on Σt for each t ∈ [0, ℓ]. Hence, +by a simple change of variable in (3.3), we have +g = dt2 + gt. +(3.6) +In particular, the t-lines are geodesics. Hence, along each leaf Σ = Σt, H = H(t) +satisfies the scalar Riccati equation, +∂H +∂t = − Ric(∂t, ∂t) − |A|2, +which, since H(t) = n − 1, implies, +Ric(∂t, ∂t) + |A|2 = 0. +By the Gauss equation, we have the standard rewriting of the left-hand side in the +above equation, +Ric(∂t, ∂t) + |A|2 = 1 +2(S − SΣ + |A|2 + H2). +Hence, since SΣ = 0, we have, +S = −|A|2 − H2 ≤ − H2 +n − 1 − H2 = −n(n − 1), +(3.7) +which establishes the inequality part in (c). If equality holds, then |A(t)|2 = n − 1, +which, together with H(t) = n−1, implies that each Σt is umbilic; in fact, A(t) = gt. +Using this in (3.6) easily implies the isometry part in (c). +Since −(K +g) is (n−1)-convex, it is not difficult to see that tr K ≤ −n. In fact, +if {e1, . . . , en} is an orthonormal basis for TpM, p ∈ M, then +(n − 1) tr K = +n +� +i=1 +� +j̸=i +K(ej, ej) ≤ − +n +� +i=1 +(n − 1) = −n(n − 1), +(3.8) + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +12 +that is, tr K ≤ −n. If tr K = −n, it follows from (3.8) that +� +j̸=i +K(ej, ej) = −(n − 1) +for each +i = 1, . . . , n. +Therefore, +−n = tr K = K(ei, ei) + +� +j̸=i +K(ej, ej) = K(ei, ei) − (n − 1), +that is, K(ei, ei) = −1 for each i = 1, . . . , n. Since {e1, . . . , en} is arbitrary, we have +K = −g. Thus, using that A(t) = −K|Σt = g|Σt in (3.7), we obtain +S = −|A(t)|2 − |H(t)|2 = −n(n − 1). +Finally, +µ = 1 +2(S − |K|2 + (tr K)2) = 0 +and +J = div(K − (tr K)g) = 0. +□ +4. Applications: closed cases +In this section, we wish to apply the results of the previous section to initial data +manifolds that are closed (compact without boundary). +These results naturally +relate to cosmological (i.e. spatially closed) spacetimes. We’ll illustrate the results +with various examples. +4.1. The spherical case. In this section, we want to apply Theorem 3.1 to the +case that M is closed. +Let M be an n-dimensional closed manifold. Suppose the (n − 1)-th homology +group Hn−1(M) is nontrivial. +Any nontrivial element of Hn−1(M) gives rise to +a smooth closed embedded non-separating orientable hypersurface Σ ⊂ M. +In +particular, Σ is two-sided in M, i.e. there is an embedding F : [−1, 1]×Σ → M such +that F(0, p) = p for each p ∈ Σ. Let U denote the open set F((−1, 1)×Σ) ⊂ M. We +say that M is retractable with respect to Σ if M \ U retracts onto some component +of ∂U. If we consider a Riemannian metric g on M, given a unit normal field ν on +Σ with respect to g, we say that M is retractable with respect to Σ towards ν if +M \ U retracts onto the component of ∂U towards which ν points. +An obvious situation where this occurs is when M is of the form M = S1 × Q, +with Q closed. +Then M is retractable with respect to Σ = {x} × Q, x ∈ S1. +Another situation of interest is when M is of the form M = T n ♯ Q. View T n as an +n-dimensional cube with opposite boundary faces identified. To obtain M, we may +assume the connected sum takes place in a bounded open set U inside the cube. Let +Σ be an (n − 1)-torus parallel to one of the faces away from the set U. Then M is +retractable with respect to Σ. More generally, if M is retractable with respect to Σ, +then so is M ♯ Q, with Q closed, provided the connect sum takes place away from Σ. +Theorem 4.1. Let (M, g, K) be a 3-dimensional closed initial data set satisfying +the energy condition µ − |J| ≥ c for some constant c > 0. Suppose that (M, g, K) + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +13 +admits a MOTS Σ, with respect to a unit normal field ν, such that the following +conditions hold: +(I) M is retractable with respect to Σ towards ν, +(II) the homology group H2(M) is generated by the class of Σ, and +(III) Σ minimizes area. +Then Σ is topologically S2 and its area satisfies, +A(Σ) ≤ 4π +c . +(4.1) +Moreover, if A(Σ) = 4π/c, then the following hold: +(a’) (M, g) is isometric to [0, ℓ] × Σ/∼ endowed with the induced metric from the +product ([0, ℓ] × Σ, dt2 + h), where ‘ ∼’ means that {0} × Σ and {ℓ} × Σ are +suitably identified and h - the induced metric on Σ - has constant Gaussian +curvature κΣ = c, +(b’) K = a dt2 on M, where a ∈ C∞(M) depends only on t, and +(c’) µ = c and J = 0 on M. +Proof. First, observe that, by making a ‘cut’ along Σ, we obtain a 3-dimensional +compact manifold M ′ with two boundary components, say Σ0 and S. Also, the initial +data (g, K) on M gives rise to data (g′, K′) on M ′ in the natural way. The boundary +components Σ0 and S are both isometric to Σ with respect to the corresponding +induced metrics. +Now, consider the initial data set (M ′, g′, K′). Observe that the boundary com- +ponents Σ0 and S of M ′ can be chosen in such a way that conditions (1)-(5) of +Theorem 3.1 are satisfied. In fact, +(1) Σ0 is a MOTS with respect to the normal that points into M ′, +(2) S is a MOTS with respect to the normal that points out of M ′, +(3) M ′ satisfies the homotopy condition with respect to Σ0, since M is retractable +with respect to Σ towards ν, +(4) the relative homology group H2(M ′, Σ0) vanishes, since H2(M) is generated +by the class of Σ, and +(5) Σ0 minimizes area in (M ′, g′) as Σ minimizes area in (M, g). +Conditions (1) and (2) above follow from the fact of Σ being a MOTS in (M, g, K) +with respect to ν and the choice of Σ0 and S. Therefore, by Theorem 3.1, Σ0 is +topologically S2 and its area satisfies A(Σ0) ≤ 4π/c. The same conclusions hold +for Σ. Moreover, if A(Σ) = 4π/c, that is, A(Σ0) = 4π/c, then conclusions (a)-(c) of +Theorem 3.1 hold for (M ′, g′, K′) and thus (M, g, K) satisfies (a’)-(c’). +□ +Remark 4.2. Initial data sets satisfying the assumptions of Theorem 4.1 arise nat- +urally in the Nariai spacetime. The Nariai spacetime is a solution to the vacuum +Einstein equations with positive cosmological constant, Λ > 0. It is a metric product +of 2-dimensional de Sitter space dS2 and S2, +¯N = (R × S1) × S2, +¯h = −dt2 + a2 cosh2(t/a) dχ2 + a2dΩ2, + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +14 +where a = +1 +√ +Λ. As described in [6, 7], the Nariai spacetime is an interesting limit +of Schwarzschild-de Sitter space, as the size of the black hole increases and its area +approaches the upper bound in (4.1), with c = Λ. +Under the transformation, cosh(t/a) = sec τ, the metric ¯h becomes, +¯h = +a2 +cos2(τ) +� +−dτ 2 + dχ2� ++ a2dΩ2, +where τ is in the range, −π +2 < τ < π +2. With this change of time coordinate, we +see that dS2 is locally conformal to the Minkowski plane. A Penrose type diagram +for ( ¯N, ¯h) is depicted in Figure 1. Each point in the diagram represents a round +2-sphere of radius a. In the diagram, M = Γ × S2, where Γ is a smooth spacelike +graph over the circle: τ = 0, 0 ≤ χ ≤ 2π in dS2. Taking Σ to be the 2-sphere +intersection of M with the totally geodesic null hypersurface H, one easily verifies +that (M, g, K), where g is the induced metric and K is the second fundamental form +of M, respectively, satisfies the assumptions of Theorem 4.1, with equality in (4.1). +We note that there are initial data sets in (spatially closed) Schwarzschild-de Sitter +that satisfy all the assumptions of Theorem 4.1, except for equality in (4.1). +AB8XicbVA9SwNBEJ2LXzF+RS1tFhPBKtylUBshaGMZwXxgcoS9zV6yZG/v2J0TQsi/sLFQxNZ/Y+e/cZNcoYkPBh7vz +TAzL0ikMOi6305ubX1jcyu/XdjZ3ds/KB4eNU2casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo9uZ3ri2ohYPeA4X5EB0qEg +lG0mO5y4aCXBO3CuW3Io7B1klXkZKkKHeK351+zFLI6QSWpMx3MT9CdUo2CSTwvd1PCEshEd8I6likbc+JP5xVNyZpU+C +WNtSyGZq78nJjQyZhwFtjOiODTL3kz8z+ukGF75E6GSFLli0VhKgnGZPY+6QvNGcqxJZRpYW8lbEg1ZWhDKtgQvOWXV0mzWv +EuKt59tVS7yeLIwmcwjl4cAk1uIM6NICBgmd4hTfHOC/Ou/OxaM052cwx/IHz+QNc2Y9n� = 0 +AB73icbVA9T8MwEL2Ur1K+CowsFi0SU5V0AMYKFs 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Nariai spacetime. +4.2. The non-spherical case. We now consider applications of Theorems 2.4, 3.2 +and 3.3 to closed initial data sets satisfying the DEC. As a consequence, we obtain +results concerning the existence and rigidity of MOTS with nontrivial (e.g. toroidal) +topology in the cosmological setting. +We first consider an example. Let ( ¯N, ¯h) be the FLRW spacetime, +¯N = (0, a) × M, +¯h = −dt2 + gt, +where gt = G2(t) dΩ2 and (M, dΩ2) is the unit 3-sphere. For each t ∈ (0, a), consider +the initial data (Mt = {t} × M, gt, Kt), where Kt, the second fundamental form, is +given by +Kt = +˙G(t) +G(t) gt. +In particular, either Kt or −Kt is 2-convex, depending on the sign of ˙G(t). One +easily verifies that the DEC holds (strictly) for any choice of scale factor G(t). + ++SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +15 +For each t ∈ (0, a), it is easy to see that (Mt, gt, Kt) contains a spherical MOTS. +Indeed, the latitudinal 2-spheres take on all mean curvature values between −∞ +and +∞. Choose the latitudinal 2-sphere Σt such that its mean curvature satisfies +Ht = − trΣt Kt = −2 +˙G(t) +G(t). +(4.2) +Then, by (2.1), Σt is a MOTS, θ+ +t = 0. +In fact, it is also the case that (Mt, gt, Kt) contains a toroidal MOTS. Here, we +rely on the one-parameter family of Clifford tori Tr in the unit 3-sphere S3. By +identifying S3 with the unit sphere centered at the origin in R4, Tr, 0 < r < 1, is +defined as +Tr = +� +(x, y, u, v) ∈ S3 : x2 + y2 = r2, u2 + v2 = 1 − r2� +. +The ‘standard’ Clifford torus is obtained by setting r = +1 +√ +2. An elementary compu- +tation shows that each Tr has constant mean curvature (see [18]), +Hr = 1 − 2r2 +r +√ +1 − r2. +In particular, the Clifford tori take on all mean curvature values between −∞ +and +∞. Thus, arguing as above in the sphere case, there exists an embedded +torus Σt in (Mt, gt, Kt) satisfying (4.2), which hence is a MOTS. +One can modify the initial data set (Mt, gt, Kt) by adding a handle from one +side of the torus Σt to the other, `a la Gromov-Lawson [17], so that Σt is no longer +homologically trivial, and such that the DEC still holds. However, the resulting +initial data manifold won’t be retractable with respect to Σt, as follows from the +next theorem. +Theorem 4.3. Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, closed initial data +set satisfying the DEC, µ ≥ |J|. Suppose that (M, g, K) admits a MOTS Σ, with +respect to a unit normal field ν, such that the following conditions hold: +(I) M is retractable with respect to Σ towards ν and +(II) Σ satisfies the cohomology condition. +Then χ+ = 0 on Σ and Σ is a flat (n − 1)-torus with respect to the induced metric. +Moreover, the following hold: +(a’) M \ Σ ∼= (0, ℓ) × Σ for some ℓ > 0. +Let Σt ∼= {t} × Σ with unit normal νt in direction of the foliation. +(b’) χ+ = 0 on Σt for every t ∈ (0, ℓ). +(c’) Σt is a flat (n−1)-torus with respect to the induced metric for every t ∈ (0, ℓ). +(d’) µ + J(νt) = 0 on Σt for every t ∈ (0, ℓ). In particular, µ = |J| on M. +If we assume further that K is (n − 1)-convex, we also have: +(e’) (M, g) is isometric to [0, ℓ] × Σ/∼ endowed with the induced metric from the +product ([0, ℓ]×Σ, dt2+h), where h is the induced metric on Σ. In particular, +(M, g) is flat. +(f’) K = a dt2, where a ∈ C∞(M) depends only on t. + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +16 +(g’) µ = 0 and J = 0 on M. +Proof. As in the proof of Theorem 4.1, let (M ′, g′, K′) be the initial data set derived +from (M, g, K) - by making a ‘cut’ along Σ - with two boundary components, Σ0 +and S, both isometric to Σ, such that Σ0 is a MOTS with respect to the normal +that points into M ′ and S is a MOTS with respect to the normal that points out +of M ′. +It is not difficult to see that (M ′, g′, K′) satisfies all the assumptions of Theo- +rem 2.4 and then all its conclusions. Thus Σ is a flat (n − 1)-torus with χ+ = 0 on +it and conclusions (a’)-(d’) of the theorem hold. +If K is (n − 1)-convex, since A(Σ0) = A(S), it follows that (M ′, g′, K′) satisfies +all the hypotheses of Theorem 3.2 for ϵ = 0. Conclusions (e’)-(g’) then follow. +□ +Remark 4.4. It follows, for example, that in a 4-dimensional spacetime which satis- +fies the DEC strictly and which has toroidal Cauchy surfaces, there cannot be any +homologically nontrivial toroidal MOTS in any Cauchy surface. This applies, in +particular, to the time slices in the toroidal (k = 0) FLRW spacetimes, that satisfy +the Einstein equations with dust (zero-pressure perfect fluid) source. +In view of property (g’), to find initial data sets satisfying the assumptions of +Theorem 4.3, one should perhaps consider vacuum spacetimes. A well-known class +of examples are the toroidal Kasner spacetimes, +¯N = (0, ∞) × M, +¯h = −dt2 + t2p1dx2 + t2p2dy2 + t2p3dz2, +where x, y, z are to be understood as periodic coordinates, and where p1 ≤ p2 ≤ p3 +must satisfy, +p1 + p2 + p3 = 1 +and +p2 +1 + p2 +2 + p2 +3 = 1. +Let M0 be the t = 1 time slice, with metric g and second fundamental form K +induced from ( ¯N, ¯h). It is not hard to show that in order for K to be 2-convex, one +must have, p1 = p2 = 0 and p3 = 1, so that ¯h becomes, +¯h = −dt2 + dx2 + dy2 + t2dz2. +This is an exceptional Kasner spacetime, known as ‘flat Kasner’. It is locally iso- +metric to Minkowski space. Taking Σ to be the torus t = 1, z = z0, we see that M0 +satisfies the assumptions of Theorem 4.3, including the 2-convexity assumption. +We mention one further example which illustrates a certain flexibility in initial +data sets satisfying (I) and (II), but not the convexity condition. It’s a small modi- +fication of Example 4.2 in [12]. +Let R3 +1 be Minkowski space with standard coordinates t, x, y, z. Consider the box +B = {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} in the t = 0 slice. Let f : B → R +be a smooth function that vanishes near the boundary of B and whose graph is +spacelike in R3 +1. By identifying opposite sides of the box, we obtain an initial data +set (M, g, K) with M ∼= T 3, where M is given by the graph of f, and where g +and K are induced from the graph of f. Let Σ be the intersection of M with the +null hyperplane t = z − 1 +2; see Figure 2. Because the null hyperplane is totally +geodesic, Σ is necessarily a MOTS. It follows that (M, g, K) satisfies (I) and (II) + +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +17 +with respect to Σ. Note also that (M, g, K) satisfies the DEC; in fact, because it +essentially sits in Minkowski space, it is a vacuum initial data set, µ = 0, J = 0. +Hence, (M, g, K) satisfies all the assumptions of Theorem 4.3, except, in general, +the convexity condition on K. The foliation by MOTS guaranteed by properties +(a’)-(d’) comes from intersecting M with the null hyperplanes t = z + c. That these +properties hold may be understood in terms of special features of totally geodesic +null hypersurfaces. +Figure 2. Initial data set satisfying (I) and (II) of Theorem 4.3. +Remark 4.5. Finally, we mention a connection to the spacetime positive mass the- +orem, specifically the approach taken by Lohkamp [21], from a perspective slightly +different from the discussion in [12]. Lohkamp reduces the proof to a stand alone +result, namely the nonexistence of ‘µ − |J| > 0 islands’, see [21, Theorem 2]. By a +standard compactification (which Lohkamp also considers), the setting of Theorem 2 +immediately gives an initial data set satisfying the DEC, with initial data manifold +M ∼= T n ♯ Q, Q closed, and a toroidal MOTS Σ, such that M is retractable with +respect to Σ (see the discussion at the beginning of Section 4.1). Theorem 4.3 then +yields that µ = |J| (among other things), which implies Lohkamp’s no µ − |J| > 0 +islands result in dimensions 3 ≤ n ≤ 7. +Lastly, we consider the following consequence of Theorem 3.3. +Corollary 4.6. Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, closed initial data +set satisfying the DEC, µ ≥ |J|. Assume that −(K + g) is (n − 1)-convex. Then +(M, g, K) cannot satisfy conditions (I)-(II) of Theorem 4.3. +Examples like those discussed at the beginning of Section 4.2 show that, while +the conditions (I) and (II) can’t be simultaneously satisfied, either one can be. +Proof. Let (M ′, g′, K′) be as in the proof of Theorem 4.3. If −(K + g) is (n − 1)- +convex, in particular, −(K′ + g′) is (n − 1)-convex, it follows from the first part of + +t +-= +1 +M = graph f/ ~ +c,ySOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS +18 +Theorem 3.3 that B(Σ0) ≤ B(S), which is a contradiction, because +B(Σ0) = A(Σ0) = A(S) > B(S). +□ +References +1. Lars Andersson, Michael Eichmair, and Jan Metzger, Jang’s equation and its applications to +marginally trapped surfaces, Complex analysis and dynamical systems IV. Part 2, Contemp. +Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45. 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Richard Schoen and Shing-Tung Yau, Positive Scalar Curvature and Minimal Hypersurface +Singularities, preprint, https://arxiv.org/abs/1704.05490 (2017). +Department of Mathematics, University of Miami, Coral Gables, FL, USA. +Email address: galloway@math.miami.edu +Instituto de Matem´atica, Universidade Federal de Alagoas, Macei´o, AL, Brazil. +Email address: abraao.mendes@im.ufal.br + diff --git a/Z9AyT4oBgHgl3EQfv_kg/content/tmp_files/load_file.txt b/Z9AyT4oBgHgl3EQfv_kg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a677e71275bf87693837646e28fb28c9ec2c7bab --- /dev/null +++ b/Z9AyT4oBgHgl3EQfv_kg/content/tmp_files/load_file.txt @@ -0,0 +1,779 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf,len=778 +page_content='SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS GREGORY J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' GALLOWAY AND ABRA˜AO MENDES Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We also obtain some extensions of results in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A number of examples are given in order to illustrate some of the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Introduction The theory of marginally outer trapped surfaces has played an important role in several areas of mathematical general relativity, for example, in proofs of the spacetime positive mass theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' [13, 20]) and in results on the topology of black holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' [14, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In [14], a local MOTS rigidity result was obtained, which implies that an outermost MOTS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' the surface of a black hole) in an initial data set satisfying the dominant energy condition (µ ≥ |J|) is positive Yamabe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' admits a metric of positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This in turn leads to well-known restrictions on the topology of 3-dimensional outermost MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Such results extend to the spacetime setting well-known results concerning Riemannian manifolds of nonnegative scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In [12], the authors, together with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Eichmair, obtained, among other results, a global version of the local MOTS rigidity result in [14], which, in particular, does not require a weakly outermost condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This result was motivated in part by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Lohkamp’s approach to the spacetime positive mass theorem in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It implies, in dimensions 3 ≤ n ≤ 7, Lohkamp’s result on the nonexistence of ‘µ − |J| > 0 islands’, [21, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 in [12] has also been applied to obtain a positive mass theorem for asymptotically hyperbolic manifolds with boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This theorem will be a useful tool in the present work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this paper, we present some further initial data rigidity results for compact initial data sets, in both the boundary and no boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In [15], the authors considered 3-dimensional initial data sets containing spherical MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It was shown, roughly speaking, that in a matter-filled spacetime, perhaps with positive cosmo- logical constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' namely, its area must be bounded above by 4π/c, where c > 0 is a lower bound on a natural energy-momentum term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We then established rigidity results for stable, or weakly outermost, marginally outer trapped 2-spheres when this bound is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, we prove a local splitting result, [15, Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2], that extends to the spacetime setting a result of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Bray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Brendle, and 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='00639v1 [gr-qc] 2 Jan 2023 SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Neves [8] concerning area minimizing 2-spheres in Riemannian 3-manifolds with positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' These spacetime results have interesting connections to the Vaidya and Nariai spacetimes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' One of the main aims of the present work is to obtain a global version of [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 in Section 3 for a statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The proof makes use of certain techniques introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this work, we have also been led to consider certain variations of [12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Here, it becomes useful to consider the so-called ‘brane action’, as well as the area functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' These results are then used to examine the question of the existence of MOTS in closed (compact without boundary) initial data sets in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The relationship to known spacetimes is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The paper is organized as follows: in Section 2, we review some background material on MOTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' in Section 3, we state and prove several global rigidity results for compact-with-boundary initial data sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' and, in Section 4, we apply the results obtained in Section 3 to prove some global rigidity statements for closed initial data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In Section 4, we also give various examples in order to illustrate the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The work of GJG was partially supported by the Simons Foundation, under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 850541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The work of AM was partially supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq, Brazil (Grant 305710/2020-6), the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - CAPES, Brazil (CAPES-COFECUB 88887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='143161/2017-0), and the Funda¸c˜ao de Amparo `a Pesquisa do Estado de Alagoas - FAPEAL, Brazil (Process E:60030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='0000002254/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The authors would like to thank Ken Baker and Da Rong Cheng for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Preliminaries All manifolds in this paper are assumed to be connected and orientable except otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' An initial data set (M, g, K) consists of a Riemannian manifold (M, g) with boundary ∂M (possibly ∂M = ∅) and a symmetric (0, 2)-tensor K on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be an initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The local energy density µ and the local current density J of (M, g, K) are given by µ = 1 2(S − |K|2 + (tr K)2) and J = div(K − (tr K)g), where S is the scalar curvature of (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We say that (M, g, K) satisfies the domi- nant energy condition (DEC for short) if µ ≥ |J| on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Consider a closed embedded hypersurface Σ ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Since, by assumption, Σ and M are orientable, we can choose a unit normal field ν on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If Σ separates M, by convention, we say that ν points to the outside of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 3 The null second fundamental forms χ+, χ− of Σ in (M, g, K) with respect to ν are given by χ+ = K|Σ + A and χ− = K|Σ − A, where A is the second fundamental form of Σ in (M, g) with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' More precisely, A(X, Y ) = g(∇Xν, Y ) for X, Y ∈ X(Σ), where ∇ is the Levi-Civita connection of (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The null expansion scalars θ+, θ− of Σ in (M, g, K) with respect to ν are given by θ+ = trΣ(K) + H and θ− = trΣ(K) − H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1) where H = tr A is the mean curvature of Σ in (M, g) with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observe that θ± = tr χ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Penrose introduced the now famous notion of a trapped surface, when both θ+ and θ− are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Restricting to one side, we say that Σ is outer trapped if θ+ < 0, weakly outer trapped if θ+ ≤ 0, and marginally outer trapped if θ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the latter case, we refer to Σ as a marginally outer trapped surface (MOTS for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Assume now that Σ is a MOTS in (M, g, K), with respect to a unit normal ν, that is a boundary in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' More precisely, assume that ν points towards a top- dimensional submanifold M + ⊂ M such that ∂M + = Σ ⊔ S, where S (possibly S = ∅) is a union of components of ∂M (in particular, if Σ separates M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We think of M + as the region outside of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then we say that Σ is outermost (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' weakly outermost) if there is no closed embedded hypersurface in M + with θ+ ≤ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' θ+ < 0) that is homologous to and different from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The notions of locally weakly outermost and locally outermost MOTS can be given in an analogous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is important to mention that initial data sets arise naturally in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In fact, let M be a spacelike hypersurface in a spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' a time-oriented Lorentzian manifold, ( ¯N, ¯h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let g be the Riemannian metric on M induced from ¯h and K be the second fundamental form of M in ( ¯N, ¯h) with respect to the future-pointing timelike unit normal u on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then (M, g, K) is an initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As before, let Σ be a closed embedded hypersurface in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this setting, χ+ and χ− are the null second fundamental forms of Σ in ( ¯N, ¯h) with respect to the null normal fields ℓ+ = u|Σ + ν and ℓ− = u|Σ − ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observe that θ± = divΣ ℓ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Physically, θ+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' θ−) measures the divergence of the outward pointing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' inward pointing) light rays emanating from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' An initial data set (M, g, K) is said to be time-symmetric or Riemannian if K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this case, a MOTS in (M, g, K) is nothing but a minimal hypersurface in (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, the energy condition µ − |J| ≥ c, for some constant c, reduces to the requirement on the scalar curvature S ≥ 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Quite generally, marginally outer SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 4 trapped surfaces share many properties with minimal hypersurfaces, which they generalize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' the survey article [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As in the minimal hypersurfaces case, an important notion for the theory of MOTS is the notion of stability introduced, in the context of MOTS, by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Andersson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Mars, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Simon [2, 3], which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σ be a MOTS in (M, g, K) with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Consider a normal variation of Σ in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' a variation t → Σt of Σ = Σ0 with variation vector field ∂ ∂t|t=0 = φ ν, φ ∈ C∞(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let θ±(t) denote the null expansion scalars of Σt with respect to νt, ν = νt|t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Computations as in [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 2] or [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 861] give, ∂θ+ ∂t ���� t=0 = Lφ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) where Lφ = −∆φ + 2⟨X, ∇φ⟩ + (Q + div X − |X|2)φ and Q = 1 2SΣ − (µ + J(ν)) − 1 2|χ+|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Here, ∆ is the negative semi-definite Laplace-Beltrami operator, ∇ the gradient, div the divergence, and SΣ the scalar curvature of Σ with respect to the induced metric ⟨ · , · ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, X is the tangent vector field on Σ that is dual to the 1-form K(ν, · )|Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is possible to prove (see [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1]) that L has a real eigenvalue λ1 = λ1(L), called the principal eigenvalue of L, such that Re λ ≥ λ1 for any other complex eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Furthermore, the corresponding eigenfunction φ1, Lφ1 = λ1φ1, is unique up to a multiplicative constant and can be chosen to be real and everywhere positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then a MOTS Σ is said to be stable if λ1(L) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This is equivalent to the existence of a positive function φ ∈ C∞(Σ) such that Lφ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It follows directly from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) with φ = φ1 that every locally weakly outermost (in particular, locally outermost) MOTS is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observe that in the Riemannian case, L reduces to the classical stability operator, also known as the Jacobi operator, for minimal hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As such, in the literature, L is known as the MOTS stability operator or the stability operator for MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The study of rigidity results for minimal surfaces in Riemannian manifolds with a lower scalar curvature bound has been, and continues to be, an active area of re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' From the point of view of initial data sets, these are time-symmetric results, as noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It has been of interest to extend some of these results to general initial data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the context of general relativity, black hole horizons within initial data sets are often modeled by MOTS, and, in particular, minimal surfaces in the time-symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' These rigidity results often shed light on properties of spacetimes with black holes, as noted in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 5 The next proposition and theorem extend to the general non-time-symmetric set- ting some results of Bray, Brendle, and Neves [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 (Infinitesimal rigidity, [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σ be a stable MOTS in a 3- dimensional initial data set (M, g, K) with respect to a unit normal field ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose there exists a constant c > 0 such that µ + J(ν) ≥ c on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then the area of Σ satisfies, A(Σ) ≤ 4π c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, if A(Σ) = 4π/c, then the following hold: (a) Σ is a round 2-sphere with Gaussian curvature κΣ = c, (b) the second fundamental form χ+ of Σ with respect to ν vanishes, and (c) µ + J(ν) = c on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The proposition above is used in the proof of the following local splitting theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' But, before stating the next result, which is also used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, let us remember the notion of an area minimizing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' With respect to a fixed Riemannian metric g on a 3-dimensional manifold M, a closed embedded surface Σ ⊂ M is said to be area minimizing if Σ is of least area in its homology class in M, that is, A(Σ) ≤ A(Σ′) for any closed embedded surface Σ′ that is homologous to Σ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this case, we also say that Σ minimizes area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Similarly, Σ is said to be locally area minimizing if A(Σ) ≤ A(Σ′) for any such Σ′ in a neighborhood of Σ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3 (Local splitting, [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be a 3-dimensional initial data set with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that (M, g, K) satisfies the energy condition µ − |J| ≥ c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σ0 be a closed connected component of ∂M such that the following conditions hold: (1) Σ0 is a MOTS with respect to the normal that points into M and (2) Σ0 is locally weakly outermost and locally area minimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then Σ0 is topologically S2 and its area satisfies, A(Σ0) ≤ 4π c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Furthermore, if A(Σ0) = 4π/c, then a collar neighborhood U of Σ in M is such that: (a) (U, g) is isometric to ([0, δ) × Σ0, dt2 + g0) for some δ > 0, where g0 - the induced metric on Σ0 - has constant Gaussian curvature κΣ0 = c, (b) K = a dt2 on U, where a ∈ C∞(U) depends only on t ∈ [0, δ), and (c) µ = c and J = 0 on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This theorem extends to the general non-time-symmetric setting the local rigidity statements in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The local rigidity obtained in [8] is then used to obtain a global rigidity result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see [8, Proposition 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 in the next section, we obtain a global version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A key improvement in this global rigidity result is that it does not require the ‘weakly outermost’ assumption, and hence parallels somewhat more closely the global result in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 6 Now, we recall two topological concepts that are important for our purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see also [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We say that M satisfies the homotopy condition with respect to Σ ⊂ M provided there exists a continuous map ρ : M → Σ such that ρ ◦ i : Σ → Σ is homotopic to idΣ, where i : Σ �→ M is the inclusion map (for example, if Σ is a retract of M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' On the other hand, a closed not necessarily connected manifold N of dimension m is said to satisfy the cohomology condition if there are m classes ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , ωm in the first cohomology group H1(N), with integer coefficients, whose cup product ω1 ⌣ · · · ⌣ ωm ∈ Hm(N) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' For example, the m-torus T m = S1 × · · · × S1 satisfies the cohomol- ogy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' More generally, the connected sums T m ♯ Q satisfy the cohomology condition for any closed m-manifolds Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A version of this condition is considered in [23, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Here, we are using the form of the condition as it appears in [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A manifold N satisfying this cohomology condition has a component that does not carry a metric of positive scalar curvature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see the discussion in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We will make use of the following theorem (mentioned in the introduction) in several situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4 ([12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, compact-with-boundary initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that (M, g, K) satisfies the domi- nant energy condition, µ ≥ |J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose also that the boundary can be expressed as a disjoint union ∂M = Σ0 ∪ S of nonempty unions of components such that the following conditions hold: (1) θ+ ≤ 0 on Σ0 with respect to the normal that points into M, (2) θ+ ≥ 0 on S with respect to the normal that points out of M, (3) M satisfies the homotopy condition with respect to Σ0, and (4) Σ0 satisfies the cohomology condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then the following hold: (a) M ∼= [0, ℓ] × Σ0 for some ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σt ∼= {t} × Σ0 with unit normal νt in direction of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (b) χ+ = 0 on Σt for every t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (c) Σt is a flat (n−1)-torus with respect to the induced metric for every t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (d) µ + J(νt) = 0 on Σt for every t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, µ = |J| on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The following is the basic existence result for MOTS due to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Andersson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Metzger in 3-dimensions, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Eichmair in dimensions 3 ≤ n ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, and is the source of the dimension restriction appearing in various results discussed herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='5 (Existence of MOTS, [4, 10, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, compact-with-boundary initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that the boundary can be expressed as a disjoint union ∂M = Σin ∪ Σout, where Σin and Σout are nonempty unions of components of ∂M with θ+ ≤ 0 on Σin with respect to the normal pointing SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 7 into M and θ+ > 0 on Σout with respect to the normal pointing out of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then there is an outermost MOTS in (M, g, K) that is homologous to Σout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The compact-with-boundary cases In this section, we obtain several global initial data results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' These results, in turn, will be applied in the next section to the case that the initial data manifold is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The first is a global version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see the comments above, after the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be a 3-dimensional compact-with-boundary initial data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that (M, g, K) satisfies the energy condition µ − |J| ≥ c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose also that the boundary can be expressed as a disjoint union ∂M = Σ0 ∪ S of nonempty unions of components such that the following conditions hold: (1) θ+ ≤ 0 on Σ0 with respect to the normal that points into M, (2) θ+ ≥ 0 on S with respect to the normal that points out of M, (3) M satisfies the homotopy condition with respect to Σ0, (4) the relative homology group H2(M, Σ0) vanishes, and (5) Σ0 minimizes area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then Σ0 is topologically S2 and its area satisfies, A(Σ0) ≤ 4π c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, if A(Σ0) = 4π/c, then the following hold: (a) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + g0) for some ℓ > 0, where g0 - the induced metric on Σ0 - has constant Gaussian curvature κΣ0 = c, (b) K = a dt2 on M, where a ∈ C∞(M) depends only on t ∈ [0, ℓ], and (c) µ = c and J = 0 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' First, observe that Σ0 is connected, since M is connected and satisfies the homotopy condition with respect to Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If Σ0 is not homeomorphic to S2, then Σ0 is homeomorphic to T 2 ♯ Q for some closed orientable surface Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, Σ0 satisfies the cohomology condition and so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4 applies to (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, 0 = µ − |J| ≥ c on M, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then Σ0 is topologically S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Claim: Σ0 is a weakly outermost MOTS in (M, g, K) of area A(Σ0) = 4π/c unless A(Σ0) < 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Assume that A(Σ0) ≥ 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If θ+ K ≤ 0 is not identically zero on Σ0, it follows from [4, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2] that there is a surface Σ ⊂ M - obtained by a small perturbation of Σ0 into M - such that θ+ K < 0 on Σ with respect to the normal pointing away from Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let W be the connected compact region bounded by Σ and S in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observe that θ+ −K ≤ 0 on S with respect to the normal that points into W and θ+ −K > 0 on Σ with respect to the normal that points out of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Applying the MOTS existence theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='5), we obtain SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 8 an outermost MOTS Σ′ in (W, g, −K) that is homologous to and disjoint from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Clearly, Σ′ is homologous to Σ0 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Without loss of generality, we may assume that each connected component of Σ′ is homologically nontrivial in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Also, because H2(M, Σ0) = 0, Σ′ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Since we are assuming that Σ0 minimizes area in its homology class, we have 4π c ≤ A(Σ0) ≤ A(Σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' On the other hand, because Σ′ is an outermost MOTS in (W, g, −K), in particular stable, the infinitesimal rigidity (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) gives that A(Σ′) = 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' There- fore, Σ′ is an area minimizing outermost MOTS in (W, g, −K) of area A(Σ′) = 4π/c and then the local splitting theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3) applies so that an outer neigh- borhood of Σ′ in W is foliated by MOTS, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This proves that Σ0 is a MOTS in (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Now, we claim that Σ0 is weakly outermost in (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If not, there is a surface Σ that is homologous to Σ0 in M and such that θ+ K < 0 on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Perturbing Σ a bit, we may assume that Σ ∩ Σ0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Also, by the strong maximum principle as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4] or [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1], Σ ∩ S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As before, without loss of generality, we may assume that each connected compo- nent of Σ is homologically nontrivial in M and, in particular, Σ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let W be the region in M bounded by Σ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Arguing with (W, g, −K) as above, we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus Σ0 is weakly outermost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We have then proved that, if A(Σ0) ≥ 4π/c, then Σ0 is a weakly outermost MOTS in (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this case, by the infinitesimal rigidity, A(Σ0) = 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This finishes the proof of the Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We have then obtained that Σ0 is homeomorphic to S2 and its area satisfies A(Σ0) ≤ 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Furthermore, if A(Σ0) = 4π/c, then Σ0 is an area minimizing weakly outermost MOTS in (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this case, by the local splitting theorem, there is a collar neighborhood U ∼= [0, δ) × Σ0 of Σ0 in M such that conclusions (a), (b), and (c) of the theorem hold on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Clearly, Σt ∼= {t} × Σ0 converges to a closed embedded MOTS Σδ of area 4π/c as t ↗ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If Σδ ∩ S ̸= ∅, by the strong maximum principle, Σδ = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If Σδ ∩ S = ∅, we can replace Σ0 by Σδ and M by the complement of U and run the process again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The result then follows by a continuity argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' □ The next two theorems make use of the notion of (n−1)-convexity of a symmetric (0, 2)-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Imposing such convexity leads to stronger rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We say that a symmetric (0, 2)-tensor P on (M, g) is (n − 1)-convex if, at every point p ∈ M, the sum of the smallest (n − 1) eigenvalues of P with respect to g is nonnegative (in particular, if P is positive semi-definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This is equivalent to the trace of P with respect to any (n − 1)-dimensional linear subspace of TpM being nonnegative, for every p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, if P is (n − 1)-convex, then trΣ P ≥ 0 for every hypersurface Σ ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This convexity condition has been used by the second-named author in [22] and by the authors, together with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Eichmair, in [12] in related contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 9 Let (M, g, K) be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4, and let Σ be a closed embedded hypersurface homologous to Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The next theorem makes use of the functional, Bϵ(Σ) = A(Σ) − (n − 1) ϵ V(Σ), ϵ = 0, 1, where A(Σ) is the area of Σ and V(Σ) is the volume of the region bounded by Σ and Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the case ϵ = 0, we are just talking about the area functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the case ϵ = 1, we are talking about the functional associated with hypersurfaces of constant mean curvature n − 1, sometimes referred to as the brane action and denoted by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The following theorem extends in a couple of directions Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Assume that (i) K + ϵ g is (n − 1)-convex, where ϵ = 0 or ϵ = 1, and (ii) Σ0 and S are such that Bϵ(Σ0) ≤ Bϵ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then the following hold: (a) Σ0 is a flat (n − 1)-torus with respect to the induced metric g0, (b) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + e2 ϵ tg0) for some ℓ > 0, (c) K = (1 − ϵ)a − ϵ g on M, where a ∈ C∞(M) depends only on t ∈ [0, ℓ], and (d) µ = 0 and J = 0 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The convexity assumption holds if, in particular, K satisfies, K ≥ −ϵ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the case ϵ = 0, this would apply to cosmological models that are expanding to the future (in all directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4, M ∼= [0, ℓ] × Σ0 for some ℓ > 0, and each leaf Σt ∼= {t} × Σ0 is a MOTS with respect to the unit normal νt in direction of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' On the other hand, since K + ϵ g is (n − 1)-convex, we have H(t) − (n − 1) ϵ ≤ H(t) + trΣt K = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1) where H(t) = divΣt νt is the mean curvature of Σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Now, the first variation of Bϵ gives that d dtBϵ(Σt) = � Σt φ (H(t) − (n − 1) ϵ) dΣt ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) where φ = ⟨νt, ∂t⟩ is the lapse function of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, Bϵ(t) = Bϵ(Σt) is a nonincreasing function on [0, ℓ] satisfying Bϵ(0) ≤ Bϵ(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus Bϵ(t) = Bϵ(Σt) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) give that H(t) = (n − 1) ϵ = − trΣt K for all t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, θ− = −2 (n − 1) ϵ on Σℓ = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The result then follows directly from [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3] (observe that our sign convention in the definition of θ− in this work is the opposite of that one in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' □ In the next theorem we consider B = B1 under a modified convexity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Assume that −(K + g) is (n − 1)-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then B(Σ0) ≤ B(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, if equality holds, we have the following: SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 10 (a) (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + gt) for some ℓ > 0, where gt is the induced metric on Σt ∼= {t} × Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (b) Each Σt is a flat (n − 1)-torus with respect to gt and has constant mean curvature H(t) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (c) The scalar curvature of (M, g) satisfies S ≤ −n(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If equality holds, (M, g) is isometric to ([0, ℓ] × Σ0, dt2 + e2 tg0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (d) For each t ∈ [0, ℓ], µ + J(νt) = 0 on Σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, µ = |J| on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (e) tr K ≤ −n on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If equality holds, K = −g, S = −n(n − 1), µ = 0, and J = 0 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The convexity assumption holds if, in particular, K satisfies, K ≤ −g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If one views K as being defined with respect to the past directed unit normal, this would apply to cosmological models that are strongly contracting to the past, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' that begin with a ‘big bang’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4, M ∼= [0, ℓ] × Σ0 with g = φ2dt2 + gt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3) where gt is the induced metric on Σt ∼= {t} × Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Each (Σt, gt) is a flat (n − 1)-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Every leaf Σt is a MOTS in (M, g, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In fact, 0 = χ+(t) = A(t) + K|Σt, where A(t) is the second fundamental form of Σt computed with respect to the unit normal νt in direction of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' For each t ∈ [0, ℓ], µ + J(νt) = 0 on Σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, µ = |J| on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Now, since −(K + g) is (n − 1)-convex, we have H(t) − (n − 1) ≥ H(t) + trΣt K = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4) where H(t) = tr A(t) is the mean curvature of Σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then the first variation of B gives that d dtB(Σt) = � Σt φ (H(t) − (n − 1)) dΣt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='5) Therefore, B(t) = B(Σt) is a nondecreasing function defined on [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, B(0) ≤ B(ℓ), that is, B(Σ0) ≤ B(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If B(Σ0) = B(S), then B(t) = B(Σt) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='5) imply that H(t) = n − 1 = − trΣt K for all t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Now, fix t ∈ [0, ℓ], p ∈ Σt, and let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , en−1} be an orthonormal basis for TpΣt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Define η(s) = cos s · en−1 + sin s · νt, s ∈ R, and let π(s) be the (n − 1)-dimensional linear subspace of TpM generated by {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , en−2, η(s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 11 Since −(K + g) is (n − 1)-convex and trΣt K = −(n − 1), we have f(s) := trπ(s) K ≤ −(n − 1) and f(0) = trΣt K = −(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, s = 0 is a critical point of f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observing that f(s) = n−2 � i=1 K(ei, ei) + K(η(s), η(s)), we obtain, 0 = f ′(0) = 2K(η′(0), η(0)) = 2K(νt, en−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Analogously, K(νt, ei) = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This gives that X♭ = K(νt, · )|Σt = 0 for all t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' On the other hand, the first variation of θ+(t) = 0 reads as ∂θ+ ∂t = −∆φ + 2⟨X, ∇φ⟩ + (Q + div X − |X|2)φ = −∆φ + Qφ, where Q = 1 2SΣt − (µ + J(νt)) − 1 2|χ+(t)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus ∆φ = 0 on Σt and then φ = φ(t) is constant on Σt for each t ∈ [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Hence, by a simple change of variable in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3), we have g = dt2 + gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='6) In particular, the t-lines are geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Hence, along each leaf Σ = Σt, H = H(t) satisfies the scalar Riccati equation, ∂H ∂t = − Ric(∂t, ∂t) − |A|2, which, since H(t) = n − 1, implies, Ric(∂t, ∂t) + |A|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By the Gauss equation, we have the standard rewriting of the left-hand side in the above equation, Ric(∂t, ∂t) + |A|2 = 1 2(S − SΣ + |A|2 + H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Hence, since SΣ = 0, we have, S = −|A|2 − H2 ≤ − H2 n − 1 − H2 = −n(n − 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='7) which establishes the inequality part in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If equality holds, then |A(t)|2 = n − 1, which, together with H(t) = n−1, implies that each Σt is umbilic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' in fact, A(t) = gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Using this in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='6) easily implies the isometry part in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Since −(K +g) is (n−1)-convex, it is not difficult to see that tr K ≤ −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In fact, if {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , en} is an orthonormal basis for TpM, p ∈ M, then (n − 1) tr K = n � i=1 � j̸=i K(ej, ej) ≤ − n � i=1 (n − 1) = −n(n − 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='8) SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 12 that is, tr K ≤ −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If tr K = −n, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='8) that � j̸=i K(ej, ej) = −(n − 1) for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, −n = tr K = K(ei, ei) + � j̸=i K(ej, ej) = K(ei, ei) − (n − 1), that is, K(ei, ei) = −1 for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Since {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' , en} is arbitrary, we have K = −g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus, using that A(t) = −K|Σt = g|Σt in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='7), we obtain S = −|A(t)|2 − |H(t)|2 = −n(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Finally, µ = 1 2(S − |K|2 + (tr K)2) = 0 and J = div(K − (tr K)g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Applications: closed cases In this section, we wish to apply the results of the previous section to initial data manifolds that are closed (compact without boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' These results naturally relate to cosmological (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' spatially closed) spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We’ll illustrate the results with various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In this section, we want to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 to the case that M is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let M be an n-dimensional closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose the (n − 1)-th homology group Hn−1(M) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Any nontrivial element of Hn−1(M) gives rise to a smooth closed embedded non-separating orientable hypersurface Σ ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, Σ is two-sided in M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' there is an embedding F : [−1, 1]×Σ → M such that F(0, p) = p for each p ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let U denote the open set F((−1, 1)×Σ) ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We say that M is retractable with respect to Σ if M \\ U retracts onto some component of ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If we consider a Riemannian metric g on M, given a unit normal field ν on Σ with respect to g, we say that M is retractable with respect to Σ towards ν if M \\ U retracts onto the component of ∂U towards which ν points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' An obvious situation where this occurs is when M is of the form M = S1 × Q, with Q closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then M is retractable with respect to Σ = {x} × Q, x ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Another situation of interest is when M is of the form M = T n ♯ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' View T n as an n-dimensional cube with opposite boundary faces identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' To obtain M, we may assume the connected sum takes place in a bounded open set U inside the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σ be an (n − 1)-torus parallel to one of the faces away from the set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then M is retractable with respect to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' More generally, if M is retractable with respect to Σ, then so is M ♯ Q, with Q closed, provided the connect sum takes place away from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be a 3-dimensional closed initial data set satisfying the energy condition µ − |J| ≥ c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that (M, g, K) SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 13 admits a MOTS Σ, with respect to a unit normal field ν, such that the following conditions hold: (I) M is retractable with respect to Σ towards ν, (II) the homology group H2(M) is generated by the class of Σ, and (III) Σ minimizes area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then Σ is topologically S2 and its area satisfies, A(Σ) ≤ 4π c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1) Moreover, if A(Σ) = 4π/c, then the following hold: (a’) (M, g) is isometric to [0, ℓ] × Σ/∼ endowed with the induced metric from the product ([0, ℓ] × Σ, dt2 + h), where ‘ ∼’ means that {0} × Σ and {ℓ} × Σ are suitably identified and h - the induced metric on Σ - has constant Gaussian curvature κΣ = c, (b’) K = a dt2 on M, where a ∈ C∞(M) depends only on t, and (c’) µ = c and J = 0 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' First, observe that, by making a ‘cut’ along Σ, we obtain a 3-dimensional compact manifold M ′ with two boundary components, say Σ0 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Also, the initial data (g, K) on M gives rise to data (g′, K′) on M ′ in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The boundary components Σ0 and S are both isometric to Σ with respect to the corresponding induced metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Now, consider the initial data set (M ′, g′, K′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Observe that the boundary com- ponents Σ0 and S of M ′ can be chosen in such a way that conditions (1)-(5) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In fact, (1) Σ0 is a MOTS with respect to the normal that points into M ′, (2) S is a MOTS with respect to the normal that points out of M ′, (3) M ′ satisfies the homotopy condition with respect to Σ0, since M is retractable with respect to Σ towards ν, (4) the relative homology group H2(M ′, Σ0) vanishes, since H2(M) is generated by the class of Σ, and (5) Σ0 minimizes area in (M ′, g′) as Σ minimizes area in (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Conditions (1) and (2) above follow from the fact of Σ being a MOTS in (M, g, K) with respect to ν and the choice of Σ0 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, Σ0 is topologically S2 and its area satisfies A(Σ0) ≤ 4π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The same conclusions hold for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, if A(Σ) = 4π/c, that is, A(Σ0) = 4π/c, then conclusions (a)-(c) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 hold for (M ′, g′, K′) and thus (M, g, K) satisfies (a’)-(c’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Initial data sets satisfying the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1 arise nat- urally in the Nariai spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The Nariai spacetime is a solution to the vacuum Einstein equations with positive cosmological constant, Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is a metric product of 2-dimensional de Sitter space dS2 and S2, ¯N = (R × S1) × S2, ¯h = −dt2 + a2 cosh2(t/a) dχ2 + a2dΩ2, SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 14 where a = 1 √ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As described in [6, 7], the Nariai spacetime is an interesting limit of Schwarzschild-de Sitter space, as the size of the black hole increases and its area approaches the upper bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1), with c = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Under the transformation, cosh(t/a) = sec τ, the metric ¯h becomes, ¯h = a2 cos2(τ) � −dτ 2 + dχ2� + a2dΩ2, where τ is in the range, −π 2 < τ < π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' With this change of time coordinate, we see that dS2 is locally conformal to the Minkowski plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A Penrose type diagram for ( ¯N, ¯h) is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Each point in the diagram represents a round 2-sphere of radius a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In the diagram, M = Γ × S2, where Γ is a smooth spacelike graph over the circle: τ = 0, 0 ≤ χ ≤ 2π in dS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Taking Σ to be the 2-sphere intersection of M with the totally geodesic null hypersurface H, one easily verifies that (M, g, K), where g is the induced metric and K is the second fundamental form of M, respectively, satisfies the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, with equality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We note that there are initial data sets in (spatially closed) Schwarzschild-de Sitter that satisfy all the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, except for equality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='AB8XicbVA9SwNBEJ2LXzF+RS1tFhPBKtylUBshaGMZwXxgcoS9zV6yZG/v2J0TQsi/sLFQxNZ/Y+e/cZNcoYkPBh7vz ' metadata={'source': 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+page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='AB6HicbVA9TwJBEJ3DL8Qv1NJm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='IzGxIncUaqyINpaQCJLAhewtc7Cyt3fZ3TMhF36BjYXG2PqT7Pw3LnCFgi+Z5OW9mczMCxLBtXHdb6ewtr6xuVXcLu3s7u0flA+P2jpOFcMWi0WsOgHVKLjEluFGYCdRSKNA4EMwvp35D0+oNI/lvZk6Ed0KHn ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='Wj27U9vkFAdsRioIEr1XCcFPyMSOBVsWvK0YimhYzJkPUNjEjHlZ/PDp/jYKAMcJtJUDHiu/pzISKTUJApMZ0RgpJa9mfif19MQXvoZj1MNLKaLRaEWGBI8SwEPuGQUxMQiU3t2I6IpJQMFmVTAju8st/Sbtec89r7l290rjO4yiQ3SETpCLlA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='D3aImaiGKNHpCL+jVerSerTfrfdFasPKZA/QL1sc3I/GRcw=⌧ = �⇡/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='𝐵𝐻 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='Eb/nlVdKsVryLindfLdVusjycAKncA4eXEIN7qAODWCg4Ble4c0xzovz7nwsWnNONnMf+B8/gB+uo9⌧ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Nariai spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The non-spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We now consider applications of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3 to closed initial data sets satisfying the DEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As a consequence, we obtain results concerning the existence and rigidity of MOTS with nontrivial (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' toroidal) topology in the cosmological setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We first consider an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let ( ¯N, ¯h) be the FLRW spacetime, ¯N = (0, a) × M, ¯h = −dt2 + gt, where gt = G2(t) dΩ2 and (M, dΩ2) is the unit 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' For each t ∈ (0, a), consider the initial data (Mt = {t} × M, gt, Kt), where Kt, the second fundamental form, is given by Kt = ˙G(t) G(t) gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, either Kt or −Kt is 2-convex, depending on the sign of ˙G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' One easily verifies that the DEC holds (strictly) for any choice of scale factor G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' +SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 15 For each t ∈ (0, a), it is easy to see that (Mt, gt, Kt) contains a spherical MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Indeed, the latitudinal 2-spheres take on all mean curvature values between −∞ and +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Choose the latitudinal 2-sphere Σt such that its mean curvature satisfies Ht = − trΣt Kt = −2 ˙G(t) G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2) Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1), Σt is a MOTS, θ+ t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In fact, it is also the case that (Mt, gt, Kt) contains a toroidal MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Here, we rely on the one-parameter family of Clifford tori Tr in the unit 3-sphere S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By identifying S3 with the unit sphere centered at the origin in R4, Tr, 0 < r < 1, is defined as Tr = � (x, y, u, v) ∈ S3 : x2 + y2 = r2, u2 + v2 = 1 − r2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The ‘standard’ Clifford torus is obtained by setting r = 1 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' An elementary compu- tation shows that each Tr has constant mean curvature (see [18]), Hr = 1 − 2r2 r √ 1 − r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, the Clifford tori take on all mean curvature values between −∞ and +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus, arguing as above in the sphere case, there exists an embedded torus Σt in (Mt, gt, Kt) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2), which hence is a MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' One can modify the initial data set (Mt, gt, Kt) by adding a handle from one side of the torus Σt to the other, `a la Gromov-Lawson [17], so that Σt is no longer homologically trivial, and such that the DEC still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' However, the resulting initial data manifold won’t be retractable with respect to Σt, as follows from the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, closed initial data set satisfying the DEC, µ ≥ |J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Suppose that (M, g, K) admits a MOTS Σ, with respect to a unit normal field ν, such that the following conditions hold: (I) M is retractable with respect to Σ towards ν and (II) Σ satisfies the cohomology condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then χ+ = 0 on Σ and Σ is a flat (n − 1)-torus with respect to the induced metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Moreover, the following hold: (a’) M \\ Σ ∼= (0, ℓ) × Σ for some ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σt ∼= {t} × Σ with unit normal νt in direction of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (b’) χ+ = 0 on Σt for every t ∈ (0, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (c’) Σt is a flat (n−1)-torus with respect to the induced metric for every t ∈ (0, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (d’) µ + J(νt) = 0 on Σt for every t ∈ (0, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, µ = |J| on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If we assume further that K is (n − 1)-convex, we also have: (e’) (M, g) is isometric to [0, ℓ] × Σ/∼ endowed with the induced metric from the product ([0, ℓ]×Σ, dt2+h), where h is the induced metric on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In particular, (M, g) is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' (f’) K = a dt2, where a ∈ C∞(M) depends only on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 16 (g’) µ = 0 and J = 0 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' As in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1, let (M ′, g′, K′) be the initial data set derived from (M, g, K) - by making a ‘cut’ along Σ - with two boundary components, Σ0 and S, both isometric to Σ, such that Σ0 is a MOTS with respect to the normal that points into M ′ and S is a MOTS with respect to the normal that points out of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is not difficult to see that (M ′, g′, K′) satisfies all the assumptions of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4 and then all its conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Thus Σ is a flat (n − 1)-torus with χ+ = 0 on it and conclusions (a’)-(d’) of the theorem hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If K is (n − 1)-convex, since A(Σ0) = A(S), it follows that (M ′, g′, K′) satisfies all the hypotheses of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 for ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Conclusions (e’)-(g’) then follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It follows, for example, that in a 4-dimensional spacetime which satis- fies the DEC strictly and which has toroidal Cauchy surfaces, there cannot be any homologically nontrivial toroidal MOTS in any Cauchy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This applies, in particular, to the time slices in the toroidal (k = 0) FLRW spacetimes, that satisfy the Einstein equations with dust (zero-pressure perfect fluid) source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' In view of property (g’), to find initial data sets satisfying the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3, one should perhaps consider vacuum spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' A well-known class of examples are the toroidal Kasner spacetimes, ¯N = (0, ∞) × M, ¯h = −dt2 + t2p1dx2 + t2p2dy2 + t2p3dz2, where x, y, z are to be understood as periodic coordinates, and where p1 ≤ p2 ≤ p3 must satisfy, p1 + p2 + p3 = 1 and p2 1 + p2 2 + p2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let M0 be the t = 1 time slice, with metric g and second fundamental form K induced from ( ¯N, ¯h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is not hard to show that in order for K to be 2-convex, one must have, p1 = p2 = 0 and p3 = 1, so that ¯h becomes, ¯h = −dt2 + dx2 + dy2 + t2dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' This is an exceptional Kasner spacetime, known as ‘flat Kasner’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It is locally iso- metric to Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Taking Σ to be the torus t = 1, z = z0, we see that M0 satisfies the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3, including the 2-convexity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' We mention one further example which illustrates a certain flexibility in initial data sets satisfying (I) and (II), but not the convexity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It’s a small modi- fication of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let R3 1 be Minkowski space with standard coordinates t, x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Consider the box B = {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} in the t = 0 slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let f : B → R be a smooth function that vanishes near the boundary of B and whose graph is spacelike in R3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By identifying opposite sides of the box, we obtain an initial data set (M, g, K) with M ∼= T 3, where M is given by the graph of f, and where g and K are induced from the graph of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let Σ be the intersection of M with the null hyperplane t = z − 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Because the null hyperplane is totally geodesic, Σ is necessarily a MOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' It follows that (M, g, K) satisfies (I) and (II) SOME RIGIDITY RESULTS FOR COMPACT INITIAL DATA SETS 17 with respect to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Note also that (M, g, K) satisfies the DEC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' in fact, because it essentially sits in Minkowski space, it is a vacuum initial data set, µ = 0, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Hence, (M, g, K) satisfies all the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3, except, in general, the convexity condition on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' The foliation by MOTS guaranteed by properties (a’)-(d’) comes from intersecting M with the null hyperplanes t = z + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' That these properties hold may be understood in terms of special features of totally geodesic null hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Initial data set satisfying (I) and (II) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Finally, we mention a connection to the spacetime positive mass the- orem, specifically the approach taken by Lohkamp [21], from a perspective slightly different from the discussion in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Lohkamp reduces the proof to a stand alone result, namely the nonexistence of ‘µ − |J| > 0 islands’, see [21, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' By a standard compactification (which Lohkamp also considers), the setting of Theorem 2 immediately gives an initial data set satisfying the DEC, with initial data manifold M ∼= T n ♯ Q, Q closed, and a toroidal MOTS Σ, such that M is retractable with respect to Σ (see the discussion at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3 then yields that µ = |J| (among other things), which implies Lohkamp’s no µ − |J| > 0 islands result in dimensions 3 ≤ n ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Lastly, we consider the following consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M, g, K) be an n-dimensional, 3 ≤ n ≤ 7, closed initial data set satisfying the DEC, µ ≥ |J|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Assume that −(K + g) is (n − 1)-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Then (M, g, K) cannot satisfy conditions (I)-(II) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Examples like those discussed at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='2 show that, while the conditions (I) and (II) can’t be simultaneously satisfied, either one can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Let (M ′, g′, K′) be as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' If −(K + g) is (n − 1)- convex, in particular, −(K′ + g′) is (n − 1)-convex, it follows from 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+page_content='org/abs/1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='05490 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Department of Mathematics, University of Miami, Coral Gables, FL, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Email address: galloway@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='miami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='edu Instituto de Matem´atica, Universidade Federal de Alagoas, Macei´o, AL, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content=' Email address: abraao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='mendes@im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='ufal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} +page_content='br' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AyT4oBgHgl3EQfv_kg/content/2301.00639v1.pdf'} diff --git a/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/2301.01348v1.pdf.txt b/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/2301.01348v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..23f6b6b016233707c584436c19345096627b2745 --- /dev/null +++ b/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/2301.01348v1.pdf.txt @@ -0,0 +1,416 @@ +DADAgger: Disagreement-Augmented Dataset +Aggregation +Akash Haridas +akashh@cs.toronto.edu +Karim Hamadeh +kar@cs.toronto.edu +Samarendra Chandan Bindu Dash +dashsam1@cs.toronto.edu +Abstract: DAGGER is an imitation algorithm that aggregates its original datasets +by querying the expert on all samples encountered during training. In order to +reduce the number of samples queried, we propose a modification to DAGGER, +known as DADAGGER, which only queries the expert for state-action pairs that +are out of distribution (OOD). OOD states are identified by measuring the variance +of the action predictions of an ensemble of models on each state, which we sim- +ulate using dropout. Testing on the Car Racing and Half Cheetah environments +achieves comparable performance to DAGGER but with reduced expert queries, +and better performance than a random sampling baseline. We also show that our +algorithm may be used to build efficient, well-balanced training datasets by run- +ning with no initial data and only querying the expert to resolve uncertainty. +Keywords: Imitation Learning, Ensemble +1 +Problem Description +The DAGGER algorithm [1] addresses the covariate shift encountered in behavioural cloning by +querying the expert again on all the states that the agent encounters during a test run, then aggregat- +ing these new samples with the existing dataset. +However, querying the expert can be costly in some situations. A drawback of DAGGER is that it +queries the expert at all collected observations without considering which actions are most valuable +towards training a good policy. The DRIL algorithm [2] handles the issue of covariance shift by +adding a regularizer term to the optimisation of the policy, implicitly favouring policies that choose +to enter states which minimise the variance of an ensemble of policies for these states. The rationale +is that the variance should be low for states that are in-distribution, so this should motivate the learner +policy to stay within this distribution, assuming it is sufficient to achieve proper performance. A +drawback of this method is that it may not be able to learn from incomplete datasets, and cannot +explore further. +We therefore design an algorithm that does not needlessly query the expert, while also selectively +querying it to learn about states which genuinely have not been encountered. This combines the +flexibility and exploration of DAGGER, while taking advantage of the ability of an ensemble to +identify out-of-distribution states as done in Brantley et al. [2]. +2 +Related Work +Prior work in this area mainly consists of estimating uncertainty in the predictions of deep neural +networks, as well as improving the sample efficiency of the DAGGER algorithm. +Blundell et al. [3] learn a probability distribution on the weights of a neural network, which al- +lows the network to make more reasonable predictions about unseen data. Gal and Ghahramani [4] +estimate model uncertainty from the dropout layers in deep neural networks. A common way to +estimate model uncertainty is with an ensemble of learners: the uncertainty is said to be high when +the predictions of the ensemble have high variance. Wen et al. [5] reduces the computational cost of +ensembles by sharing some weights across the networks. Brantley et al. [2] includes the ensemble +variance in an additional loss term which implicitly trains the imitation policy to avoid states for +which it has not seen demonstrations. +arXiv:2301.01348v1 [cs.LG] 3 Jan 2023 + +Several works have attempted to improve the sampling efficiency of the DAGGER imitation learning +algorithm. Kim and Pineau [6] propose a query metric that determines how close an encountered +state is from the distribution of data gathered so far, and query the expert only if this metric is above +some threshold. They use Maximum Mean Discrepancy as the metric, which was originally used to +determine whether two sets of data are from different probability distributions. Zhang and Cho [7] +achieve a similar goal by employing a separate network, called the safety policy, to predict whether +the current policy being trained is likely to make an error, and subsequently use this information to +determine which visited states need to be included in the next DAGGER iteration. Similarly, Laskey +et al. [8] uses a Support Vector Machine classifier to determine the risk level of a particular state. +Menda et al. [9] uses ensembling to estimate model uncertainty to build a probabilistic variant of +DAGGER, where they use the model’s uncertainty over its predicted actions to determine when to +query the expert. They use this to enforce model safety, rather than to select when to augment the +dataset in training as in our work. +3 +Methodology +We propose a new algorithm called DADAGGER (Disagreement-Augmented Dataset Aggregation +algorithm), which is based on DAGGER, and borrows from [2] the principle that out-of-distribution +states induce higher disagreement among an ensemble of policies. We use this to modify DAGGER +to only query the expert on states with high disagreement (a particular percentage of visited states), +in an attempt to gain maximal information and resolve the most uncertainty in a smaller number of +queries. This improves on DAGGER by reducing the number of expert calls, while still being able +to handle incomplete datasets unlike the entirely offline DRIL. Importantly, this method does not +query the expert to determine when the learner has encountered a state that it cannot handle, but +rather relies entirely on the disagreement of an ensemble of learners. +Disagreement can be defined here as the variance of the output, if the outputs admit a distance +between themselves, which is keeping with the implementation in [2] or possibly the entropy, if dis- +crete and non-transitive. Since training an ensemble of neural networks is prohibitively expensive, +in order to efficiently obtain the variance we instead use dropout layers to approximate uncertainty +of our estimators. By passing the same input M times through a single network with dropout en- +abled after every layer, we obtain a Gaussian Process sampling theoretically different networks [10]. +DADAGGER is formally stated below. Tunable hyperparameters include the percentage of states to +save, denoted by α, according to their variance, and M, the number of policies present in our en- +semble. Since we are using dropout to simulate the ensemble, the choice of M does not significantly +increase training time, as only one network has its weights updated. The network is called M times +in order to determine the uncertainty associated with a particular action. Smaller values of α query a +smaller number of samples, but in turn make convergence more difficult due to less augmentation of +the dataset. Note that during evaluation of the policy (i.e at test time), we sample the network once +to predict actions, rather than obtain and ensemble mean, to ensure a fair comparison to DAGGER. +Figure 1 shows a simplified schematic of different models agreeing over an area of the number line +R centered on the origin (representing the in-distribution states), but disagreeing in areas beyond it +(the out-of-distribution that is not being explicitly learned). +Algorithm 1 DADAgger Algorithm with Ensemble +Initialise D ← ∅ +Initialise ˆπ1,1, ˆπ1,2..ˆπ1,M +for i = 1 to i = n do +Sample T-step trajectories using ˆπi,1 +Compute Variance of ˆπi,1(s), ˆπi,2(s)..ˆπi,M(s) for all visited s. +Keep the α percent with the highest variance. +Get dataset Di = {s, π∗(s)} for all filtered s. +Aggregate datasets D ← Di ∪ D +Train classifiers ˆπi+1,1, ˆπi+1,2..ˆπi+1,M on D +end for +return best ˆπi,1 on validation +2 + +Figure 1: Simplified diagram of different functions agreeing over an area of R and disagreeing +beyond. +Algorithm 2 DADAgger Algorithm with Dropout +Initialise D ← ∅ +Initialise ˆπ1 with dropout layers +for i = 1 to i = n do +Sample T-step trajectories using ˆπi +Generate ˆyi,j = ˆπ‘i,j(s) where ˆπ‘i,j is a variation of ˆπi for which a subset of neurons are +turned off randomly using the dropout layer. +Compute Variance of ˆyi,1, ˆyi,2, ..., ˆyi,M for all visited s. +Keep the α percent with the highest variance. +Get dataset Di = {s, π∗(s)} for all filtered s. +Aggregate datasets D ← Di ∪ D +Train classifiers ˆπi+1 on D +end for +return best ˆπi on validation +4 +Experiments +4.1 +Baselines and Parameter Tuning +We implement our method on the Car Racing environment available through OpenAI Gym [11]. For +the policy network we use a CNN with dropout layers in the fully-connected section, that takes in +an image of the current state and predicts a steering action. +Our first experiment consists of varying the hyper-parameters M and α in order to gauge their effect +on the convergence of DADAGGER. Note that there are no “optimal parameters”. Increasing M +and α should hypothetically reduce the number of iterations required to reach convergence (to be +verified in testing), however these increases come at the cost of training time for more models in the +ensemble and more expert queries. We compare the performance and convergence properties of our +algorithm to DAGGER (obtained by setting both α and M to 1), as well as to an agent which queries +the expert at observed states selected at random (obtained by setting M to 1 for varying α) to test +whether the quality of sampling plays a role in achieving convergence. We run the algorithm for 10 +iterations with 5 different random seeds, and measure the proportion of runs that converged. We also +measure the distribution of actions accumulated in the dataset, to obtain a quantitative assessment of +the efficiency of the algorithm in exploring the action space. Reported standard deviation is that of +a p = 0.5 binomial estimator, which is guaranteed to be an upper-bound of error. +4.2 +Half Cheetah +After implementing our algorithm on a simple environment, we test it on the Half Cheetah MuJoCo +[12] environment. Compared to Racer, Half Cheetah involves more complex dynamics, and a higher +dimensional and continuous-valued action space. The observation is a state vector rather than an +image, therefore we swap out the CNN for an MLP (with dropout layers), keeping the rest of the +3 + +algorithm the same. We repeat each of the values of M and α used in the previous experiment, and +measure the Reward obtained by each policy at the end of each DADAGGER iteration. We compare +the performance and convergence properties with base DAGGER, as well as an agent querying the +expert at observed states selected at random. +4.3 +Dataset Construction +Our final experiment exploits the efficiency of our choice of expert queries to construct a compact, +and representative dataset that allows rapid, one-shot training. Instead of using an initial training +dataset as in previous experiments, we run DADAGGER with an empty initial dataset, and conse- +quently create the dataset over several iterations, querying the expert only to resolve uncertainty. +Our hypothesis is that the final dataset upon convergence will be fairly small, due to the inclusion +of only points which the models disagree on at different stages of training. We use α = 0.1, as +well as 50 iterations, which is the time we found needed to obtain a proper convergence using this +technique. We then measure the distribution of actions in our final dataset and compare it to the +dataset produced by DAGGER and DADAGGER when supplied with an initial dataset. +5 +Results and Discussion +5.1 +Baselines and Parameter Tuning +The results are shown in Table 1. For α = 0.4, all trained models converge in all runs, including +random querying. We hypothesise that this is due to the relatively small difference between consec- +utive frames, which means skipping a frame, on average, will not be overly detrimental to training. +With a lower α of 0.2, our algorithm, regardless of the choice of M, maintains 100% convergence, +while random querying only succeeds 40% of the time. Our algorithm thus outperforms the random +baseline on this metric (p < 0.003). Finally, on the lowest setting of α, random sampling fails to +converge entirely, while our algorithm only sometimes converges. This convergence often happens +in the 9th or 10th iteration, so it is entirely possible that convergence could follow in the cases where +it did not terminate, however we are restricting testing to 10 iterations to control across all methods. +Note that the difference between M = 25 and M = 10 is not statistically significant. +To verify the mechanism by which our improvement is obtained, we obtain the histogram of saved +actions in different settings in Figure 3 . DAGGER is seen to heavily sample straight-line actions +(close to the middle). Our algorithms are seen to substantially increase the proportion of sampled +turning actions with respect to the size of the dataset, but also use less sampling overall. Interestingly, +the starting dataset is more balanced than any of the ones resulting from DAGGER / DADAGGER +iterations, yet fails to converge, meaning the quality of a dataset is not entirely determined by how +balanced it is. +5.2 +Half Cheetah +Figure 2 shows that all variants of DADAGGER converge to a similar performance as base DAGGER +on Half Cheetah, which demonstrates that our uncertainty measurement and sampling technique is +also capable of handling a multi-dimensional and continuous action space. It achieves this with as +little as 10% of queries to the expert (α = 0.1) as base DAGGER, again suggesting that it samples +only the most important datapoints to learn a policy and resolve uncertainty. Interestingly, neither of +the methods (incl. base DAGGER) matched the expert’s performance, possibly due to the increased +complexity of the environment. +5.3 +Dataset Construction +Convergence is obtained at roughly iteration 50 (hence our decision to stop). We are more interested +in examining the nature of the constructed dataset over the performance of the policy. The final +dataset consists of 746 samples, which is markedly less than even the initial dataset we were using +for DAGGER and DADAGGER, which contained 1139 samples. It is thus far smaller than datasets +generated by the previous experiments, regardless of the choice of α. This is notable because a +policy trained on this dataset is able to converge with 0 augmentation (ie, 1 DAGGER iteration, or +in one-shot), indicating a high quality of samples. We once again look to the histogram (Figure 3e), +4 + +0 +5 +10 +15 +20 +25 +30 +DADAgger iteration +2000 +2500 +3000 +3500 +4000 +Mean return +M=1, alpha=1.0 +M=10, alpha=0.1 +M=10, alpha=0.2 +M=10, alpha=0.4 +M=25, alpha=0.1 +M=25, alpha=0.2 +M=25, alpha=0.4 +Expert return +Figure 2: All variants of DADAGGER converge to a similar performance as base DAGGER (M = 1 +and α = 1) on the Half Cheetah environment, with significantly less queries to the expert. However, +none of the methods achieve a performance comparable to the expert +M / α +0.1 +0.2 +0.4 +10 +0 ± 20 +100 ± 20 +100 ± 20 +25 +60 ± 20 +100 ± 20 +100 ± 20 +Random +40 ± 20 +40 ± 20 +100 ± 20 +Table 1: Percentage of trials that led to a successful lap on the Car Racing environment. +and find it to be the most balanced of all datasets, especially with respect to extreme sharp turns +which were relatively undersampled even in DADAGGER experiments. DADAGGER thus acts as a +way to create small and efficient datasets for one-shot training. +6 +Limitations and Conclusion +While DADAGGER is able to improve on the sampling efficiency of DAGGER, a few weaknesses +arise. Primarily, the agent is still required to engage in the environment as many times as DAGGER, +meaning there is no improvement on the number of episodes required to converge, should that be +expensive. This is particularly true in the case where the dataset was initialised to zero, as many +more episodes were required to achieve convergence. Another weakness we occasionally observed +in our tests was high confidence predictions for wrong outcomes, which could lead to skipping +essential sampling and repeating the same error. It is therefore absolutely essential for our algorithm +to function, that a certain degree of independence exist between predictors. Any bias will create +agreement in out-of-distribution states that may lead to the algorithm getting stuck in a cycle of not +sampling, failing because of that error, and not sampling the requisite states again. Indeed, 30 of +50 iterations in the dataset construction experiment were failures of relatively similar form, which +could hint at the existence of this problem in sparse datasets. +In conclusion, this paper introduces DADAGGER, a novel method to increase querying efficiency +for the DAGGER algorithm without compromising convergence. Our method matches DAGGER +performance on both the Car Racing and Half Cheetah environments. Furthermore, our technique is +efficient in its execution due to the use of dropout, and could also find use as a dataset generator due +to its selectivity. Further avenues of exploration involve exploration of alternate methods to assess +uncertainty that could allow the use of an absolute metric rather than the fraction α, and would not +require several network evaluations for each action. Another uncovered area is treating the sample +space without regard to the action, and attempting to devise a measure of how out-of-distribution a +particular sample is that is completely independent of the predictor which generates the action. +5 + +0.15 +0.10 +0.05 +0.00 +0.05 +0.10 +0.15 +Expert steering command +0 +100 +200 +300 +400 +500 +(a) Starting dataset +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Expert steering command +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +(b) Dataset after a DAGGER run +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Expert steering command +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +(c) Ours: Dataset after a DADAGGER run +with M = 10, α = 0.4 +(d) Ours: Dataset after a DADAGGER run +with M = 10, α = 0.1 +(e) Ours: Efficient dataset after a DADAG- +GER run starting with an empty intial +dataset. +Figure 3: Comparison of sampling distributions on the Car Racing environment. While the initial +dataset mostly contains straight-line examples, DAGGER samples at least some points at turns. Our +algorithm DADagger overall samples far fewer examples by being less wasteful with sampling at +straight lines (notice the difference in scale in the y-axes). Starting DADAGGER with a null initial +datset produces the most balanced of all datasets, especially with respect to extreme sharp turns +which are relatively undersampled in other experiments. +Contributions +KH was responsible for the algorithm design and Experiment 3. Experiment 1 was split among +all members. AH and SCBD conducted Experiment 2. SCBD and AH jointly wrote/adapted and +optimized most of the code used to perform all experiments. +References +[1] S. Ross, G. Gordon, and D. Bagnell. A reduction of imitation learning and structured predic- +tion to no-regret online learning. In Proceedings of the fourteenth international conference +6 + +800 +700 +600 +500 +400- +300 +200 +100 +0 +1.0 +0.8 +-0.6 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +Expert steering command300 +250 +200 +150 +100 +50 +0 +1.0 +-0.8 +-0.6 +0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +Expertsteeringcommandon artificial intelligence and statistics, pages 627–635. JMLR Workshop and Conference Pro- +ceedings, 2011. +[2] K. Brantley, W. Sun, and M. Henaff. Disagreement-regularized imitation learning. In Interna- +tional Conference on Learning Representations, 2019. +[3] C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra. Weight uncertainty in neural +network. In International conference on machine learning, pages 1613–1622. PMLR, 2015. +[4] Y. Gal and Z. Ghahramani. Dropout as a bayesian approximation: Representing model uncer- +tainty in deep learning. In international conference on machine learning, pages 1050–1059. +PMLR, 2016. +[5] Y. Wen, D. Tran, and J. Ba. Batchensemble: an alternative approach to efficient ensemble and +lifelong learning. arXiv preprint arXiv:2002.06715, 2020. +[6] B. Kim and J. Pineau. Maximum mean discrepancy imitation learning. In Robotics: Science +and systems, 2013. +[7] J. Zhang and K. Cho. Query-efficient imitation learning for end-to-end autonomous driving. +arXiv preprint arXiv:1605.06450, 2016. +[8] M. Laskey, S. Staszak, W. Y.-S. Hsieh, J. Mahler, F. T. Pokorny, A. D. Dragan, and K. Gold- +berg. Shiv: Reducing supervisor burden in dagger using support vectors for efficient learning +from demonstrations in high dimensional state spaces. In 2016 IEEE International Conference +on Robotics and Automation (ICRA), pages 462–469. IEEE, 2016. +[9] K. Menda, K. Driggs-Campbell, and M. J. Kochenderfer. Dropoutdagger: A bayesian approach +to safe imitation learning. arXiv preprint arXiv:1709.06166, 2017. +[10] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: a simple +way to prevent neural networks from overfitting. The journal of machine learning research, 15 +(1):1929–1958, 2014. +[11] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, and W. Zaremba. +Openai gym, 2016. +[12] E. Todorov, T. Erez, and Y. Tassa. Mujoco: A physics engine for model-based control. In 2012 +IEEE/RSJ international conference on intelligent robots and systems, pages 5026–5033. IEEE, +2012. +7 + diff --git a/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/load_file.txt b/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3b57727008dd86e3c7ee0f838fb386795b107ee --- /dev/null +++ b/Z9AzT4oBgHgl3EQfZPxu/content/tmp_files/load_file.txt @@ -0,0 +1,294 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf,len=293 +page_content='DADAgger: Disagreement-Augmented Dataset Aggregation Akash Haridas akashh@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='edu Karim Hamadeh kar@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='edu Samarendra Chandan Bindu Dash dashsam1@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='edu Abstract: DAGGER is an imitation algorithm that aggregates its original datasets by querying the expert on all samples encountered during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' In order to reduce the number of samples queried, we propose a modification to DAGGER, known as DADAGGER, which only queries the expert for state-action pairs that are out of distribution (OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' OOD states are identified by measuring the variance of the action predictions of an ensemble of models on each state, which we sim- ulate using dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Testing on the Car Racing and Half Cheetah environments achieves comparable performance to DAGGER but with reduced expert queries, and better performance than a random sampling baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We also show that our algorithm may be used to build efficient, well-balanced training datasets by run- ning with no initial data and only querying the expert to resolve uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Keywords: Imitation Learning, Ensemble 1 Problem Description The DAGGER algorithm [1] addresses the covariate shift encountered in behavioural cloning by querying the expert again on all the states that the agent encounters during a test run, then aggregat- ing these new samples with the existing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' However, querying the expert can be costly in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' A drawback of DAGGER is that it queries the expert at all collected observations without considering which actions are most valuable towards training a good policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' The DRIL algorithm [2] handles the issue of covariance shift by adding a regularizer term to the optimisation of the policy, implicitly favouring policies that choose to enter states which minimise the variance of an ensemble of policies for these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' The rationale is that the variance should be low for states that are in-distribution, so this should motivate the learner policy to stay within this distribution, assuming it is sufficient to achieve proper performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' A drawback of this method is that it may not be able to learn from incomplete datasets, and cannot explore further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We therefore design an algorithm that does not needlessly query the expert, while also selectively querying it to learn about states which genuinely have not been encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' This combines the flexibility and exploration of DAGGER, while taking advantage of the ability of an ensemble to identify out-of-distribution states as done in Brantley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 2 Related Work Prior work in this area mainly consists of estimating uncertainty in the predictions of deep neural networks, as well as improving the sample efficiency of the DAGGER algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [3] learn a probability distribution on the weights of a neural network, which al- lows the network to make more reasonable predictions about unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Gal and Ghahramani [4] estimate model uncertainty from the dropout layers in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' A common way to estimate model uncertainty is with an ensemble of learners: the uncertainty is said to be high when the predictions of the ensemble have high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [5] reduces the computational cost of ensembles by sharing some weights across the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Brantley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [2] includes the ensemble variance in an additional loss term which implicitly trains the imitation policy to avoid states for which it has not seen demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='01348v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='LG] 3 Jan 2023 Several works have attempted to improve the sampling efficiency of the DAGGER imitation learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Kim and Pineau [6] propose a query metric that determines how close an encountered state is from the distribution of data gathered so far, and query the expert only if this metric is above some threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' They use Maximum Mean Discrepancy as the metric, which was originally used to determine whether two sets of data are from different probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Zhang and Cho [7] achieve a similar goal by employing a separate network, called the safety policy, to predict whether the current policy being trained is likely to make an error, and subsequently use this information to determine which visited states need to be included in the next DAGGER iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Similarly, Laskey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [8] uses a Support Vector Machine classifier to determine the risk level of a particular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Menda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' [9] uses ensembling to estimate model uncertainty to build a probabilistic variant of DAGGER, where they use the model’s uncertainty over its predicted actions to determine when to query the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' They use this to enforce model safety, rather than to select when to augment the dataset in training as in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 3 Methodology We propose a new algorithm called DADAGGER (Disagreement-Augmented Dataset Aggregation algorithm), which is based on DAGGER, and borrows from [2] the principle that out-of-distribution states induce higher disagreement among an ensemble of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We use this to modify DAGGER to only query the expert on states with high disagreement (a particular percentage of visited states), in an attempt to gain maximal information and resolve the most uncertainty in a smaller number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' This improves on DAGGER by reducing the number of expert calls, while still being able to handle incomplete datasets unlike the entirely offline DRIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Importantly, this method does not query the expert to determine when the learner has encountered a state that it cannot handle, but rather relies entirely on the disagreement of an ensemble of learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Disagreement can be defined here as the variance of the output, if the outputs admit a distance between themselves, which is keeping with the implementation in [2] or possibly the entropy, if dis- crete and non-transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Since training an ensemble of neural networks is prohibitively expensive, in order to efficiently obtain the variance we instead use dropout layers to approximate uncertainty of our estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' By passing the same input M times through a single network with dropout en- abled after every layer, we obtain a Gaussian Process sampling theoretically different networks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' DADAGGER is formally stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Tunable hyperparameters include the percentage of states to save, denoted by α, according to their variance, and M, the number of policies present in our en- semble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Since we are using dropout to simulate the ensemble, the choice of M does not significantly increase training time, as only one network has its weights updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' The network is called M times in order to determine the uncertainty associated with a particular action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Smaller values of α query a smaller number of samples, but in turn make convergence more difficult due to less augmentation of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Note that during evaluation of the policy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='e at test time), we sample the network once to predict actions, rather than obtain and ensemble mean, to ensure a fair comparison to DAGGER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Figure 1 shows a simplified schematic of different models agreeing over an area of the number line R centered on the origin (representing the in-distribution states), but disagreeing in areas beyond it (the out-of-distribution that is not being explicitly learned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Algorithm 1 DADAgger Algorithm with Ensemble Initialise D ← ∅ Initialise ˆπ1,1, ˆπ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='.ˆπ1,M for i = 1 to i = n do Sample T-step trajectories using ˆπi,1 Compute Variance of ˆπi,1(s), ˆπi,2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='.ˆπi,M(s) for all visited s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Keep the α percent with the highest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Get dataset Di = {s, π∗(s)} for all filtered s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Aggregate datasets D ← Di ∪ D Train classifiers ˆπi+1,1, ˆπi+1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='.ˆπi+1,M on D end for return best ˆπi,1 on validation 2 Figure 1: Simplified diagram of different functions agreeing over an area of R and disagreeing beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Algorithm 2 DADAgger Algorithm with Dropout Initialise D ← ∅ Initialise ˆπ1 with dropout layers for i = 1 to i = n do Sample T-step trajectories using ˆπi Generate ˆyi,j = ˆπ‘i,j(s) where ˆπ‘i,j is a variation of ˆπi for which a subset of neurons are turned off randomly using the dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Compute Variance of ˆyi,1, ˆyi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=', ˆyi,M for all visited s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Keep the α percent with the highest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Get dataset Di = {s, π∗(s)} for all filtered s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Aggregate datasets D ← Di ∪ D Train classifiers ˆπi+1 on D end for return best ˆπi on validation 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 Baselines and Parameter Tuning We implement our method on the Car Racing environment available through OpenAI Gym [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' For the policy network we use a CNN with dropout layers in the fully-connected section, that takes in an image of the current state and predicts a steering action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our first experiment consists of varying the hyper-parameters M and α in order to gauge their effect on the convergence of DADAGGER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Note that there are no “optimal parameters”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Increasing M and α should hypothetically reduce the number of iterations required to reach convergence (to be verified in testing), however these increases come at the cost of training time for more models in the ensemble and more expert queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We compare the performance and convergence properties of our algorithm to DAGGER (obtained by setting both α and M to 1), as well as to an agent which queries the expert at observed states selected at random (obtained by setting M to 1 for varying α) to test whether the quality of sampling plays a role in achieving convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We run the algorithm for 10 iterations with 5 different random seeds, and measure the proportion of runs that converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We also measure the distribution of actions accumulated in the dataset, to obtain a quantitative assessment of the efficiency of the algorithm in exploring the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Reported standard deviation is that of a p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='5 binomial estimator, which is guaranteed to be an upper-bound of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2 Half Cheetah After implementing our algorithm on a simple environment, we test it on the Half Cheetah MuJoCo [12] environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Compared to Racer, Half Cheetah involves more complex dynamics, and a higher dimensional and continuous-valued action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' The observation is a state vector rather than an image, therefore we swap out the CNN for an MLP (with dropout layers), keeping the rest of the 3 algorithm the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We repeat each of the values of M and α used in the previous experiment, and measure the Reward obtained by each policy at the end of each DADAGGER iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We compare the performance and convergence properties with base DAGGER, as well as an agent querying the expert at observed states selected at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='3 Dataset Construction Our final experiment exploits the efficiency of our choice of expert queries to construct a compact, and representative dataset that allows rapid, one-shot training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Instead of using an initial training dataset as in previous experiments, we run DADAGGER with an empty initial dataset, and conse- quently create the dataset over several iterations, querying the expert only to resolve uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our hypothesis is that the final dataset upon convergence will be fairly small, due to the inclusion of only points which the models disagree on at different stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We use α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1, as well as 50 iterations, which is the time we found needed to obtain a proper convergence using this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We then measure the distribution of actions in our final dataset and compare it to the dataset produced by DAGGER and DADAGGER when supplied with an initial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 5 Results and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 Baselines and Parameter Tuning The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' For α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='4, all trained models converge in all runs, including random querying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We hypothesise that this is due to the relatively small difference between consec- utive frames, which means skipping a frame, on average, will not be overly detrimental to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' With a lower α of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2, our algorithm, regardless of the choice of M, maintains 100% convergence, while random querying only succeeds 40% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our algorithm thus outperforms the random baseline on this metric (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Finally, on the lowest setting of α, random sampling fails to converge entirely, while our algorithm only sometimes converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' This convergence often happens in the 9th or 10th iteration, so it is entirely possible that convergence could follow in the cases where it did not terminate, however we are restricting testing to 10 iterations to control across all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Note that the difference between M = 25 and M = 10 is not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' To verify the mechanism by which our improvement is obtained, we obtain the histogram of saved actions in different settings in Figure 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' DAGGER is seen to heavily sample straight-line actions (close to the middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our algorithms are seen to substantially increase the proportion of sampled turning actions with respect to the size of the dataset, but also use less sampling overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Interestingly, the starting dataset is more balanced than any of the ones resulting from DAGGER / DADAGGER iterations, yet fails to converge, meaning the quality of a dataset is not entirely determined by how balanced it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2 Half Cheetah Figure 2 shows that all variants of DADAGGER converge to a similar performance as base DAGGER on Half Cheetah, which demonstrates that our uncertainty measurement and sampling technique is also capable of handling a multi-dimensional and continuous action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' It achieves this with as little as 10% of queries to the expert (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1) as base DAGGER, again suggesting that it samples only the most important datapoints to learn a policy and resolve uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Interestingly, neither of the methods (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' base DAGGER) matched the expert’s performance, possibly due to the increased complexity of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='3 Dataset Construction Convergence is obtained at roughly iteration 50 (hence our decision to stop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We are more interested in examining the nature of the constructed dataset over the performance of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' The final dataset consists of 746 samples, which is markedly less than even the initial dataset we were using for DAGGER and DADAGGER, which contained 1139 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' It is thus far smaller than datasets generated by the previous experiments, regardless of the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' This is notable because a policy trained on this dataset is able to converge with 0 augmentation (ie, 1 DAGGER iteration, or in one-shot), indicating a high quality of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' We once again look to the histogram (Figure 3e), 4 0 5 10 15 20 25 30 DADAgger iteration 2000 2500 3000 3500 4000 Mean return M=1, alpha=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='0 M=10, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 M=10, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2 M=10, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='4 M=25, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 M=25, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2 M=25, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='4 Expert return Figure 2: All variants of DADAGGER converge to a similar performance as base DAGGER (M = 1 and α = 1) on the Half Cheetah environment, with significantly less queries to the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' However, none of the methods achieve a performance comparable to the expert M / α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='4 10 0 ± 20 100 ± 20 100 ± 20 25 60 ± 20 100 ± 20 100 ± 20 Random 40 ± 20 40 ± 20 100 ± 20 Table 1: Percentage of trials that led to a successful lap on the Car Racing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' and find it to be the most balanced of all datasets, especially with respect to extreme sharp turns which were relatively undersampled even in DADAGGER experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' DADAGGER thus acts as a way to create small and efficient datasets for one-shot training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 6 Limitations and Conclusion While DADAGGER is able to improve on the sampling efficiency of DAGGER, a few weaknesses arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Primarily, the agent is still required to engage in the environment as many times as DAGGER, meaning there is no improvement on the number of episodes required to converge, should that be expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' This is particularly true in the case where the dataset was initialised to zero, as many more episodes were required to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Another weakness we occasionally observed in our tests was high confidence predictions for wrong outcomes, which could lead to skipping essential sampling and repeating the same error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' It is therefore absolutely essential for our algorithm to function, that a certain degree of independence exist between predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Any bias will create agreement in out-of-distribution states that may lead to the algorithm getting stuck in a cycle of not sampling, failing because of that error, and not sampling the requisite states again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Indeed, 30 of 50 iterations in the dataset construction experiment were failures of relatively similar form, which could hint at the existence of this problem in sparse datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' In conclusion, this paper introduces DADAGGER, a novel method to increase querying efficiency for the DAGGER algorithm without compromising convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our method matches DAGGER performance on both the Car Racing and Half Cheetah environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Furthermore, our technique is efficient in its execution due to the use of dropout, and could also find use as a dataset generator due to its selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Further avenues of exploration involve exploration of alternate methods to assess uncertainty that could allow the use of an absolute metric rather than the fraction α, and would not require several network evaluations for each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Another uncovered area is treating the sample space without regard to the action, and attempting to devise a measure of how out-of-distribution a particular sample is that is completely independent of the predictor which generates the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='15 Expert steering command 0 100 200 300 400 500 (a) Starting dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} 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3000 3500 (c) Ours: Dataset after a DADAGGER run with M = 10, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='4 (d) Ours: Dataset after a DADAGGER run with M = 10, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content='1 (e) Ours: Efficient dataset after a DADAG- GER run starting with an empty intial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Figure 3: Comparison of sampling distributions on the Car Racing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' While the initial dataset mostly contains straight-line examples, DAGGER samples at least some points at turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Our algorithm DADagger overall samples far fewer examples by being less wasteful with sampling at straight lines (notice the difference in scale in the y-axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Starting DADAGGER with a null initial datset produces the most balanced of all datasets, especially with respect to extreme sharp turns which are relatively undersampled in other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Contributions KH was responsible for the algorithm design and Experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Experiment 1 was split among all members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' AH and SCBD conducted Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' SCBD and AH jointly wrote/adapted and optimized most of the code used to perform all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Ross, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Gordon, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9AzT4oBgHgl3EQfZPxu/content/2301.01348v1.pdf'} +page_content=' Bagnell.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Halain14, 7, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Heerlein4, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Hochedez15, 16, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Gyo5, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Poedts17, 18, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Rochus14 1 Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium, Ringlaan -3- Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Circulaire, 1180 Brussels, Bel- gium 2 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK 3 Institut d’Astrophysique Spatiale, CNRS, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Paris-Sud, Université Paris-Saclay, Bât.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 121, 91405 Orsay, France 4 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany 5 Physikalisch-Meteorologisches Observatorium Davos, World Radiation Center, 7260, Davos Dorf, Switzerland 6 ETH Zürich, Institute for Particle Physics and Astrophysics , Wolfgang-Pauli-Strasse 27, 8093 Zürich 7 European Space Agency, ESTEC, Keplerlaan 1, PO Box 299, NL-2200 AG Noordwijk, The Netherlands 8 UCL-Mullard Space Science Laboratory, Holmbury St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Mary, Dorking, Surrey, RH5 6NT, UK 9 Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA 10 Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia 11 Sorbonne Université, Observatoire de Paris - PSL, École Polytechnique, Institut Polytechnique de Paris, CNRS, Laboratoire de physique des plasmas (LPP), 4 place Jussieu, F-75005 Paris, France 12 Rosseland Centre for Solar Physics, University of Oslo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Box 1029, Blindern, NO-0315 Oslo, Norway 13 Laboratoire Charles Fabry, Institut d’Optique Graduate School, Université Paris-Saclay, 91127 Palaiseau Cedex, France 14 Centre Spatial de Liège, Université de Liège, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' du Pré-Aily B29, 4031 Angleur, Belgium 15 AESTER INCOGNITO, 75008 Paris, France 16 LATMOS, CNRS - UVSQ - Sorbonne Université, 78280, Guyancourt, France 17 Centre for mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium 18 Institute of Physics, University of Maria Curie-Skłodowska, Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Curie-Skłodowskiej 5, 20-031 Lublin, Poland Received ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' accepted ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The Extreme Ultraviolet Imager (EUI), onboard Solar Orbiter consists of three telescopes: the two High Resolution Imagers in EUV (HRIEUV) and in Lyman-α (HRILya), and the Full Sun Imager (FSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter/EUI started its Nominal Mission Phase on 2021 November 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUI images from the largest scales in the extended corona off limb, down to the smallest features at the base of the corona and chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUI is therefore a key instrument for the connection science that is at the heart of the Solar Orbiter mission science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The highest resolution on the Sun is achieved when Solar Orbiter passes through the perihelion part of its orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On 2022 March 26, Solar Orbiter reached for the first time a distance to the Sun close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' No other coronal EUV imager has been this close to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We review the EUI data sets obtained during the period 2022 March-April, when Solar Orbiter quickly moved from alignment with the Earth (2022 March 6), to perihelion (2022 March 26), to quadrature with the Earth (2022 March 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We highlight the first observational results in these unique data sets and we report on the in-flight instrument performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUI has obtained the highest resolution images ever of the solar corona in the quiet Sun and polar coronal holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Several active regions were imaged at unprecedented cadences and sequence durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We identify in this paper a broad range of features that require deeper studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Both FSI and HRIEUV operate at design specifications but HRILya suffered from performance issues near perihelion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We conclude emphasising the EUI open data policy and encouraging further detailed analysis of the events highlighted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Sun: UV radiation – Sun: transition region – Sun: corona – Instrumentation: high angular resolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Introduction The launch of the Atmospheric Imaging Assembly (AIA) on- board the Solar Dynamics Observatory (SDO, Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2012)) in 2010 heralded the era of continuous full disc coro- ⋆ Corresponding author: David Berghmans e-mail: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='berghmans@oma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='be nal imaging at high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In normal mode, AIA images are produced with a cadence of 12 s at a spatial reso- lution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5′′, over a field of view (FOV) of (41′)2 (Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Recent developments in coronal imagers have included increased fields of view and higher spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' SWAP on PROBA2 (Seaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Halain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2013) and SUVI on GOES (Darnel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022) have boosted observations with FOVs Article number, page 1 of 19 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='05616v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='SR] 13 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main of (54′)2 and (53′)2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Thanks to these larger FOVs, both SWAP and SUVI image the EUV structures and dynam- ics well beyond the AIA FOV, into what has become known as the Middle Corona (Seaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Chitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Meanwhile, the sounding rocket Hi-C (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2014) pushed the limits in terms of spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In its second successful flight (Rachmeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2019), Hi-C took (subfield) images of an active region at a cadence of 4 s and a spatial resolution better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='46′′ (330 km on the Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) is in a highly elliptical or- bit with perihelia below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 au and, in later years of the nominal mission, well out of the ecliptic, beyond 30◦ solar latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The EUI instrument (Rochus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) onboard Solar Orbiter will use this unique orbit to observe the Sun from different vantage points through three separate telescopes, imaging the outer solar atmosphere at an even higher spatial resolution than Hi-C, and over wider fields of view than SUVI and SWAP, further extend- ing the middle corona discovery space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The first EUI telescope, the Full Sun Imager (FSI) is a one- mirror telescope taking alternating images in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For a coronal EUV imager, FSI has an un- precedented large FOV: (228′)2, which has a significant overlap (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020a) with the Solar Orbiter coronagraph Metis (Antonucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At perihelion, this FOV corresponds to (4 R⊙)2 such that the full solar disc is always seen, even at max- imal offpoint (1 R⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This FOV is significantly wider than the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='34 R⊙)2 of EUVI (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2008) or the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='38 R⊙)2 of SWAP (Seaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2013b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' When at 1 au (near aphelion), this FOV corresponds to (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 R⊙)2 providing unique opportunities to image the Middle Corona and eruptions that transit through this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The other EUI telescopes are the two High Resolution Im- agers (HRIs), HRIEUV and HRILya, which are two-mirror opti- cal systems imaging through EUV and hydrogen Lyman-α pass- bands respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRIEUV images the corona at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm, which corresponds to the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm channel of FSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya, which im- ages in the Lyman-α line, shares its resonance formation process for hydrogen with the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm channel of FSI for helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV plate scale is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='492′′, the HRILya plate scale is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='514′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At the 2022 March 26 perihelion, Solar Orbiter reached a dis- tance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='323 au from the Sun, giving (single) pixel values on the Sun of (115 km)2 for HRIEUV, and (120 km)2 for HRILya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The actual spatial resolution of the telescopes is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Both HRI cameras are capable of operating at cadences in the 1 s range, over 2048×2048 pixel arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIs image through a 17′ × 17′ FOV, corresponding to (1 R⊙)2, when observing at 1 au, and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='28 R⊙)2 at perihelion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Following its launch on 2020 February 10, Solar Orbiter spent 4 months in the Near Earth Commissioning Phase, fol- lowed by 17 months of Cruise Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the Cruise Phase only the in-situ instruments were collecting science grade data, while the remote sensing instruments were undergoing extended testing in preparation for the science phase of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The Nominal Mission Phase of Solar Orbiter started on 2021 Novem- ber 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the Nominal Mission Phase, the remote sensing instruments run a non-stop synoptic observation program inter- leaved three times per orbit with 10 days periods of enhanced observational activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These periods are called “Remote Sensing Windows” (RSWs) and are typically scheduled at the perihelion of the orbit of Solar Orbiter and at the times of minimum and maximum latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In this paper we present an overview of the unique data sets collected by EUI during the very first close perihelion RSWs, covering the period from 2022 March 2 to 2022 April 6 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Trajectory of Solar Orbiter in Geocentric Solar Ecliptic (GSE) coordinates in black, starting at 2021 December 27 (square symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The Remote Sensing Windows (RSW) correspond to the orange, red, and blue parts of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In this paper we cover the period from 2022 March 02 (beginning of RSW1, orange) till 2022 April 06 (end of RSW3, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The perihelion occurred at the end of the RSW2 (red), around 2022 March 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The trajectories of the ESA mission Bepi Colombo and the NASA missions STEREO-A, and Parker Solar Probe are indicated brown, green, and purple respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' reproduced from the ESA Solar Orbiter website1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' As compared to Earth, Solar Orbiter observed from a perspective of increasing solar longitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' East to West), and transitioned from near- alignment with Earth on 2022 March 6, to perihelion on 2022 March 26, and then in a quadrature formation with the Earth, observing the Sun above the West limb, on 2022 March 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Dur- ing that period, the distance to the Sun from Solar Orbiter varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='32 au to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='55 au, closer to the Sun than any coronal EUV imager before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Given the variable angle with the Earth, the vari- able distance to the Sun, and the constraints of the low teleme- try bandwidth, the instrument operations of the Remote Sensing Payload on Solar Orbiter was highly non-synoptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The aim of the paper is to guide the EUI data user through the unique but very variable EUI observations that were collected in the period 2022 March 2 to April 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In Section 2 we present the various EUI observation campaigns that were taken during the perihelion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In Section 3 we give an overview of the observational highlights that were completed in these campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' After that, in Section 4, we describe for each EUI telescope the instrument performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Finally, in Section 5, we give an outlook for upcoming orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' From Science Goals to EUI Data sets As Solar Orbiter is a deep space, non-synoptic probe, the activ- ity scheduling of its instruments is coordinated well in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is done through Solar Orbiter Observing Plans (SOOPs), which bring a group of Solar Orbiter instruments in a specific mode to target a certain science goal at appropriate times during 1 https://issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='int/solarorbiterwiki/display/SOSP Article number, page 2 of 19 SolarOrbiter-ParkerSolarProbe STEREO-Ahead BepiColombo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='25 Y (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='00 GSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 GSE X (AU)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These science goals have been generically described in Zouganelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2020) and updates are maintained on the ESA Solar Orbiter website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In order to document the intended purpose and context of the collected observations, we here summarise the SOOPs that the EUI instrument was involved in during the 2020 March 2 to April 6 period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Grouping the SOOPs by science target and by increasing operational complexity, we arrive at three themes: (1) observing the Middle Corona off limb, (2) observing the build- ing blocks of the solar atmosphere and, (3) making the connec- tion from high resolution on-disk observations to in-situ mea- surements of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The full technical details of the re- sulting EUI data sets can be found in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 in Appendix 5, where entries are linked to corresponding SOOP names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Observing the off limb Middle Corona The first set of SOOPs focused on the off limb corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Most of these SOOPs are led by the Solar Orbiter coronagraph Metis, which requires Sun center pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This makes the SOOPs op- erationally simpler as no last-minute pointing corrections are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For these SOOPs, EUI FSI images are prioritized due to the large overlap with Metis (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020b) with Metis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Optional HRI images are necessarily pointed near disc center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The aim of the L_FULL_HRES_MCAD_Coronal-He- Abundance SOOP on 2022 March 7 was to support observations during the second launch of the Herschel sounding rocket, whose first flight provided the first helium abundance maps of the so- lar corona (Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The abundance is deduced from the ratio of the resonantly scattered intensities of neutral hydro- gen and singly ionized helium, as imaged by two coronagraphs: SCORE (Romoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2003) and HeCOR (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The method is model-dependent, and requires an independent knowledge of the temperature of the scattering ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This can be constrained by simultaneous EUV observations, which was the purpose of the FSI observations at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In order to minimize stray-light at large distances from the solar limb, the instrument was used in coronagraph mode, with a movable disk masking direct sunlight (see Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Rochus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' One of the returned images is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2, composited with the closest in time disk image taken before the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The sounding rocket payload failed, but the FSI data are still very useful to study the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm emission of the extended corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The composite EUV image allows to link the magnetic structures on the disk (coronal holes, plumes, active regions) to magnetic field expansions in the extended corona, which appear always open but far from being simply radially aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The L_FULL_HRES_HCAD_CoronalDynamics SOOP is designed to observe structures in the outer corona with the aim of linking them to the solar wind observed in-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The SOOP was run twice, on 2022 March 22 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 au) and March 27 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='32 au) just after perihelion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Given the outer corona focus, the main contribution of EUI was through FSI observations in both wavelengths which provide a large overlap with Metis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Never- theless, HRI images of the quiet Sun at disc center were also taken at relative low cadence (30 s to 60 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' While these HRI im- ages are perhaps not directly useful for the SOOP goal, they are unique observations as, given the close solar approach, they were the sharpest quiet Sun EUV images ever taken during the first perihelion passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The goal of the L_FULL_HRES_HCAD_Eruption-Watch SOOP is to observe eruptive events and to contribute to the un- derstanding of Coronal Mass Ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The SOOP was carried out in two campaigns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' the first one on 2022 March 22-23, and the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' FSI image taken in coronagraph mode (outer FOV) on 2022 March 7 at 16:00:05 UT, composited with a regular disk image (cen- ter) taken at 11:29:45 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The images were enhanced using the WOW algorithm (Auchère et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022) to reveal faint features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' second one between 2022 March 29-30 (close to Solar Orbiter in quadrature at the west side of Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' There was long term mon- itoring with the FSI (every 6 min) while the HRIs operated in 30 min bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Observed active regions on 2022 March 2 (left), March 17 (mid- dle) and April 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Observing the building blocks of the on-disc solar atmosphere A second set of SOOPs targeted various features in the on- disc solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These SOOPs are typically led by SPICE (SPICE Consortium et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) and/or EUI and require point- ing the spacecraft to targeted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Pointing corrections were made a few days in advance, commanded through the so-called "Pointing-Very Short Term Planning" (pVSTP, Zouganelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2020)) and based on Low Latency FSI images that are brought to the ground as a high priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These are typically less than a day old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The observation of small-scale EUV brightenings in the quiet Sun was an early success of EUI (Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Therefore, particular attention was paid to scheduling the R_BOTH_HRES_HCAD_Nanoflares SOOP that is focused on surveying small impulsive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' There are many open ques- tions on their origin and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Are they also located in ac- tive regions and coronal holes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Similar events were observed in active regions by the Hi-C, but it is not clear yet if they Article number, page 3 of 19 5000" Helioprojective Latitude (Solar-Y) O" 5000" 5000" 0" 5000" Helioprojective Longitude (Solar-x)12967 12975 12956 12958 12957 12976A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main are the same phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' If this is the case, do they have the same properties everywhere?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Recent work suggests that a sub- population of the small-scale EUV brightenings does not reach the 1 MK (Dolliou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022, (this volume, in revision);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022, (this volume, in prep)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Answering these questions will help us understand, for instance, their origin and relationship with the small scale magnetic field structuring and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The R_BOTH_HRES_HCAD_Nanoflares SOOP was sched- uled several times near the Sun-Earth line (2022 March 7), at an angle with the Earth of roughly 30◦ and near quadrature (2022 March 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The spacecraft pointed alternatively to an active re- gion and to the quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This SOOP resulted in thousands of HRIEUV and HRILya images obtained with a cadence of usually 3 s (HRIEUV) and 5 s (HRILya) for a duration of approximately 30 min (co-temporal in both channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV initial plan was to run at 2s cadence, however the first run on 2022 March 6 failed and the cadence was decreased to 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This very high cadence period was, in general, followed by longer periods at lower cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This choice came from the allocated telemetry and the need for long high cadence temporal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These campaigns were widely supported by other instruments on Solar Orbiter but also by IRIS (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2014), Hinode (Ko- sugi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2007), PROBA2/LYRA (Dominique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2013) and SDO/AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The latter instrument ran in a restricted configuration (subfield, few wavelengths) to image at an enhanced 6s cadence in 4 wavelengths (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm chan- nels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The R_SMALL_HRES_MCAD_Polar-Observations SOOP, as described in Zouganelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2020), aims at observing the polar magnetic fields by PHI (Solanki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In this early phase of the mission however such observations are not ideal yet, as the spacecraft is still at low solar latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Instead, the focus of the SOOP was to observe the polar coronal holes with the high resolution imagers of EUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Such observations are timely as polar coronal holes will soon disappear with the rising solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This SOOP was carried out three times, once when Solar Orbiter was close to the Sun-Earth line (2022 March 6), once when Solar Orbiter was close to quadrature with Earth (2022 March 30) and once more when Solar Orbiter was at roughly 120◦ with the Earth (2022 April 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In the first two campaigns, the solar south pole coronal hole was the target but in the last campaign the solar north pole coronal hole was more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In particular the 2022 March 30 observation of the south pole returned the ‘best ever’ EUV images of a polar coronal hole as they were taken from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 au with an imaging cadence of 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The R_BOTH_HRES_MCAD_Bright-Points SOOP is SPICE-focused and aims at observing coronal bright points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This SOOP was carried out between 2022 March 8 08:10 and 14:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During that time, FSI operated with a relatively high time cadence (5 min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRIEUV and HRILya observed during 2 hours at 1 minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The EUI telescopes pointed at disc centre (quiet Sun) and bright points were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The R_SMALL_MRES_MCAD_AR-Long-Term was car- ried out between 2022 March 31 and 2022 April 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The goal was to track the decay phase of an active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Two good candidates appeared on the solar disk a few days before the SOOP started: NOAA AR2 12975 and NOAA AR 12976 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Com- paring both regions, the leading polarity of NOAA AR 12976 provided a good target and the EUI FOV was chosen centered on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' There was long term monitoring with FSI (every 30 min) and burst image sequences with both HRIs, typically with a time cadence of 10 s and lasting 47 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The regions observed dur- 2 Active region numbering by https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='swpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='noaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='gov ing these few days produced several flares, including an M-class flare on 2022 April 2 that is discussed further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On 2022 March 7, Solar Orbiter crossed the Sun-Earth line, at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='49 au, allowing for cross-calibration with sim- ilar Earth-bound instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For a complete inter-comparison, a full range of scenes (quiet Sun area, an active region or a coro- nal hole) was to be targeted within the small FOV of PHI/HRT, HRIEUV, HRILya and SPICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' However, by having Solar Orbiter point in a 5x5 pattern, these high resolution telescopes could however make a Full Disc Mosaic of the whole solar disc, thereby avoiding upfront guessing the position of the various scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter followed the 5x5 pointing pattern from north-east (top left) to south-west (bottom right) in columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Im- ages in subsequent pointing positions are 10 min apart in the ver- tical direction and 50 min apart in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' To en- sure a maximum overlap between image panels, the HRIEUV im- ages were commanded to be 2368x2368 in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On average, the images between dwells overlapped by 600 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV telescope took high-gain and low-gain image pairs every 30 s within each dwell period, resulting in 9 such image pairs per pointing position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The high- and low-gain images were taken 5 s apart and calibrated and combined on-ground into high dy- namic range images (15-bit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' To create a high resolution mo- saic of the full Sun, these high dynamic range images from each dwell position were aligned and stitched together using affine image transformation making use of the spacecraft attitude in- formation available in the source FITS files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The resulting panels were then blended together manually in photo editing software, minimizing image artifacts in the mosaic caused by changing views in neighboring panels due to dynamic events and solar ro- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The panels were blended together preferentially in quiet Sun areas, avoiding the faster changing active regions where pos- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The final mosaic has more than 83 million pixels, making it the highest resolution image of the Sun’s full disc and corona ever taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' An interactive version of the image can be found in Kraaikamp (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Making the connection from high resolution on disk to in-situ A third set of SOOPs aimed at finding the connection between the smallest features imaged on disc to the corresponding in-situ measurements of the out flowing solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The goal of the L_SMALL_MRES_MCAD_Connection- Mosaic SOOP is to identify, with SPICE and the high resolution imagers of EUI and PHI, the connection point on the solar disc for the solar wind observed in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In order to increase the proba- bility of successfully catching the connection point, this SOOP is implemented in combination with a mosaic of spacecraft point- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The SOOP was run twice, once when Solar Orbiter was near the Sun-Earth line (2022 March 2-3, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='56 au) and once when Solar Orbiter was in quadrature with Earth (2022 March 30, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In the first instance, the spacecraft made a mo- saic of 3 vertically aligned pointings, in the second instance, a mosaic of 3x2 pointings was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Each of these pointings was maintained for several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' It was later discovered that solar rotation was not correctly compensated, so the EUI FOV slightly shifts over this duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The location of the mosaics was decided a few days in advance with the help of various models and tools (Rouillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the first instance of this SOOP, EUI observed an M2 flare in the high resolution FOV, see sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The L_SMALL_HRES_HCAD_Slow-Wind-Connection SOOP (Yardley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' et al, 2022, this issue, in preparation) was Article number, page 4 of 19 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission HRIEUV | 2022-03-04 |10:48:50 0 100 200 300 Mm 0 100 200 300 Mm 0 100 200 300 0 100 200 300 aaaa NOAA 12957 S1 S1 S1 S1 S2 S2 S2 S2 S3 S3 S3 S3 20 Mm 20 Mm 20 Mm 20 Mm bbbb Slit-S1 0 10 20 30 40 50 Time (min) 0 5 10 15 Mm cccc Slit-S2 0 10 20 30 40 50 Time (min) 0 2 4 6 8 10 12 14 dddd Slit-S3 0 10 20 30 40 50 Time (min) 0 5 10 15 eeee Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Example of decayless kink waves observed in AR12957 on 2022 March 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Panel b shows a zoom into the loops in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The boxes S1 to S3 mark slits along which the temporal evolution is shown in the form of space-time diagrams in panels c to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' To enhance the appearance of the oscillating threads in these space-time diagrams, a smooth version of the map (boxcar-smoothed in the vertical direction) is subtracted from each original map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The black lines represent fits for the oscillation of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' designed to combine the remote-sensing and in-situ capabilities of Solar Orbiter, observing the source of solar wind connected to the spacecraft and then detecting the plasma released from the Sun as it passed over the spacecraft several days later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Although the primary instruments used in the SOOP are SPICE and SWA (Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020), the EUI HRI observations provided high resolution context images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Due to the need to identify where material passing over the spacecraft was originally ejected from the Sun, the SOOP coordinator relied heavily on the connectivity tool developed by IRAP (Rouillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) to identify the origin of the magnetic field predicted to be connected to Solar Orbiter during the observing window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For the first observing window (2022 March 3-6), this was the boundary between NOAA AR 12957 and a nearby equatorial coronal hole (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For the second window (2022 March 17-22), two different targets were selected due to a change in the connectivity of the spacecraft with respect to the solar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The boundary of the southern polar coronal hole was selected as the target for the first part of the observing window, with the positive polarity of NOAA AR 12967 in the northern hemisphere selected as the target for the second part of the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The L_BOTH_HRES_LCAD_CH-Boundary-Expansion SOOP is similar to L_SMALL_HRES_HCAD_Slow-Wind- Connection discussed above, but it specifically aims to study coronal holes boundaries as possible sources of the slow solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The SOOP was active between 2022 March 25 19:40 and 2022 March 27 00:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' FSI acquired 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm images at 10 min cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The quiet Sun at disc centre was observed in both campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Observational highlights In the previous section, the EUI observations of the 2022 March 2–April 6 period were presented from a science planning per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar activity however seldom follows the science plan so we additionally review the actual observational highlights here that were identified so far in the collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' While this 35-day period is longer than a solar rotation period, its sub-solar Article number, page 5 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main point ranged from 64◦ Carrington longitude in the beginning of the period to 95◦ Carrington longitude at the end of the period, due to the intrinsic motion of the spacecraft over the same pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The longitudinal angle Earth-Sun-spacecraft ranged from 8◦ to 122◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In what follows we use EUI Data Release 5 (Mam- paey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Active region dynamics Several active regions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3) were observed in the HRIEUV and HRILya FOVs with imaging cadences sometimes as fast as 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Below we highlight a sample of some particular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Decayless kink oscillations NOAA AR 12957 was observed first as part of the mosaic pat- tern of L_SMALL_MRES_MCAD_ConnectionMosaic (2022 March 2, 3) and then as part of the daily high resolution bursts of L_SMALL_HRES_HCAD_Slow-Wind-Connection on 2022 March 3, 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At this time, Solar Orbiter was at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='544 au from the Sun, resulting in an HRIEUV pixel footprint of 194 km on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The core of the active region showed a myriad of counter- streaming loops, of which some exhibited decayless kink oscil- lations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This event is of particular interest due to the fact that these oscillating loops are rooted inside sunspots which are generally devoid of supergranular flows, the commonly assumed driver of such decayless kink oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Moreover, these de- cayless oscillations were only observed during specific time in- tervals although the loop environment remained more-or-less similar throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Further details on the magnetic configura- tion of those loop footpoints, as well as the existence of other possible wave drivers are presented in Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Braiding loops As a part of the R_BOTH_HRES_HCAD_Nanoflares SOOP on 2022 March 17, HRIEUV observed active region AR12965 at a cadence of 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These are among the first highest cadence EUV images of an active region ever observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During this period, Solar Orbiter was at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='38 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Thus the 2-pixel foot- print of HRIEUV was about 270 km on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' An overview of the observed active region is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These high- resolution, high-cadence observations of this active region re- vealed a number of impulsive EUV brightenings on timescales of a few minutes or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A closer look at some of the brighten- ings revealed that they are associated with untangling of braided coronal strands or loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Most of these events are observed in shorter, low-lying loop features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRIEUV also observed untan- gling of coronal braids in a more conventional loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A sequence of this untangling of braided loops is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5b– j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' More details on examples of braided structures observed by HRIEUV and the implications for coronal heating are discussed in Chitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Highly dynamic cooler loops The region of interest of R_SMALL_MRES_MCAD_AR-Long- Term on 2022 April 1 was on the east limb of the Sun from the vantage point of Solar Orbiter, which means that the same re- gion appeared in the western hemisphere from the viewpoint of Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6a shows the full FOV of HRIEUV, with NOAA AR 12975 on the western side and NOAA AR 12976 on the east- ern side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Besides imaging instances of coronal braids in low- 2022-03-17 UT 03:32:03 (a) 20 Mm UT 03:31:33 (b) 2 Mm UT 03:32:03 (c) UT 03:32:33 (d) UT 03:33:03 (e) UT 03:33:33 (f) UT 03:34:03 (g) UT 03:34:33 (h) UT 03:35:03 (i) UT 03:35:33 (j) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Example of a relaxation of braided coronal loops observed on 2022 March 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Panels (b) to (j) display a zoom into the loop in panel (a) (white box) and show the evolution of the untangling within the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' lying loop systems in both these active regions (see discussion in Chitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022a), HRIEUV captured a highly dynamical sys- tem of cooler loops that appear darker in EUV due to absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These dynamic loops were observed in the core of AR12975 (see white box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These are likely related to chromo- spheric arch-filament systems associated with emerging flux re- gions (van Driel-Gesztelyi & Green 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Individual strands or loops within this system exhibited intermittent brightenings in EUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In addition, there are also repeated compact brightenings on one end of this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The morphology of these compact EUV brightenings appear akin to the transition region ultraviolet bursts that are often observed in emerging flux regions (Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The evolution of this region over a period of 1 hour is displayed in a sequence of images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6b–j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Brightening on border of dark material During the R_SMALL_MRES_MCAD_AR-Long-Term SOOP on 2022 April 1, HRIEUV observed brightening events at the top of dark jet-like structures The angular separation between the Earth and Solar Orbiter was 104◦ which allowed for simultane- ous observation from SDO/AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='35 au from the Sun, resulting in a spatial resolution of HRIEUV images (two Article number, page 6 of 19 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission (a) 2022-04-01 UT 10:28:51 20 Mm UT 09:24:42 (b) 2 Mm UT 09:33:02 (c) UT 09:41:21 (d) UT 09:49:41 (e) UT 09:58:02 (f) UT 10:06:21 (g) UT 10:14:41 (h) UT 10:23:01 (i) UT 10:31:21 (j) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Example of a highly dynamic cool loop system in the core of active region AR12975 observed by HRIEUV on 2022 April 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 pixel size) of 248 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The dark jet-like structures appear to be the so-called light walls (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2016) or fan-shaped jets (Robustini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2016), which are field-aligned long chromo- spheric jets thought to be produced by magnetic reconnection in the photosphere (Bharti 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Two kinds of brightening events can be distinguished (see time-distance map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The first kind is continuously present at the chromosphere-corona interface of the jets and is seen to oscillate up and down with a ballistic motion, strongly suggesting that this corresponds to the upward motion of the transition region observed by HRIEUV (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Its narrow thickness (as small as 200 km or less) and strong brightness is probably due in part to the passage from high to low density and cool to hot plasma, to which HRIEUV is particularly sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The second kind of brightening is far more impulsive, with life times on the order of 10 s or less, and appears on top of the first kind as pertur- bances propagating upwards at speeds of ≈ 100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Both brightening events are therefore likely due to slow mode shocks generated from the reconnection events lower down, first propa- gating in the chromosphere and then into the corona at the tube speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The second kind is seen to originate when the dark struc- ture is at the lower end of the oscillation range, as expected from chromospheric shocks leading to spicule-like events (Heggland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2007, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='!!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='"#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='"$%"&\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content="('#!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' )*%)+,-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='/01 " 2 $" $2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 3" 4)5678 " 2 $" $2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 3" 9)5678 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' :# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' :( 3:" 3:!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3:# 3:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' <=>+?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' @/A@BC/9)5DEFB81 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRIEUV observations of the brightening on a border of dark ma- terial in an active region on 2022 April 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The white arrows in the top panel mark the brightening location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The bottom panel shows a time- distance map along one of the jets observed propagating from the bright structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Note the existence of 2 kinds of brightening events, the bright edge of the dark structure oscillating up and down, and jets propagating upwards (both indicated by the arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Coronal rain Coronal rain appears ubiquitous on-disk in AR NOAA 2975 and 2976 observed on 2022 March 30, April 1 and 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In HRIEUV, coronal rain can be clearly distinguished in EUV absorption by its dynamics (with velocities close to 100 km s−1 in the plane-of-the-sky) and its clumpy and multi-stranded mor- phology (Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Antolin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At the spatial res- olution of ≈250 km, individual coronal rain clumps only a few pixels wide can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Their morphology is strongly rem- iniscent of Hα high-resolution observations (Antolin & Rouppe van der Voort 2012), a similarity that has been predicted but so far only observed at larger loop scales (Anzer & Heinzel 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Coronal rain showers (composed of clumps) can be observed in loop bundles rooted to moss, but both clumps and showers (despite the large shower widths above 10 Mm) ap- pear mostly unresolved in AIA passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This picture therefore constitutes a major difference to previous EUV observations of active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The on-disk observation at high resolution pro- vides a connection to the chromosphere (and photosphere with PHI), thus providing a unique insight of the heating events at the footpoints that lead to thermal non-equilibrium and instability associated with this phenomenon (Antolin & Froment 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A Article number, page 7 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Top: Field-of-view of the 2022 March 30 dataset observed by HRIEUV showing an active region at the South-East limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The solid white curves follow several trajectories of coronal rain clumps seen in EUV absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Bottom panels: Snapshots at various instances of a coronal rain shower, corresponding to the small white rectangle shown in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The black arrows indicate the head of the coronal rain shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' full paper reporting coronal rain observed with HRIEUV is avail- able in Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2023 (this volume, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Hints of torsional Alfvén waves in twisted coronal loops In AR NOAA 2975 observed on 2022 April 2nd twisted, in- tertwined coronal strands can be observed in the plane-of-the- sky, appearing and disappearing on a timescale of 5 − 10 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The strands disappear by the end of the sequence with hints of untwisting, ending in coronal rain falling onto bright foot- points rooted in moss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The entire event is strongly reminiscent of the coronal loop model of Díaz-Suárez & Soler (2021) in which, torsional Alfvén waves propagate along a twisted flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The radially varying magnetic field due to the twist pro- vides an Alfvén continuum that allows phase mixing to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The Kelvin-Helmholtz instability is then generated due to the ve- locity shear at the phase mixing layers, which leads to compres- sion of the plasma and the generation of coronal strands that fol- low the twisted flux tube (a process also observed for kink waves, Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The time-scale of appearance/disappearance of strands, their morphology and change in orientation during the oscillation is seen to match the one observed in this event with HRIEUV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Top: Field-of-view of the 2022 April 2 dataset observed by HRIEUV showing an active region at the South-East limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The solid white rectangle shows a twisted coronal loop where hints of torsional Alfvén waves may be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Bottom panels: snapshots over 30 min evolution of the loop observed in the field-of-view corresponding to the white rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Note the change in orientation of the twisted EUV strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Large-scale reconfiguration of coronal loop sub-structure In AR NOAA 2975 observed on 2022 April 1, a large scale loop bundle rooted in moss is composed of various strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Without the presence of any flare in the vicinity, the strands undergo a co- herent reconfiguration akin to contraction following a flare (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The strands also exhibit continuous kink motions dur- ing the global contracting motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The overall event is accompa- nied by coronal rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Coronal strands associated with coronal rain In AR NOAA 2975 observed on 2022 April 1, coronal strands rooted in moss are observed to appear and disappear on a short time-scale of tens of minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Contrary to usual coronal strands, these appear first near the loop apex, and are seen to extend dy- namically towards the footpoints in a flow-like manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is followed by localised dark or bright features at the loop apex with the appearance and dynamics of coronal rain (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The entire event strongly resembles the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5D MHD numerical modelling of coronal rain by Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 8 of 19 20 Mm 2022-03-30UT00:28:00 00:28:39 00:28:54 00:29:09 00:29:24 00:29:39 00:29:54 00:30:09 00:30:2420 Mm 2022-04-02 UT 09:19:15 09:35:55 09:40:05 09:44:15 09:48:25D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Top: A sub-field of the 2022 April 1 dataset observed by HRIEUV (located on the bottom-right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A loop bundle is seen (inverted in this figure in order to have the apex on top), where a large scale reconfiguration is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Bottom: Time-distance plot across the apex of the loop (dashed white curve on top panel), revealing a down- ward (inward) motion of many loops (akin to contraction), indicated by the black arrows, accompanied by transverse oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The time of the snapshot on the top panel corresponds to the vertical white dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' M2 flare: 2022 March 2 The chances to observe flares in the small FOVs of the HRIs, which are only operated a small fraction of an orbit, are not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Nevertheless, on 2022 March 2, during the mosaic pat- tern of L_SMALL_HRES_HCAD_Slow-Wind-Connection, an M2 flare was observed in active region NOAA 12958 by HRIEUV and HRILya (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is the largest flare seen so far in the HRI subfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Since the HRIs FOV only covered the lower part of this active region, we have in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12 also used AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm images to show the context of the evolution of the entire active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter was at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='55 au from the Sun, meaning that the spatial resolution of the HRIEUV images (two pixel size) is 397 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The imaging cadence was unfortunately only 2 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12 shows the two ribbons shortly before the flare peaked at 17:34 and HRIEUV saturated with a front filter diffraction pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV images show several small-scale brightenings during the pre-flare time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Views from HRIEUV and HRILya are shown within the white boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Thanks to the high resolution of HRIEUV (left hand side panels), we can observe in detail the brightening structures in this active region, such as the double J-shaped brightening in the core region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In the right hand side panels, the lower resolution and the saturated signal of the HRILya images only allow distinguishing the outline of the brightening structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' After the peak time, some brightenings can be found at the upper edge of the HRIs FOV appearing at the source region and propagating from west to east, forming a long thin bright band of several tens Mm, as shown in the right hand side panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Flare loops are clearly visible in the HRIEUV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Top: A sub-field of the 2022 April 1 dataset observed by HRIEUV (located on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A loop bundle rooted on moss is seen composed of various strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The strands appear first near the apex and exhibit bright and dark flows with dynamics characteristic of coro- nal rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Bottom: Time-distance plot along one of the observed strands (dashed white curve on top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Note the fuzzy brightening events along the middle of the strand followed by bright or dark flows toward either footpoint of the strand (indicated by the white arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The time of the snapshot on the top panel corresponds to the vertical white dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' image, while the flare ribbons (and flare-driven rain) are visible as two parallel bright structures in the HRILya image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 13 shows that the temporal variation of the intensities observed by HRIEUV and HRILya are in qualitative agreement with comparable observations by SDO/AIA and GOES/LYA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV intensity corresponds well with the SDO/AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm, only the emission observed in SDO/AIA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm occurs few minutes after the rest of the presented lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRILya and GOES/LYA intensity curves corresponds well too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Quiet Sun features In the current phase of the solar cycle, most active regions appear beyond 15◦ solar latitude North or South, meaning that when- ever Solar Orbiter was not off pointed away from disk center, the HRIs had the quiet Sun in the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Below we present a small sample of quiet Sun features and events that illustrate the quality of the obtained data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 9 of 19 HRIEUV - 20220401 - UT09:55:55 60 50 40 [Mm] 30 20 10 0 0 20 40 60 80 100 120 [Mm] 40 [Mm] 30 20 Distance 10 0 0 20 40 60 Time [min]HRIEUV - 20220401 - U109:44:15 25 20 15 [Mm] 10 5 0 0 10 20 30 40 [Mm] 40 「Mm] along path 30 20 Distance 10 0 0 10 20 30 40 50 Time [min]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main AIA 171 & EUI-HRIEUV 174 0 100 200 300 400 Solar Orbiter: 2022-03-02 17:34:01 UT AIA 304 & EUI-HRILYA Solar Orbiter: 2022-03-02 17:34:01 UT AIA 171 & EUI-HRIEUV 174 0 100 200 300 400 X [Mm] 0 100 200 300 400 Y [Mm] Solar Orbiter: 2022-03-02 17:44:01 UT AIA 304 & EUI-HRILYA 0 100 200 300 400 Solar Orbiter: 2022-03-02 17:44:01 UT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' M2 flare on 2022 March 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Left hand side panels: combination of AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm images and HRIEUV 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm, the latter being shown within the white box (only few seconds apart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Right hand side panels: Same, but for the combination of AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm and HRILya images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The bottom row is taken 10 min later than the top row images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The white boxes delineate the boundaries of the areas showing data from EUI- HRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 17:28 17:30 17:32 17:34 17:36 17:38 Start Time (02−Mar−22 17:28:00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 normalized count rate (AIA, HRI) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='65 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='70 GOES/LYA at 1 AU [W/m 2]x10 −3 AIA 304 AIA 171 AIA 94 HRI/LYA HRI/EUV GOES/LYA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Temporal evolution of intensity of the M2 flare on 2022 March 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The colour-coded lines present average intensities at the flare region observed with HRIEUV, HRILya, SDO/AIA and GOES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Small-scale EUV brightenings On 2022 March 26, Solar Orbiter reached its first perihelion dur- ing the Nominal Mission Phase at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='323 au from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV observations closest to perihelion were taken on March 27, starting at 19:40 (distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='324 au) and were part of the L_FULL_HRES_HCAD_Coronal-Dynamics SOOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On this day, HRIEUV had a pixel footprint on the sun of (115 km)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Small EUV brightenings observed by HRIEUV, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' camp- fires, were first identified by (Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2021) in data taken at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='556 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The observed campfires were typically elon- gated structures from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 Mm to 4 Mm with aspect ratios be- tween 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 14 we show a subfield, particularly rich in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Examples of a group of campfires observed near perihelion on 2022 March 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A long and slender campfire observed on 2022 March 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The left image shows the calibrated images (L2) with the original sensor pixelisation, each pixel corresponds to (115 km)2) on the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On the right, we show various cross-cuts through the campfire demonstrating that the FWHM of the feature is 1-2 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' campfires, taken on 2022 March 27 at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='324 au from the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The 2022 March 27 L_FULL_HRES_HCAD_Coronal- Dynamics dataset has a cadence of 1 min but several of the R_BOTH_HRES_HCAD_Nanoflares data sets have cadences down to 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The visually identified sample of campfires in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 14 appear somewhat smaller (none is larger than 2 Mm) but this needs to be confirmed by objective algorithmic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 15 shows an example of particular long and slender campfire demonstrating that loop-like features are present in the quiet Sun with a width of the order of 200 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' It also demon- strates that the spatial resolution of HRIEUV is pixel-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUV network flares Also at larger scales than campfires (say ≳ 10 Mm), flare-like brightenings are frequently seen in the quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 16 shows (top-right) the location of 4 such flare-like brightenings in a typi- cal quiet Sun scenery observed by HRIEUV on March 27/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The time evolution of the 4th event is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 17 with corre- sponding SDO/AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm imagery in quadrature showing the off-limb evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In X-rays, such events have been called Net- work Flares (Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Attie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV extreme high resolution EUV images of the quiet Sun confirm that these brightenings do indeed show many of the usual flare at- tributes such as pre-flare sigmoids, dimmings, ribbons and post- flare loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At least for some of these (like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 17) we can also confirm jets and filament eruptions, making them candidate sites for mini-CMEs (Innes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Sterling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 10 of 19 20:51 20:52 20:53 20:54 20:55 20:56 20:57 20:58 1 Mmprofile across feature 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 1-2 pixels FWHM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 00 0 10 15 20 25 30 pixelsD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUV network flares observed in the quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The top-left panel shows an FSI 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm image with the white rectangle indicat- ing the HRIEUV FOV that is zoomed in on the top-right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On this HRIEUV image, taken at a time without network flares, the location of four network flares is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Three of these network flares are shown in the bottom row of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The 4th event is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV pixels correspond to (115 km)2 on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Time evolution of an EUV network flare seen by HRIEUV on 2022 March 28 at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm (left) near disc center and by SDO/AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 nm near the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This event corresponds to location 4 in the top- right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The off limb data confirm the eruptive character of this EUV network flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Polar coronal holes Fine-scale structure and dynamics of coronal holes is of partic- ular interest for studies of the fast solar wind origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Cir- tain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Poletto 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Due to the progression of the ascending phase of the solar cycle, polar coronal holes were shrinking in 2022, so it was important to observe these struc- tures at high spatio-temporal resolution early in the Solar Orbiter mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the first perihelion passage in 2022, this was done on three occasions: on March 6, March 30, and April 4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The R_SMALL_HRES_MCAD_Polar-Observations SOOP was used (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On March 30 Solar Orbiter was situated at the distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 au from the Sun, and HRIEUV reached the two-pixel spa- tial resolution of around 240 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This was the highest ever spa- tial resolution reached in coronal hole observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' An excellent Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Examples of filaments and prominences observed in HRIEUV in 2022 March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' South polar coronal hole imaged by HRIEUV on 2022 March 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar north is up, west is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' cadence of 3 s allowed observing numerous dynamic fine-scale structures in the south polar coronal hole (see Figure 19), includ- ing “bright points”, plumes, plumelets, jetlets, and jets (Chitta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022, submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Numerous campfires were visible in the ad- jacent quiet Sun area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Similar datasets were taken for the north polar coronal hole on March 6 and April 4–5, when the distance from the space- craft to the Sun was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 au and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='37 au respectively (Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRIEUV spatial resolution of around 270 km was reached on April 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Even if 7 HRIEUV images were taken at the very high cadence of 2 s on March 6, the typical cadence was 30 s in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 11 of 19 2 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 10 Mm 2022-03-28T07:59 2022-03-28T07:59 3 10 Mm 2022-03-28T07:35 2022-03-27T23:17 2022-03-27T21:4407:05 07:27 10 Mm 07:09 07:31 07:15 07:35 07:19 07:39 07:23 07:43 10 Mm2022-03-30T04:39:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='046A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Filaments and prominences observations The first perihelion passage of Solar Orbiter has also permitted EUI to take close up observations of filaments and prominences at high cadence and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' On the disk, the width of their core fine structure has been resolved by HRIEUV down to the limit of the Hα instrument resolution (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In Hα the threads have a width distribution centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3′′, which means ≈ 225 km on the Sun (for a review of the observations see Parenti 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18 shows some example of filaments seen mostly in absorption by HRIEUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Similar absorption features are also ob- served by HRIEUV in coronal rain events (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The top left panel shows a filament which was followed for about half of an hour on 2022 March 18 at 10:30UT at a cadence of 5 s, allowing to detect fast intensity variation at small scales along and across the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Full Sun images show local activity on 2022 March 17, which led to the formation of the filament and the opening of a small coronal hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The merging of the polar coronal hole and the dimming region formed by the eruption of the filament is discussed in Ngampoopun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' in prepa- ration, this issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The second panel on the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18 shows a filament close to the limb, that was observed during the Full Disc Mo- saic campaign on 2022 March 7 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Both the on disk and off disk part of the filament shows a complex fine struc- ture, highlighted by dark and bright alternating wavy shaped fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the same campaign, we also observed the promi- nence shown on the bottom left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This has a tornado-like shape, with thin bright and dark threads and a bright extension at the base of the coronal cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These observations are very promis- ing for future quiescent prominence and filament studies, as they provide elements to derive the fine scale morphology, and char- acterize the dynamics from possible injected plasma and waves activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The bottom-right panel shows part of an AR filament observed in 2022 March 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' During the 30 min HRIEUV high ca- dence sequence, the filament was quite active, with brightenings in threads and fine dark features levitating on top of the main dark body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The cadence of 3 – 5 s that was chosen for all the se- quences shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18 appear to be adequate for studying the dynamics at such a small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Eruptions Due to varying distance to the Sun, the FSI FOV (from disc cen- ter to edge) changed from 4 R⊙ on 2022 March 2, to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 R⊙ at per- ihelion and back to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 R⊙ on 2022 April 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This large FOV, un- precedented for an EUV imaging telescope, allowed us to mon- itor the early evolution of eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Table 1 lists the approxi- mate starting time, the shape and the greatest height reached by the eruptions observed by FSI during the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Some exam- ple prominence eruptions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' They appear in a multitude of shapes: surge-like, loop-like, curled-like eruptions and have different kinematic behavior, from slow rising to fast eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In the following subsections, we highlight a number of eruptions with particularly good coverage by other instruments or with space weather relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 flare: 2022 March 10 The C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 flare from active region NOAA 12962 on 2022 March 10 (GOES X-ray peak at 20:33) was observed by FSI as a clas- sical two-ribbon flare, and later a post-flare arcade, with a ca- dence of 10 min in the 304 channel and 30 min in the 174 chan- nel (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' It was also observed by STIX (Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022-03-11T01:00 2022-03-04T19:30 2022-03-10T01:00 2022-03-16T15:30 2022-03-19T12:00 2022-03-19T07:00 2022-03-21T07:30 2022-03-25T05:20 2022-03-30T17:30 Text 2022-03-31T03:30 2022-03-28T12:20 2022-03-26T21:18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Mosaic of prominence eruptions observed by FSI in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband in 2022 March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The ’enhance off limb’ functionality of JHe- lioviewer (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2017) was used when creating these graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) and EPD (Rodríguez-Pacheco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020) on Solar Or- biter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The Earth was separated by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8◦ from Solar Orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' From the Earth’s perspective the flare was near the central meridian and associated with a partial halo coronal mass ejection (CME) that eventually led to a moderate geomagnetic storm (Kp = 6) on March 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The evolution of the CME shock and its ef- fect on ion acceleration was studied by Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', 2022 (this volume, in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 12 of 19 20 30 +15 15 0 30 45 60 0 20 40 60 60 80 80 2022-03-04T19:30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='36700 CF- 20 40 60 80 2022-03-10T01:00:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='27760 40 20 60-45 B0 2022-03-11T01:00:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='17860 40 20 601-45 30 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='11 15 0 0 20 2022-03-16T15:30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='76530 45 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='175 15 0 20 40 60 80 2022-03-19T07:00:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='184-45 B0 ±15 0 20 40 60 80 2022-03-19T12:00:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='215-30 15 30 60 0 20 40 60 60 80 80 2022-03-21T07:30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='234-60-45 30 15 0 20 40 60 2022-03-25T05:20:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='212 0880 80 60 40 20 20215-26T21:18:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='Q8 15 30 45 607540 20 60 45 30 15 0 0 20 40 2022-03-28T12:10:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='21640 20 60 45 30 +15 0 20 2022-03-30T17:30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30230 0 20 40 60 80 2022-03-31T03:30:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='242D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Eruptions observed by FSI between 2022 March 2 and 2022 April 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Start Date Position FSI channel Comments 2022 March 04 SW and E 304 two prominences at SW (18:00, loop-like opening, up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='65 R⊙) and E (21:00, jet-like, up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='25 R⊙) 2022 March 05 SE 304, 174 small prominence (12:30, along a loop, up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='24 R⊙) 2022 March 06 NE 304 small prominence (03:00, loop-like, up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='31 R⊙) 2022 March 08 NE 174 NE (08:10, jet-like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 R⊙) 2022 March 08 SE 174 to the outer FOV (21:00, fan-like, extended concave-out, up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='12 R⊙) 2022 March 09 SE 304 far in the FOV (19:30, twisted, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='45 R⊙) 2022 March 10 NW quadrant 174 on-disk (18:30 dimming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 21:30 post-eruption arcade) 2022 March 10 E 304 end FOV (19:00, jet-like, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='24 R⊙) 2022 March 10 E 304 end FOV (23:30, fan-like, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='15 R⊙) 2022 March 12 W 174 to the end FOV (06:00, elongated sinusoidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='85 R⊙) 2022 March 13 E 304, 174 two small (00:30, 05:00, loop opening, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='50 R⊙) 2022 March 14 SW 174 big (17:20, loop-like opening, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='24 R⊙) 2022 March 16 E 304 2 curled prominences (13:00, 14:30, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='65 R⊙) 2022 March16 SE, NE, SW 174 elongated curved SE (08:00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='60 R⊙) 2 eruptions, 14:10 NE and SW 2022 March17 W 174 small (06:30, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='10 R⊙) 2022 March 18 W 174 small (11:00, concave-out, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30 R⊙) 2022 March 19 W, SE 304 prominence W (06:00, fan-like, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6 R⊙) and SE (10:30, curled, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='81 R⊙) 2022 March 19 SE 174 (10:00, curled, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 R⊙) 2022 March 20 NE 304, 174 prominence (08:00, curled, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='32 R⊙) 2022 March 21 SW 304, 174 (05:30 UT, fan-like, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='25 R⊙) 2022 March 24 SE 174 small (11:30, loop-like, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='50 R⊙) 2022 March 25 SE 304, 174 (05:00, loop-like, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='77 R⊙) 2022 March 26 NW 304, 174 (19:30, loop-like, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='27 R⊙) 2022 March 27 E 304, 174 2 small eruptions (13:00, 19:00, loop-like, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 R⊙) 2022 March 28 E 304, 174 prominence (11:20, loop-like + fan, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 R⊙) M4 flare and halo CME arriving at Earth on 2022 March 31 2022 March 30 NW, E 304, 174 at NW (05:30, loop-like, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='07 R⊙), and E (14:00, ragged, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9 R⊙) 2022 March31 SW 304, 174 (02:30, loop-like, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30 R⊙) 2022 April 02 NE, SE 304, 174 prominences (13:00, ragged, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7 R⊙) 2022 April 03 SE 304 prominence (15:00, untwisting to loop-like, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7 R⊙) 2022 April 04 SE 304 big filament (10:30, faint elongated off-limb, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='11 R⊙) 2022 April 05 SW 174 (13:00, fan-like, 2 R⊙) 2022 April 06 SW 304, 174 prominence (22:00, ragged, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7 R⊙) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Limb CME: 2022 March 21 Starting around 05:30 UT on 2022 March 21, an eruption was observed by FSI at the SW limb (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 22) that led to a partial halo CME observed from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The CME was associated with a Type-II radio burst (measured by RPW, Maksimovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020), with X-ray emission (observed by STIX) and with a wide SEP event measured by EPD, SOHO and STEREO-A (34◦ to the east of the Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter was at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='34 au from the Sun, and 44◦ west of the Sun-Earth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The source location of the CME was located close to the west limb as seen from Solar Orbiter, at least partially occulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The CME was fast, with speeds above 1000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At that time EUI was executing the Slow-Wind- Connection SOOP and observed in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband of FSI with a cadence of 30 min and in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband of FSI with 10 min cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A time sequence of the eruption can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' East limb eruption: 2022 March 30 On 2022 March 30, at around 14:00 UT, FSI observed in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm channel a prominence erupting at the East limb (see lower left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20) which was further observed by SolOHI (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Prominence material is still vis- ible in the FSI FOV at around 20:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At around 17:30 a flare was observed on the disk (N15W30 Stonyhurst coordinates) and a bright loop is visible off-limb overlapping with the prominence but not disturbing its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This indicates that the promi- nence was situated far away from the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' By inspecting the Article number, page 13 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main 2022−03−10T22:05:20 UT (Earth) EUI/FSI174 0 100 200 300 400 500 x [Mm] 0 100 200 300 400 500 y [Mm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 log(Intensity [DN/s]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' FSI observations in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband of the C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='8 flare on 2022 March 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The images present the arcades of the post-flare loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Time sequence of the eruption on 2022 March 21, as seen by FSI in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The ’enhance off limb’ functionality of JHelioviewer (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2017) was used when creating these graph- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' SDO/AIA304 and STEREO-A/EUVI304 (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2008) movies one could see an extended filament erupting at the East of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' FSI observed in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm passband faint material erupting at around 14:00 at the East limb followed by a big dimming off- limb at around 17:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Material moving out is still observed at 20:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A large EUV wave is observed on-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The eruption was associated with a flux-rope like coronal mass ejection observed by STEREO-A/COR2 and SOHO/LASCO-C2 (Brueckner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1995) coronagraphs at West limb at 18:23 and 18:12 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' North-east limb eruption: 2022 April 2 On 2022 April 2, FSI observed in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm channel a filament erupting at the NE limb (as seen from Solar Orbiter) between 13:00 and 13:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' It was associated with an M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='9-class flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The event was also captured by several other remote-sensing in- struments on Solar Orbiter such as SPICE, STIX, and SoloHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Interestingly, the erupting filament was also monitored a few days prior and during its eruption by Earth-based assets such as Solar Dynamics Observatory, IRIS and Hinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Its position from the Sun-Earth line was N12W68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The flare recorded by GOES soft-X ray observations indicates a start at 12:56:00, with a peak at 13:55:00 and followed by a long duration event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This event is particularly interesting for several reasons: first, the large coverage available with different instruments allows us to follow the pre-flare phase, during which the filament slowly rises and pushes overlying coronal arcades away, as modelled in 3D numerical simulations of eruptive flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This may be linked to the observed large-scale reconfiguration reported in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Second, Doppler velocity and intensity changes in several lines are reported between the upper chromosphere and transition regions (SPICE diagnostics) as well as coronal lines (Hinode/EIS diagnostics) with different view points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is the first time that such an event is seen stereoscopically with dif- ferent spectrometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Finally, the extended coverage, from spec- troscopy to EUV and X-Ray imaging allows us to understand the evolution of the magnetic field changes during the different phases of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A dedicated study of this event is available in Janvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022 (this volume, in prep) and in Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022 (this volume, in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Instrument Performance at perihelion The pre-flight instrument characterisation is discussed in tele- scope specific papers in this issue (Auchère and EUI consor- tium partners 2022 (this volume, in prep);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Aznar Cuadrado and EUI consortium partners 2022 (this volume, in prep);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Gissot and EUI consortium partners 2022 (this volume, in prep)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The pe- riod around the 2022 March 27 perihelion was the first time the instrument was operated in the environment for which it was pri- marily designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In this section we review how the instrument was operated technically and the resulting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Over- all, FSI and HRIEUV performed largely nominally while HRILya suffered from a temporary degradation in throughput and resolu- tion (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Sensors The three EUI telescopes share the same CMOS sensor design (Rochus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The sensors consist of two parts of each 1536 x 3072 pixels, stitched together as a 3072 x 3072 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The HRI sensors are used sub-fielded to 2048 x 2048 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Careful inspection reveals that the stitching line remains visible in all three telescopes but most noticeably in HRILya (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 25 c) at x=180 Mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Each pixel has a high-gain and low-gain read-out which can be brought to the ground independently or selected per pixel per intensity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Onboard electronics then re-scales the high- gain and low-gain signals from different pixels into one coher- ent intensity range over all pixels in a ’recombined’ image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In Article number, page 14 of 19 2022-03-21T05:30 2022-03-21T06:00 2022-03-21T06:30 2022-03-21T07:00D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission the 2022 March/April period, FSI and HRIEUV have been nom- inally operated in the combined gain mode, resulting in 15-bit images (an intensity range in Level 1 files of 0–32767 DN for FSI and 0–25600 DN for HRIEUV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The selection threshold be- tween low-gain and high-gain read-out happens near a Level 1 intensity level of 1097 DN for FSI and 1118 DN for HRIEUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This transition is weakly visible in the FSI and HRIEUV images as a band of enhanced noise, which is to be expected given the different photon statistics in the low-gain and high-gain read-out channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In contrast, HRILya has been operated exclusively in low-gain read-out resulting in 12-bit images (an intensity range of 0–4095 DN in the Level 1 image files) and which do not show such transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Other sensor artefacts, affecting FSI, are dark vertical bands in very faint areas, aligned with the brightest on-disk features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This effect is assumed to be caused by saturation of the high-gain read-out of pixels in the same column and is still under investi- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A post-processing semi-empirical fix is being developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Onboard Processing EUI is equipped with software controlled onboard calibration electronics to correct the images pixel-wise for offset and flat field before compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For FSI, pre-flight offset and flat field maps are available on board and have been applied until 2022 March 16 when it was discovered that the flat field map was not applied correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Correction for the flat field was turned off at this time and subsequent FSI images have only the offset map applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For HRIEUV, only a synthetic 4-column pattern is subtracted that mimics the observed offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For HRILya no onboard correc- tion is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Despite the close solar proximity, radiation hits on the sen- sors have been very limited and the onboard cosmic ray corrector has therefore not been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Enhanced radiation hits were observed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' following the 2022 March 21 event (see subsec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Image resolution During commissioning, the FWHM of the FSI Point Spread Function (PSF) was estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 pixels, or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='66".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' As shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 23, there is no sign of changes so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At closest approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 au, this corresponds to a resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5" (two pixels) as seen from 1 au, similar to that of STEREO/EUVI (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4", Wülser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The excellent resolving quality of HRIEUV was confirmed during perihelion through identification of point-like features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 24), and slender features (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 15), with a FWHM width of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is consistent with the spatial resolution of the HRIEUV telescope being equal the Nyquist sampling limit of 2 pixels, 2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='492′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' At perihelion just inside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 au this corre- sponds to about 2 × 100 km on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' From the beginning of the mission, the HRILya spatial reso- lution was found to be lower than expected, with a first estimate placing it at around 3′′ (see Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' However, during the perihelion approach of Solar Orbiter the telescope has shown a further substantial degradation of spatial resolution, contrast and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 25 shows three images of quiet sun regions taken on 2022 March 8 (where Solar Orbiter was at a dis- tance to the Sun of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='49 au), March 22 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 au), and March 30 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='34 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' All targets were selected to be near disk centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The loss in performance can be clearly seen in panels (b) and 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Enlargements of selected compact features observed by FSI at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm on 2022 March 7 at 06:20:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Assuming that the sources are unresolved, the green and red (horizontal and vertical respectively) profiles indicate a FWHM width of the effective Point Spread Function of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Enlargements of selected compact features observed by HRIEUV on 2022 March 7 at 00:41:55 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Assuming that the sources are unre- solved, the green and red (horizontal and vertical respectively) profiles indicate a FWHM width of the effective Point Spread Function of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (c), immediately before and after the closest approach to the Sun on 2022 March 26 (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='32 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Most obvious is the resolution degradation which may be a result of a heat effect on the en- trance filter of the HRILya telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In addition, as Solar Orbiter approaches the Sun, both the contrast and throughput degrade by Article number, page 15 of 19 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main approximately 37 % with respect to data taken before the perihe- lion (around mid-February 2022), and these recover slowly after perihelion passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Filters and light leaks The reasons for the observed overall loss of performance of HRILya during perihelion passage are currently under investiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Experience from ground testing of the entrance filter with heating to 200 ◦C has revealed a non-linear decline of its trans- mission of up to 40 % as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Part of the loss of the channel’s throughput may be associated with the temperature dependency of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Consistent with this, some throughput and resolution recovery was observed further from the Sun, on 2022 June 12, which was the first HRILya obser- vations after 2022 April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' A full assessment of the evolution of throughput since launch and of its comparison with the expec- tations from ground calibration will be the subject of a separate publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In contrast to HRILya, the EUV channels may be affected by light leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Two issues are known to affect the FSI filters: – A faint light leak, likely caused by a pinhole in the front filter, affects the images from both channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Due to the specific design of FSI, its visibility depends on the distance to the Sun and pointing of the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' There is no quantitative correction for this yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This has no impact for morphological studies, but care must be taken for photo-metric analysis off- disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' – A very faint light-leak, invisible in regular images, affects the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='4 nm data taken in coronagraph mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Its origin is still unknown, but only a small number of images are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Pointing error and jitter The pointing information in the World Coordinate System (WCS) keywords of the EUI Level 1 FITS files are based on the as-flown Solar Orbiter spacecraft kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' As such, these key- words capture most of Solar Orbiter pointing instabilities, but unfortunately not all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Even after correcting for the known point- ing variation, occasional jitter remains visible from image to image (as well as slower trends) in high cadence HRIEUV se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' But in general, HRIEUV images do not seem to be af- fected much by jitter blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' For FSI images, in which the solar limb is always visible, the WCS pointing keywords are updated in the EUI Level 2 FITS files with much more precise information from a procedure that fits a circle to the solar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This is unfortunately not possible for HRIEUV and HRILya image sequences and the data user is advised to use alignment methods to remove the remaining jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Conclusions During the Solar Orbiter perihelion passage of 2022 March 26, and the weeks before and after, EUI collected more than 35000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Solar Orbiter reached a distance to the Sun as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='32 au, closer to the Sun than any other coronal imager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Both FSI and HRIEUV operated at design specifications during the per- ihelion passage but HRILya suffered from an unexpected (but re- versible) performance degradation near perihelion that needs to be studied further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' EUI has achieved the highest resolution images ever of the solar corona in the quiet Sun and polar coronal holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Ubiquitous EUV brightenings (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' campfires) and small scale jets were re- covered down to the resolution limit of HRIEUV of about 200 km on the solar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' These smallest features require further in- vestigations for their relevance to the heating of the corona and the powering of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Whereas the Hi-C sounding rocket (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Rachmeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2019) achieved comparable resolution in active regions, HRIEUV imaged active regions at much longer sequence durations (hours) at high cadence (3 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Known phenomena such as coronal braiding, decayless oscillations, coronal rain and flar- ing activity were observed in unprecedented details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The highest resolution full disc image ever was constructed as a mosaic of 25 high resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Together with the PHI and SPICE instruments onboard Solar Orbiter, this full disc mo- saic will be repeated twice per year when Solar Orbiter crosses a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 au from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Meanwhile, the big novelty of FSI, namely its very extended FOV, allowed the imaging of eruptions off limb further than ever before, with in particular the prominence eruption showing a be- wildering variety in structural appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Future perihelia will go another 10 % closer to the Sun, to a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='29 au from the Sun, and as the mission progresses, Solar Orbiter/EUI will also observe from increasing solar lati- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Many of the SOOPs and EUI observations presented in this paper will be repeated from these upcoming vantage points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Special attention will be paid to deepening joint observations with other instruments on Solar Orbiter but also with Earth- bound observatories in space and on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' This paper presented how the EUI observations contributed to the various Solar Orbiter Observations Programs (SOOPs) that implement cross-instrument science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' By highlighting particular features and events, many of which require further study, this paper intended to demonstrate the potential of the EUI data and to inspire external users to take part in the EUI data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The EUI dataset presented in this paper has been distributed as part of the EUI Data Release 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='0 (Mampaey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2022) and is freely accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We encourage EUI data users to read the release notes and get in contact with the EUI team for specific support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The building of EUI was the work of more than 150 indi- viduals during more than 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' We gratefully acknowledge all the efforts that have led to a successfully operating instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The authors thank the Bel- gian Federal Science Policy Office (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the European Space Agency (ESA) under contract numbers 4000112292, 4000134088, 4000134474, and 4000136424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' The French contribution to the EUI instrument was funded by the French Centre National d’Études Spatiales (CNES);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' the UK Space Agency (UKSA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' the Deutsche Zentrum für Luft- und Raumfahrt e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' (DLR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' and the Swiss Space Office (SSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' PA and DML acknowledge funding from STFC Ernest Rutherford Fellowships No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' ST/R004285/2 and ST/R003246/1, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' SP acknowledges the funding by CNES through the MEDOC data and op- erations center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' gratefully acknowledges funding by the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Views and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', De Groof, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', Walsh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 2020, A&A, 642, A3 Appendix A: EUI Data set characteristics Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya observations of a set of quiet sun regions located near disk centre, obtained on 2022 March 8 (panel a), 2022 March 22 (panel b), and 2022 March 30 (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Article number, page 17 of 19 (a) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='49 AU) 300 100 2022-03-08T00:43:24 UT 100 200 300 X [Mm] 250 (b) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='33 AU) 200 150 Y [Mm] 100 50 2022-03-22T16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='26:10 UT 0 50 100 150 200 250 X [Mm] 250 (c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='34 AU) 200 150 [Mm] Y 100 50 2022-03-30T22:00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='10 UT 0 50 100 150 200 250 X [Mm]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' main Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Summary of SOOPs and corresponding EUI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In between the SOOPs, additional FSI synoptic images have been taken that are not listed in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Some specific calibration datasets have also been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='# images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='cadence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='comment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='L_SMALL_MRES_MCAD_Connection-Mosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-01 18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-03 03:21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='3 pointings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRIEUV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='810 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-02 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-03 03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRILya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='809 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-02 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-03 03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='15 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-02 00:01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-03 02:45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI304 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='15 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-02 00:01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-03 02:46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='L_SMALL_MRES_MCAD_Connection-Mosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 07:55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-31 17:40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6 pointings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRIEUV & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='576 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 11:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-31 15:47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='6 bursts of 48 min at 11:00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18:00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya 288 60 s 22:00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 03:30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 09:00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 15:00 FSI174 188 10 min 2022-03-30 08:00 2022-03-31 17:30 FSI304 64 30 min 2022-03-30 08:00 2022-03-31 17:30 L_SMALL_HRES_HCAD_Slow-Wind-Connection 2022-03-03 06:00 2022-03-06 16:45 various pointings HRIEUV & 3x720 5 s 2022-03-03 09:40 2022-03-05 16:20 1h bursts starting 3th 09:40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya 3x720 5 s 4th 10:45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5th 15:20 FSI174 479 10 min 2022-03-03 06:00 2022-03-06 15:11 FSI304 160 30 min 2022-03-03 06:00 2022-03-06 15:01 L_SMALL_HRES_HCAD_Slow-Wind-Connection 2022-03-17 06:00 2022-03-22 00:00 various pointings HRIEUV & 5x720 5 s 2022-03-17 09:47 2022-03-21 12:36 1h bursts: 17th 09:47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 18th 10:10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya 5x720 5 s 19th 10:36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20th 11:27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 21th 11:36 FSI174 630 10 min 2022-03-17 06:00 2022-03-21 23:51 FSI304 210 30 min 2022-03-17 06:00 2022-03-21 23:30 R_SMALL_HRES_MCAD_Polar-Observations 2022-03-06 16:45 2022-03-06 21:50 pointing: North pole HRIEUV 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 149 2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 30 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-06 17:34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-06 18:51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRILya ' metadata={'source': 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='R_SMALL_HRES_MCAD_Polar-Observations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 03:30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 07:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='pointing: South Pole ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRIEUV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='600 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 05:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='10 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 03:30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 07:01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} 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2022-04-05 23:50 5 min cadence in last 3h FSI304 51 30 min 2022-04-04 16:30 2022-04-05 20:30 R_BOTH_HRES_HCAD_Nanoflares 2022-03-06 21:50 2022-03-07 03:00 pointing: active region HRIEUV 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1188,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 60 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20 s 2022-03-07 00:29 2022-03-07 03:00 variable cadence HRILya 783,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 60 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20 s 2022-03-07 00:00 2022-03-07 03:00 variable cadence FSI174 354 30 s 2022-03-07 00:00 2022-03-07 03:00 R_BOTH_HRES_HCAD_Nanoflares 2022-03-08 00:00 2022-03-08 03:00 pointing: disc center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' quiet Sun HRIEUV 588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 1188,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 60 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20 s 2022-03-08 00:00 2022-03-08 03:00 variable cadence HRILya 783,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 60 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' 20 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='variable cadence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='355 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30 s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='R_BOTH_HRES_HCAD_Nanoflares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-17 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-17 02:55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='quiet Sun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRIEUV ' metadata={'source': 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03:03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-17 04:03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='gap between 03:16 and 03:50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='R_BOTH_HRES_HCAD_Nanoflares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 03:24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='pointing: active region ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='03:10 missing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI304 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='30 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 01:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-30 03:20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' page 18 of 19 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Berghmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' : First Perihelion of EUI on the Solar Orbiter mission Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Summary of SOOP instances and corresponding EUI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' In between the SOOPs, additional FSI synoptic images have been taken that are not listed in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' Some specific calibration datasets have also been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' # images cadence start end comment Full Disc Mosaic 2022-03-07 07:00 2022-03-07 11:30 25 pointings HRIEUV 450 2022-03-07 07:01 2022-03-07 11:30 18 images/pointing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HG/LG HRILya 200 2022-03-07 07:04 2022-03-w07 11:29 8 images/pointing FSI174 25 11 min 2022-03-07 07:05 2022-03-07 11:30 FSI304 25 11 min 2022-03-07 07:05 2022-03-07 11:30 L_FULL_HRES_MCAD_Coronal-He-Abundance 2022-03-07 16:00 2022-03-07 20:00 FSI174 8 30 min 2022-03-07 16:00 2022-03-07 19:31 HG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' occulted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' exptime 1000s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='R_BOTH_HRES_MCAD_Bright-Points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 08:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 16:45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='pointing: disc center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRIEUV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 08:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 10:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='HRILya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='1 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 08:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 10:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 08:10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='2022-03-08 16:41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 min cadence till 14:05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='FSI304 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='5 min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=', 15:00-16:00, 16:30 -19:30 L_FULL_HRES_HCAD_Eruption-Watch 2022-03-22 19:40 2022-03-23 16:30 pointing: disc center HRIEUV & 60 30 s 2022-03-22 03:30 2022-03-23 16:30 30 min burst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya 30 60 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} 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2022-03-29 23:53 gap 15:00-16:00 R_SMALL_MRES_MCAD_AR-Long-Term 2022-03-31 17:45 2022-04-04 20:26 pointing: Active Region HRIEUV & 4x450 10 s 2022-04-01 09:19 2022-04-04 10:34 75 min burst at 09:19 on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content=' HRILya 4x150 30 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} +page_content='April 1, 2, 3, 4 FSI174 538 10 min 2022-03-31 17:50 2022-04-04 20:26 gap April 4 18:30-20:00 FSI304 179 30 min 2022-03-31 18:00 2022-04-04 20:01 30 min cadence till April 4 18:30 Article number, page 19 of 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE5T4oBgHgl3EQfeA-9/content/2301.05616v1.pdf'} diff --git a/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/2301.00177v1.pdf.txt b/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/2301.00177v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f8b26053514376c46c11ca126e78c57cbff9e5f --- /dev/null +++ b/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/2301.00177v1.pdf.txt @@ -0,0 +1,1895 @@ +arXiv:2301.00177v1 [math.OC] 31 Dec 2022 +On the Arrow–Hurwicz differential system for linearly +constrained convex minimization +Simon K. Niederländer +Institute for Systems Theory and Automatic Control, University of Stuttgart, +Stuttgart, Germany +ARTICLE HISTORY +Compiled January 3, 2023 +Abstract +In a real Hilbert space setting, we reconsider the classical Arrow–Hurwicz differential +system in view of solving linearly constrained convex minimization problems. We in- +vestigate the asymptotic properties of the differential system and provide conditions +for which its solutions converge towards a saddle point of the Lagrangian associated +with the convex minimization problem. Our convergence analysis mainly relies on a +‘Lagrangian identity’ which naturally extends on the well-known descent property of +the classical continuous steepest descent method. In addition, we present asymptotic +estimates on the decay of the solutions and the primal-dual gap function measured +in terms of the Lagrangian. These estimates are further refined to the ones of the +classical damped harmonic oscillator provided that second-order information on the +objective function of the convex minimization problem is available. Finally, we show +that our results directly translate to the case of solving structured convex minimiza- +tion problems. Numerical experiments further illustrate our theoretical findings. +KEYWORDS +Arrow–Hurwicz differential system; Lyapunov analysis; asymptotic properties; +exponential stabilization; convex minimization; saddle-point problem +2010 MATHEMATICS SUBJECT CLASSIFICATIONS +37N40; 46N10; 49M30; 65K05; 90C25 +1. Introduction +Let X and Y be real Hilbert spaces endowed with inner products ⟨ · , · ⟩X, ⟨ · , · ⟩Y and +induced norms ∥ · ∥X, ∥ · ∥Y . Consider the minimization problem +inf {f(x) | Ax − b = 0Y }, +(P) +where f : X → R is a convex and continuously differentiable function, A : X → Y a +linear and continuous operator, and b ∈ Y . We associate with (P) the Lagrangian +L : X × Y −→ R +(x, λ) �−→ f(x) + ⟨λ, Ax − b⟩Y +CONTACT Simon K. Niederländer. Email: niederlaender@ist.uni-stuttgart.de + +which, by construction, is a convex-concave and continuously differentiable bifunction. +A pair (¯x, ¯λ) ∈ X × Y is a saddle point of the Lagrangian L if +L(¯x, λ) ≤ L(¯x, ¯λ) ≤ L(x, ¯λ), +∀(x, λ) ∈ X × Y. +It is well known that (¯x, ¯λ) ∈ X ×Y is a saddle point of L if and only if ¯x is a minimizer +of (P), ¯λ is a maximizer of the Lagrange dual to (P), that is +sup {−f ∗(−A∗λ) − ⟨λ, b⟩Y | λ ∈ Y }, +(D) +and the optimal values of (P) and (D) coincide; see, e.g., Ekeland and Témam [1]. +Here, f ∗ : X → R ∪ {+∞} denotes the Fenchel conjugate of f defined by f ∗(u) = +sup {⟨u, x⟩X − f(x) | x ∈ X}, and A∗ : Y → X refers to the adjoint operator of A. +Equivalently, (¯x, ¯λ) ∈ X ×Y is a saddle point of L if and only if (¯x, ¯λ) solves the system +of primal-dual optimality conditions +� +∇f(x) + A∗λ = 0X +Ax − b = 0Y +with ∇f denoting the gradient of f. Throughout the text, we denote by S ×M ⊂ X ×Y +the (possibly empty) set of saddle points of L. We recall that a saddle point of L exists +whenever (P) admits a minimizer and, for instance, the constraint qualification +b ∈ sri A(X) +is verified1. Here, for a convex set C ⊂ Y , we denote by +sri C = {x ∈ C | +� +µ>0 +µ(C − x) is a closed linear subspace of Y } +its strong relative interior; see, e.g., Bauschke and Combettes [3]. We further recall that +(P) admits a minimizer whenever its feasible set is non-empty and, for instance, f is +coercive, that is, lim∥x∥X→+∞ f(x) = +∞. On the other hand, if the feasible set of (P) +is non-empty and f is strongly convex, then (P) admits a unique minimizer. +In this work, we reconsider the classical Arrow–Hurwicz differential system +� +˙x + ∇f(x) + A∗λ = 0X +˙λ + b − Ax = 0Y +(AH) +relative to the convex minimization problem (P). The (AH) differential system was in +essence originated by Arrow and Hurwicz [4] (see also Kose [5], Arrow et al. [6]) and is +known to be intimately related to the mini-maximization of the Lagrangian L associ- +ated with (P). Indeed, given the above system of primal-dual optimality conditions, we +immediately observe that the zeros of the operator +T : X × Y −→ X × Y +(x, λ) �−→ (∇xL(x, λ), −∇λL(x, λ)), +1We remark that, in the finite-dimensional case, the condition amounts to b ∈ A(X) which is commonly re- +ferred to as Slater assumption; see, e.g., Hiriart-Urruty and Lemaréchal [2]. +2 + +that is, the ‘generator’ of the (AH) differential system, are precisely the saddle points +of the Lagrangian L, i.e., +(¯x, ¯λ) ∈ S × M +⇐⇒ +T(¯x, ¯λ) = (0X, 0Y ). +Moreover, the operator T is maximally monotone on X × Y as it is both monotone and +continuous; cf. Minty [7]. Therefore, S × M can be interpreted as the set of zeros of +the maximally monotone operator T and, as such, it is a closed and convex subset of +X × Y . The latter may also be deduced more elementary from the convexity-concavity +properties of the ‘saddle function’ L; cf. Rockafellar [8]. +1.1. Preliminary facts +As emphasized by Rockafellar [9], the general theory for semi-groups of contractions +generated by maximally monotone operators (see, e.g., Crandall and Pazy [10], Brézis +[11]) applies to the Arrow–Hurwicz differential system (AH). These results, dating back +to the works of Kato [12] and K¯omura [13] (see also Browder [14]), imply that the +Cauchy problem associated with (AH) is well posed and that its (classical) solutions +(x, λ), (y, η) : [0, +∞) → X × Y verify the ‘non-expansiveness property’ +d +dt +�∥x(t) − y(t)∥2 +X + ∥λ(t) − η(t)∥2 +Y +� ≤ 0, +∀t ≥ 0. +If, in addition, the set S × M is non-empty, then the solutions (x(t), λ(t)) of (AH) +remain bounded and, in fact, weakly converge in average, as t → +∞, towards a saddle +point of the Lagrangian L (see Baillon and Brézis [15]), i.e., there exists (¯x, ¯λ) ∈ S × M +such that +1 +t +� t +0 +(x(τ), λ(τ)) dτ ⇀ (¯x, ¯λ) as t → +∞. +The (asymptotic) stability properties of the solutions of (AH) (in the sense of Lya- +punov) were further investigated by Venets [16] (see also Flåm and Ben-Israel [17]). +These results suggest that the solutions of (AH) tend towards a saddle point of L giv- +en that, for any (x, λ) ∈ X × Y with x /∈ S, it holds that +L(¯x, λ) ≤ L(¯x, ¯λ) < L(x, ¯λ), +∀(¯x, ¯λ) ∈ S × M. +The above condition is, of course, trivially satisfied whenever f is strictly convex. In the +respective works, the authors further noted that the solutions (x, λ) : [0, +∞) → X ×Y +of (AH) obey the ‘Lagrangian identity’ +d +dtL(x(t), λ(t)) + ∥ ˙x(t)∥2 +X = ∥ ˙λ(t)∥2 +Y , +∀t ≥ 0. +(1) +The identity, however, was not pursued any further due to its indefinite character. We +remark that, in the unconstrained case of (P), the above identity reduces to the well- +known ‘descent property’ +d +dtf(x(t)) + ∥ ˙x(t)∥2 +X = 0, +∀t ≥ 0 +3 + +associated with the classical continuous steepest descent method; see, e.g., Brézis [11], +Aubin and Cellina [18]. +Finally, the exponential decay properties of the solutions of (AH) were investigated +by Polyak [19]. Using spectral arguments, the work provides conditions for which the +solutions (x(t), λ(t)) of (AH) converge at an exponential rate, as t → +∞, towards a +saddle point (¯x, ¯λ) of L, i.e., for which there exists ρ > 0 such that +∥x(t) − ¯x∥2 +X + ∥λ(t) − ¯λ∥2 +Y = O +�e−ρt� as t → +∞. +The decay rate estimates are, however, not derived in an explicit form. +In this work, our objective is to recover, unify and extend some of the previous results +on the classical Arrow–Hurwicz differential system (AH) in view of solving the linearly +constrained convex minimization problem (P). Using tools from monotone operator +theory, we focus our attention on the convergence properties of the solutions of (AH) and +further aim to characterize their limit within the set of saddle points of the Lagrangian. +We also intend to make a contribution to the issue of finding (explicit) decay rate esti- +mates on the solutions of (AH). +1.2. Presentation of the results +The mini-maximizing properties of the solutions (x, λ) : [0, +∞) → X × Y of (AH) +with respect to the convex minimization problem (P) and its associated Lagrange dual +(D) are conveniently measured in terms of the ‘primal-dual gap function’ +t �−→ L(x(t), · ) − L( · , λ(t)) +relative to the set S × M. Whenever the function f is convex, we observe that the +solutions (x(t), λ(t)) of (AH) may fail to converge as t → +∞ even though the set of +saddle points of L is comprised of a single element. As a consequence, it is natural to +first study the average behavior of a solution of (AH). Using the notion of the Cesàro +average (σ, ω) : (0, +∞) → X × Y of a solution of (AH), viz., +(σ(t), ω(t)) = 1 +t +� t +0 +(x(τ), λ(τ)) dτ, +we find that the solutions of (AH) obey in average, for any (ξ, η) ∈ S ×M, the estimate +L(σ(t), η) − L(ξ, ω(t)) = O +�1 +t +� +as t → +∞. +In this case, the Cesàro average (σ(t), ω(t)) of a solution of (AH) weakly converges, as +t → +∞, towards a saddle point of L. This result is in line with the work by Nemirovski +and Yudin [20] on the classical Arrow–Hurwicz method and may also be deduced more +elementary by the results of Baillon and Brézis [15]. +Whenever f is strongly convex, we obtain more stringent mini-maximizing properties +of the solutions of (AH) relative to the primal-dual gap function. More precisely, we +show that the solutions (x, λ) : [0, +∞) → X×Y of (AH) evolve, for any (ξ, η) ∈ S×M, +according to the estimate +L(x(t), η) − L(ξ, λ(t)) = o +� 1 +√ +t +� +as t → +∞. +4 + +Moreover, the solutions (x(t), λ(t)) of (AH) are proven to converge weakly, as t → +∞, +towards an element of the set of saddle points of L. In particular, we characterize the +weak limit of a solution of (AH) as the orthogonal projection of its initial data (x0, λ0) ∈ +X × Y onto the (closed and convex) set S × M, i.e., +(x(t), λ(t)) ⇀ projS×M(x0, λ0) as t → +∞. +If, in addition, the linear operator A∗ is bounded from below, we observe that the so- +lutions of (AH) obey, for (ξ, η) ∈ S × M, the refined estimate +L(x(t), η) − L(ξ, λ(t)) = o +�1 +t +� +as t → +∞. +In this case, it is proven that the solutions (x(t), λ(t)) of (AH) strongly converge, as +t → +∞, towards the unique saddle point of L. +Finally, we show that the solutions of (AH) decay asymptotically at an exponential +rate provided that f is twice continuously differentiable, satisfying +⟨∇2f(x)(x − y), x − y⟩X ≤ 2Df(y, x), +∀x, y ∈ X. +Here, Df denotes the Bregman distance associated with f, cf. Bregman [21], and ∇2f +refers to the Hessian of f. In particular, we show that under the above condition there +exists ρ > 0 such that the solutions (x, λ) : [0, +∞) → X × Y of (AH) verify, for any +(ξ, η) ∈ S × M, either one of the following exponential estimates: +L(x(t), η) − L(ξ, λ(t)) = O +�e−2ρt� as t → +∞; +L(x(t), η) − L(ξ, λ(t)) = O(t2e−2ρt) as t → +∞. +This result complements the decay rate estimates obtained earlier by Polyak [19]. +1.3. Organization +We begin our discussion by reviewing some basic properties of the solutions of (AH) in +the case of a convex objective function of the minimization problem (P). In Section 3, +we then investigate their asymptotic properties under the more stringent assumption of +a strongly convex objective function. In Section 4, we show that the solutions of (AH) +decay at an exponential rate provided that second-order information on the objective +function is available. In Section 5, we highlight that the results on the Arrow–Hurwicz +differential system (AH) may directly be conveyed to the case of solving structured +minimization problems. Finally, Section 6 is devoted to numerical experiments. +2. Basic properties +Let X ×Y be equipped with the Hilbertian product structure ⟨ · , · ⟩ = ⟨ · , · ⟩X +⟨ · , · ⟩Y +and induced norm ∥ · ∥. Throughout the text, we assume that +(A1) f : X → R is convex and continuously differentiable; +(A2) ∇f : X → X is Lipschitz continuous on bounded sets; +(A3) A : X → Y is linear and continuous, and b ∈ Y . +5 + +Consider the Arrow–Hurwicz differential system +� +˙x + ∇f(x) + A∗λ = 0X +˙λ + b − Ax = 0Y +(AH) +with initial data (x0, λ0) ∈ X × Y , and recall that (x, λ) : [0, +∞) → X × Y is a +(classical) solution of (AH) if (x, λ) ∈ C1([0, +∞); X × Y ) and (x, λ) satisfies (AH) on +[0, +∞) with (x(0), λ(0)) = (x0, λ0). The following result is an immediate consequence +of the monotonicity of the operator +T : X × Y −→ X × Y +(x, λ) �−→ (∇f(x) + A∗λ, b − Ax) +associated with the (AH) differential system; cf. Brézis [11, Theorem 3.1], Aubin and Cel- +lina [18, Theorem 3.2.1]. The existence and uniqueness of the (classical) solutions of +(AH) thereby follow at once from the Cauchy–Lipschitz theorem2; see, e.g., Haraux [22, +Proposition 6.2.1]. +Theorem 2.1. For any (x0, λ0) ∈ X × Y , there exists a unique solution (x, λ) : +[0, +∞) → X × Y of (AH). Moreover, +(i) t �→ ∥( ˙x(t), ˙λ(t))∥ is non-increasing and +∥( ˙x(t), ˙λ(t))∥ ≤ ∥T(x0, λ0)∥, +∀t ≥ 0; +(ii) limt→+∞∥( ˙x(t), ˙λ(t))∥ exists. +Remark 2.2. We note that the assertions of Theorem 2.1 essentially remain valid even +under the assumption that f : X → R∪{+∞} is a proper convex lower semi-continuous +function. In this case, the (AH) dynamics generalize to the evolution system +� +˙x + ∂f(x) + A∗λ ∋ 0X +˙λ + b − Ax = 0Y +with ∂f denoting the convex subdifferential of f. The existence and uniqueness of the +(strong) solutions of the above differential system are then deduced from the general +theory for semi-groups of contractions generated by maximally monotone operators; cf. +Brézis [11] (see also Pazy [23], Peypouquet and Sorin [24]). +In the following, let S × M denote the set of saddle points of the Lagrangian +L : X × Y −→ R +(x, λ) �−→ f(x) + ⟨λ, Ax − b⟩Y +associated with the convex minimization problem (P). Using the convexity of f, we +immediately observe that, for any (x, λ), (y, η) ∈ X × Y , it holds that +⟨T(x, λ), (x, λ) − (y, η)⟩ ≥ L(x, η) − L(y, λ). +(2) +2Given the above assumptions, it is easy to verify that (x, λ) �→ T(x, λ) is Lipschitz continuous on the bounded +subsets of X × Y . +6 + +Anchoring the above inequality to the set S ×M yields the following integrability result +for the primal-dual gap function. +Proposition 2.3. Let S × M be non-empty and let (x, λ) : [0, +∞) → X × Y be a +solution of (AH). Then, for any (ξ, η) ∈ S × M, +(i) limt→+∞∥(x(t), λ(t)) − (ξ, η)∥ exists; +(ii) it holds that +� ∞ +0 +L(x(τ), η) − L(ξ, λ(τ)) dτ < +∞. +Proof. (i) Let (ξ, η) ∈ S × M. Using (AH) together with (2), we have for any t ≥ 0, +d +dt +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +�/2 + L(x(t), η) − L(ξ, λ(t)) ≤ 0. +(3) +Since (ξ, η) belongs to S ×M, it follows that L(x(t), η)−L(ξ, η) ≥ 0 as well as L(ξ, η)− +L(ξ, λ(t)) ≥ 0 and thus, +L(x(t), η) − L(ξ, λ(t)) ≥ 0. +Consequently, t �→ ∥(x(t), λ(t)) − (ξ, η)∥ is non-increasing and bounded from below on +[0, +∞) so that +lim +t→+∞∥(x(t), λ(t)) − (ξ, η)∥ exists. +(ii) Integration of (3) over [0, t] yields +∥x(t) − ξ∥2 +X/2 + ∥λ(t) − η∥2 +Y /2 + +� t +0 +L(x(τ), η) − L(ξ, λ(τ)) dτ +≤ ∥x(0) − ξ∥2 +X/2 + ∥λ(0) − η∥2 +Y /2. +(4) +Taking into account that ∥(x(t), λ(t)) − (ξ, η)∥2 ≥ 0, we obtain +� t +0 +L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ ∥x(0) − ξ∥2 +X/2 + ∥λ(0) − η∥2 +Y /2. +This majorization being valid for any t ≥ 0, taking the supremum gives +� ∞ +0 +L(x(τ), η) − L(ξ, λ(τ)) dτ < +∞. +Remark 2.4. Proposition 2.3(i) asserts that the solutions of (AH) remain bounded +whenever the set S × M is non-empty. Conversely, it can be shown that the set S × M +is non-empty whenever (AH) admits a bounded solution; cf. Pazy [23]. +Let us next focus on the asymptotic properties of the solutions of (AH). To this end, +define the Cesàro average (σ, ω) : (0, +∞) → X × Y of a solution of (AH) by +(σ(t), ω(t)) = 1 +t +� t +0 +(x(τ), λ(τ)) dτ. +7 + +The following result asserts the weak convergence of the Cesàro average of a solution of +(AH) and further provides an estimate on the decay of the primal-dual gap function. +Proposition 2.5. Let S × M be non-empty and let (σ, ω) : (0, +∞) → X × Y be the +Cesàro average of a solution of (AH). Then, for any (ξ, η) ∈ S × M, it holds that +L(σ(t), η) − L(ξ, ω(t)) = O +�1 +t +� +as t → +∞. +Moreover, there exists (¯x, ¯λ) ∈ S × M such that (σ(t), ω(t)) ⇀ (¯x, ¯λ) weakly in X × Y +as t → +∞. +Proof. Let (ξ, η) ∈ S × M and recall from (4) that for any t ≥ 0, +∥x(t) − ξ∥2 +X/2 + ∥λ(t) − η∥2 +Y /2 + +� t +0 +L(x(τ), η) − L(ξ, λ(τ)) dτ +≤ ∥x(0) − ξ∥2 +X/2 + ∥λ(0) − η∥2 +Y /2. +Dividing the above inequality by t > 0 and taking into account that ∥(x(t), λ(t)) − +(ξ, η)∥2 ≥ 0, we obtain +1 +t +� t +0 L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ 1 +2t +�∥x(0) − ξ∥2 +X + ∥λ(0) − η∥2 +Y +�. +By Jensen’s inequality, as L( · , η) and −L(ξ, · ) are both convex, it follows +L(σ(t), η) − L(ξ, ω(t)) ≤ 1 +2t +�∥x(0) − ξ∥2 +X + ∥λ(0) − η∥2 +Y +� +and thus, +lim sup +t→+∞ t +�L(σ(t), η) − L(ξ, ω(t)) +� < +∞. +The weak convergence of (σ(t), ω(t)) as t → +∞ is an immediate consequence of the +Opial–Passty lemma applied to the set S × M; cf. Passty [25]. +Remark 2.6. We note that the weak ergodic convergence of the solutions of (AH) may +also be deduced from the general theory for semi-groups of contractions generated by +maximally monotone operators; cf. Baillon and Brézis [15] (see also Brézis [26] for a +localization result of the weak limit). +3. The strongly monotone case +In this section, we investigate the asymptotic properties of the solutions of (AH) under +the more stringent assumptions that +(A4) ∇f : X → X is α-strongly monotone, i.e., +∃α > 0 ∀x, y ∈ X +⟨∇f(x) − ∇f(y), x − y⟩X ≥ α∥x − y∥2 +X; +(A5) S × M ⊂ X × Y is non-empty. +8 + +We recall that the latter assumption is verified whenever (P) admits a minimizer and, +for instance, the constraint qualification +b ∈ sri A(X) +holds; see, e.g., Bauschke and Combettes [3]. In turn, the former assumption implies +that S is reduced to a singleton. +3.1. Weak convergence +Let us begin with a result on the weak convergence of the solutions of (AH). Since f is α- +strongly convex (∇f being α-strongly monotone), we have for any (x, λ), (y, η) ∈ X ×Y , +⟨T(x, λ), (x, λ) − (y, η)⟩ ≥ L(x, η) − L(y, λ) + α∥x − y∥2 +X/2. +(5) +Utilizing this inequality relative to the set S × M gives the following asymptotic prop- +erties of the solutions of (AH) and the primal-dual gap function. +Theorem 3.1. Let ∇f : X → X be α-strongly monotone, let S × M be non-empty, +and let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for any (ξ, η) ∈ S × M, +it holds that +L(x(t), η) − L(ξ, λ(t)) = o +� 1 +√ +t +� +as t → +∞; +∥( ˙x(t), ˙λ(t))∥ = o +� 1 +√ +t +� +as t → +∞. +Moreover, there exists (¯x, ¯λ) ∈ S × M such that (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y +as t → +∞. +Proof. Let (ξ, η) ∈ S × M. Using the ‘Lagrangian identity’ (1), we have for any t ≥ 0, +d +dt +�L(ξ, η) − L(x(t), λ(t)) +� + ∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y = 2∥ ˙x(t)∥2 +X. +Integration over [0, t] yields +L(ξ, η) − L(x(t), λ(t)) + +� t +0 ∥ ˙x(τ)∥2 +X + ∥ ˙λ(τ)∥2 +Y dτ += 2 +� t +0 +∥ ˙x(τ)∥2 +X dτ + L(ξ, η) − L(x(0), λ(0)). +Since ∇f is α-strongly monotone, we readily deduce from Theorem 2.1(i) that +∥ ˙x(t)∥2 +X/2 + ∥ ˙λ(t)∥2 +Y /2 + α +� t +0 ∥ ˙x(τ)∥2 +X dτ ≤ ∥ ˙x(0)∥2 +X/2 + ∥ ˙λ(0)∥2 +Y /2. +(6) +Using this inequality together with the above equation, we obtain +L(ξ, η) − L(x(t), λ(t)) + ∥ ˙x(t)∥2 +X/α + ∥ ˙λ(t)∥2 +Y /α + +� t +0 +∥ ˙x(τ)∥2 +X + ∥ ˙λ(τ)∥2 +Y dτ ≤ C, +9 + +where C = L(ξ, η)−L(x(0), λ(0))+∥ ˙x(0)∥2 +X/α +∥ ˙λ(0)∥2 +Y /α. Moreover, from (AH) and +the fact that T(ξ, η) = (0X, 0Y ), we infer +L(ξ, η) − L(x(t), λ(t)) = f(ξ) − f(x(t)) + ⟨ ˙x(t) + ∇f(x(t)), x(t) − ξ⟩X += d +dt∥x(t) − ξ∥2 +X/2 + Df(ξ, x(t)) +(7) +with Df denoting the Bregman distance associated with f. Since f is α-strongly convex, +we have Df(ξ, x(t)) ≥ α∥x(t) − ξ∥2 +X/2 and thus, +L(ξ, η) − L(x(t), λ(t)) ≥ d +dt∥x(t) − ξ∥2 +X/2 + α∥x(t) − ξ∥2 +X/2. +Using this inequality together with the fact that +α∥x(t) − ξ∥2 +X/2 − α∥x(0) − ξ∥2 +X/2 = α +� t +0 +d +dτ ∥x(τ) − ξ∥2 +X/2 dτ, +we obtain +L(ξ, η) − L(x(t), λ(t)) ≥ d +dt∥x(t) − ξ∥2 +X/2 + α +� t +0 +d +dτ ∥x(τ) − ξ∥2 +X/2 dτ ++ α∥x(0) − ξ∥2 +X/2. +In view of the above derivations, we deduce +d +dt∥x(t) − ξ∥2 +X/2 + α +� t +0 +d +dτ ∥x(τ) − ξ∥2 +X/2 dτ + ∥ ˙x(t)∥2 +X/α + ∥ ˙λ(t)∥2 +Y /α ++ α +� t +0 +∥ ˙x(τ)∥2 +X/α + ∥ ˙λ(τ)∥2 +Y /α dτ ≤ C − α∥x(0) − ξ∥2 +X/2. +Multiplying the above inequality by eαt and integrating over [0, t] yields +α∥x(t) − ξ∥2 +X/2 + +� t +0 ∥ ˙x(τ)∥2 +X + ∥ ˙λ(τ)∥2 +Y dτ ≤ ˜C +for some sufficiently large constant ˜C. Taking into account that ∥x(t) − ξ∥2 +X ≥ 0 and +subsequently passing to the limit as t → +∞ gives +� ∞ +0 +∥ ˙x(τ)∥2 +X + ∥ ˙λ(τ)∥2 +Y dτ < +∞. +Moreover, since t �→ ∥( ˙x(t), ˙λ(t))∥2 is non-increasing on [0, +∞), cf. Theorem 2.1(i), we +have for any t ≥ 0, +� t +t/2∥ ˙x(τ)∥2 +X + ∥ ˙λ(τ)∥2 +Y dτ ≥ t +�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +�/2. +Observing that ∥( ˙x, ˙λ)∥2 belongs to L1([0, +∞); R) entails +lim +t→+∞ t +�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +� = 0. +10 + +Finally, from inequality (5), we have for any t ≥ 0, +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ ⟨T(x(t), λ(t)), (x(t), λ(t)) − (ξ, η)⟩. +In view of (AH) and the Cauchy–Schwarz inequality, we get +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ ∥(x(t), λ(t)) − (ξ, η)∥∥( ˙x(t), ˙λ(t))∥. +(8) +Using that (x, λ) remains bounded on [0, +∞), cf. Proposition 2.3(i), there exists ˆC ≥ 0 +such that +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ ˆC∥( ˙x(t), ˙λ(t))∥. +Multiplying the above inequality by +√ +t gives +√ +t +�L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 +� ≤ ˆC +√ +t∥( ˙x(t), ˙λ(t))∥. +Observing that limt→+∞ +√ +t∥( ˙x(t), ˙λ(t))∥ = 0, passing to the limit as t → +∞ yields +lim +t→+∞ +√ +t +�L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 +� = 0. +The weak convergence of (x(t), λ(t)) as t → +∞ is an immediate consequence of the +Opial lemma applied to the set S × M; cf. Opial [27]. +Remark 3.2. We note that the weak convergence of the solutions of (AH) may also +be deduced from the graph closedness property of the maximally monotone operator T +with respect to the weak-strong topology; see, e.g., Bauschke and Combettes [3]. Yet +another tool to establish the weak convergence of the solutions of (AH) is the concept of +demipositivity, first developed by Bruck [28] for monotone operators, and later extended +by Chbani and Riahi [29] to monotone bifunctions. However, the maximally monotone +operator T associated with the Lagrangian L of the convex minimization problem (P) +need, in general, not be demipositive. We leave the details to the reader. +To further localize the weak limit of a solution of (AH), recall that the set S × M (if +non-empty) is of the form {¯x} × M, where ¯x ∈ X is the unique minimizer of (P) and +M ⊂ Y refers to the closed affine subspace of Lagrange multipliers, viz., +M = {¯λ ∈ Y | ∇f(¯x) + A∗¯λ = 0X}. +The following result characterizes the weak limit of a solution of (AH) as the orthogonal +projection of its initial data onto the (closed and convex) set S × M. +Corollary 3.3. Under the hypotheses of Theorem 3.1, let (¯x, ¯λ) ∈ S × M be such that +(x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞. Then, (¯x, ¯λ) = projS×M(x0, λ0). +Proof. Let (¯x, ¯λ) ∈ S × M be such that (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as +t → +∞ and let (ξ, η) ∈ S × M be arbitrary. Using (AH) and the fact that T(ξ, η) = +(0X, 0Y ), we have for any t ≥ 0, +⟨(¯x, ¯λ) − (ξ, η), ( ˙x(t), ˙λ(t))⟩ + ⟨T(x(t), λ(t)) − T(ξ, η), (¯x, ¯λ) − (ξ, η)⟩ = 0. +11 + +Observing that +⟨T(x(t), λ(t)) − T(ξ, η), (¯x, ¯λ) − (ξ, η)⟩ = ⟨∇f(x(t)) − ∇f(ξ), ¯x − ξ⟩X ++ ⟨∇f(¯x) − ∇f(ξ), x(t) − ξ⟩X, +and noticing that the right-hand side of the above equation vanishes (as S is reduced +to a singleton), it follows that +⟨(¯x, ¯λ) − (ξ, η), ( ˙x(t), ˙λ(t))⟩ = 0. +Integration over [0, t] yields +⟨(¯x, ¯λ) − (ξ, η), (x(t), λ(t)) − (x(0), λ(0))⟩ = 0. +Since (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞, we infer +⟨(¯x, ¯λ) − (ξ, η), (¯x, ¯λ) − (x(0), λ(0))⟩ = 0. +The above equality being true for any (ξ, η) ∈ S × M, we conclude by virtue of the pro- +jection theorem; see, e.g., Bauschke and Combettes [3]. +Finally, as an immediate consequence of Theorem 3.1, we have the following refined +asymptotic estimates whenever A : X → Y is bounded from below3, i.e., +∃β > 0 ∀x ∈ X +∥Ax∥2 +Y ≥ β∥x∥2 +X. +Corollary 3.4. Under the hypotheses of Theorem 3.1, let A : X → Y be bounded from +below. Then, for any (ξ, η) ∈ S × M, it holds that +L(x(t), η) − L(ξ, λ(t)) = o +�1 +t +� +as t → +∞; +∥x(t) − ξ∥X = o +� 1 +√ +t +� +as t → +∞. +Proof. Let (ξ, η) ∈ S × M. Since A is bounded from below, there exists β > 0 such +that for any t ≥ 0, +∥ ˙x(t)∥2 +X + β∥x(t) − ξ∥2 +X ≤ ∥ ˙x(t)∥2 +X + ∥A(x(t) − ξ)∥2 +Y . +Using (AH) together with the fact that Aξ − b = 0Y , we obtain +∥ ˙x(t)∥2 +X + β∥x(t) − ξ∥2 +X ≤ ∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y . +Multiplying the above inequality by t and using limt→+∞ t∥( ˙x(t), ˙λ(t))∥2 = 0, cf. The- +orem 3.1, we infer +lim +t→+∞ t +�∥ ˙x(t)∥2 +X + β∥x(t) − ξ∥2 +X +� = 0. +3We recall that A : X → Y is bounded from below if and only if it is injective with closed range; see, e.g., +Brézis [30]. +12 + +Moreover, from (5) together with T(ξ, η) = (0X, 0Y ), we observe that for any t ≥ 0, +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ ⟨T(x(t), λ(t)) − T(ξ, η), (x(t), λ(t)) − (ξ, η)⟩ += ⟨∇f(x(t)) − ∇f(ξ), x(t) − ξ⟩X. +Using that ∇f is Lipschitz continuous on bounded sets, there exists γ ≥ 0 such that +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ γ∥x(t) − ξ∥2 +X. +Multiplying this inequality by t and using limt→+∞ t∥x(t) − ξ∥2 +X = 0, we conclude +lim +t→+∞ t +�L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 +� = 0. +3.2. Strong convergence +Let us now complement the previous discussion with a result on the strong convergence +of the solutions of (AH). To this end, we assume that A∗ : Y → X is bounded from +below4, i.e., +∃β > 0 ∀y ∈ Y +∥A∗y∥2 +X ≥ β∥y∥2 +Y . +This clearly implies that the set S×M is reduced to {(¯x, ¯λ)}, where ¯x ∈ X is the unique +minimizer of (P) and ¯λ ∈ Y refers to the corresponding Lagrange multiplier given by +¯λ = −(AA∗)−1A∇f(¯x). +Proposition 3.5. Let ∇f : X → X be α-strongly monotone, let A∗ : Y → X be +bounded from below, and let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for +(ξ, η) ∈ S × M, it holds that +L(x(t), η) − L(ξ, λ(t)) = o +�1 +t +� +as t → +∞; +∥(x(t), λ(t)) − (ξ, η)∥ = o +� 1 +√ +t +� +as t → +∞. +Consequently, (x(t), λ(t)) converges strongly, as t → +∞, to the unique element in +S × M. +Proof. Let (ξ, η) be the unique element in S ×M. Using (AH) together with T(ξ, η) = +(0X, 0Y ) and the fact that ∇f is α-strongly monotone, we have for any t ≥ 0, +d +dt +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +�/2 + α +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� +≤ α∥λ(t) − η∥2 +Y . +(9) +Since A∗ is bounded from below, there exists β > 0 such that +β∥λ(t) − η∥2 +Y ≤ ∥A∗(λ(t) − η)∥2 +X. +4We note that A∗ : Y → X is bounded from below if and only if A is surjective; see, e.g., Brézis [30]. +13 + +Using again (AH) together with ∇f(ξ) + A∗η = 0X, we get +β∥λ(t) − η∥2 +Y ≤ ∥ ˙x(t) + ∇f(x(t)) − ∇f(ξ)∥2 +X +and thus, +β∥λ(t) − η∥2 +Y /2 ≤ ∥ ˙x(t)∥2 +X + ∥∇f(x(t)) − ∇f(ξ)∥2 +X. +Since (x, λ) remains bounded on [0, +∞), cf. Proposition 2.3(i), and owing to the fact +that ∇f is Lipschitz continuous on bounded sets, there further exists γ ≥ 0 such that +β∥λ(t) − η∥2 +Y /2 ≤ ∥ ˙x(t)∥2 +X + γ2∥x(t) − ξ∥2 +X. +In view of the above derivations, we obtain +d +dt +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +�/2 + α +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� +≤ 2α∥ ˙x(t)∥2 +X/β + 2αγ2∥x(t) − ξ∥2 +X/β, +which, by applying (9) again, reads +(β/2 + γ2) d +dt +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� ++ αβ +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� ≤ 2α∥ ˙x(t)∥2 +X. +Integration over [0, t] yields +(β/2 + γ2) +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� + αβ +� t +0 ∥x(τ) − ξ∥2 +X + ∥λ(τ) − η∥2 +Y dτ +≤ 2α +� t +0 ∥ ˙x(τ)∥2 +X dτ + (β/2 + γ2) +�∥x(0) − ξ∥2 +X + ∥λ(0) − η∥2 +Y +�. +Combining the above inequality with (6) gives +∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y + (β/2 + γ2) +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� ++ αβ +� t +0 +∥x(τ) − ξ∥2 +X + ∥λ(τ) − η∥2 +Y dτ ≤ C, +where C = ∥ ˙x(0)∥2 +X + ∥ ˙λ(0)∥2 +Y + (β/2 + γ2) +�∥x(0) − ξ∥2 +X + ∥λ(0) − η∥2 +Y +�. Taking into +account that ∥( ˙x(t), ˙λ(t))∥2 ≥ 0 and ∥(x(t), λ(t)) − (ξ, η)∥2 ≥ 0, and subsequently +passing to the limit as t → +∞ yields +� ∞ +0 +∥x(τ) − ξ∥2 +X + ∥λ(τ) − η∥2 +Y dτ < +∞. +Since t �→ ∥(x(t), λ(t))−(ξ, η)∥2 is non-increasing on [0, +∞), cf. Proposition 2.3(i), we +have for any t ≥ 0, +� t +t/2 +∥x(τ) − ξ∥2 +X + ∥λ(τ) − η∥2 +Y dτ ≥ t +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +�/2. +14 + +Noticing that ∥(x, λ) − (ξ, η)∥2 belongs to L1([0, +∞); R), we classically deduce +lim +t→+∞ t +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� = 0. +Finally, recall from (8) that for any t ≥ 0, +L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 ≤ ∥(x(t), λ(t)) − (ξ, η)∥∥( ˙x(t), ˙λ(t))∥. +Multiplying the above inequality by t and using limt→+∞ +√ +t∥( ˙x(t), ˙λ(t))∥ = 0, cf. The- +orem 3.1, together with limt→+∞ +√ +t∥(x(t), λ(t)) − (ξ, η)∥ = 0, we infer +lim +t→+∞ t +�L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 +X/2 +� = 0, +concluding the desired estimates. +4. Exponential decay rate estimates +In this section, we provide decay rate estimates of exponential type on the solutions +of (AH) under the additional assumption that f is twice continuously differentiable. In +particular, we presuppose that +(A6) f : X → R satisfies condition (C), i.e., +2Df(y, x) − ⟨∇2f(x)(x − y), x − y⟩X ≥ 0, +∀x, y ∈ X; +(A7) ∇2f( · ) : X → X is γ-bounded, i.e., +∃γ > 0 ∀x, y ∈ X +⟨∇2f(x)y, y⟩X ≤ γ∥y∥2 +X. +Here, Df denotes again the Bregman distance associated with f, cf. Bregman [21], and +∇2f refers to the Hessian of f. We remark that condition (C) is verified whenever f is +minorized by its second-order Taylor approximations. +4.1. ‘Primal exponential estimates’ +Let us first establish exponential decay rate estimates on the solutions of (AH) in the +case when A : X → Y is bounded from below, i.e., +∃β > 0 ∀x ∈ X +∥Ax∥2 +Y ≥ β∥x∥2 +X. +Theorem 4.1. Let ∇2f( · ) : X → X be α-elliptic and γ-bounded, and suppose that +A : X → Y is bounded from below with constant β. Let f : X → R satisfy condition +(C) and set +ρ = +� +α/2, +if γ2 ≤ 4β, +min{α, γ − +� +γ2 − 4β}/2, +if γ2 > 4β. +Let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for any (ξ, η) ∈ S × M, the +following assertions hold: +15 + +(i) If ρ2 − γρ + β > 0, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�e−2ρt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�e−2ρt� as t → +∞; +∥x(t) − ξ∥2 +X = O +�e−2ρt� as t → +∞; +(ii) If ρ2 − γρ + β = 0, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�t2e−2ρt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�t2e−2ρt� as t → +∞; +∥x(t) − ξ∥2 +X = O +�t2e−2ρt� as t → +∞. +Proof. Let (ξ, η) ∈ S × M and let ρ > 0 to be chosen. Using again the ‘Lagrangian +identity’ (1) together with (AH), we have for any t ≥ 0, +d +dt +�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +�/2 + ⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩X ++ ρ d +dt +�L(ξ, η) − L(x(t), λ(t)) +� + ρ +�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +� = 2ρ∥ ˙x(t)∥2 +X. +Moreover, from equation (7), we obtain +d +dt +�L(ξ, η) − L(x(t), λ(t)) +� = d +dt⟨x(t) − ξ, ˙x(x)⟩X ++ ⟨∇2f(x(t)) ˙x(t), x(t) − ξ⟩X. +Combining the above expressions yields +d +dt +�∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X + ∥ ˙λ(t)∥2 +Y +�/2 ++ ⟨(∇2f(x(t)) − 2ρ Id) ˙x(t), ˙x(t) + ρ(x(t) − ξ)⟩X ++ ρ +�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +� = 0. +Developing the term in the second line gives +d +dt +�∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 +Y +�/2 ++ ρ +�∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 +Y +� ++ ⟨(∇2f(x(t)) − 2ρ Id)( ˙x(t) + ρ(x(t) − ξ)), ˙x(t) + ρ(x(t) − ξ)⟩X ++ ρ2�2Df(ξ, x(t)) − ⟨∇2f(x(t))(x(t) − ξ), x(t) − ξ⟩X +� = 0. +Since ∇2f( · ) is α-elliptic and f satisfies condition (C), we obtain +d +dt +�∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 +Y +�/2 ++ ρ +�∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 +Y +� ++ (α − 2ρ)∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X ≤ 0. +16 + +An immediate integration over [0, t] shows that there exists C ≥ 0 such that +∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ρ2∥x(t) − ξ∥2 +X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 +Y ++ 2(α − 2ρ) +� t +0 +e−2ρ(t−τ)∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 +X dτ ≤ Ce−2ρt. +(10) +Using that ∇2f( · ) is γ-bounded and that A is bounded from below with constant β, +we have both Df(ξ, x(t)) ≤ γ∥x(t) − ξ∥2 +X/2 and ∥A(x(t) − ξ)∥2 +Y ≥ β∥x(t) − ξ∥2 +X. In +view of (AH) and Aξ − b = 0Y , we infer +∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + (ρ2 − γρ + β)∥x(t) − ξ∥2 +X ++ 2(α − 2ρ) +� t +0 e−2ρ(t−τ)∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 +X dτ ≤ Ce−2ρt. +(11) +Let us now determine the largest value for ρ ∈ (0, α/2] such that ρ2 − γρ + β ≥ 0. +Clearly, if γ2 ≤ 4β, then ρ2 − γρ + β ≥ 0 holds for any ρ > 0. On the other hand, if +γ2 > 4β, then ρ2−γρ+β ≥ 0 is attained whenever ρ ≤ γ/2− +� +γ2 − 4β/2. Consequently, +we may take +ρ = +� +α/2, +if γ2 ≤ 4β, +min{α, γ − +� +γ2 − 4β}/2, +if γ2 > 4β. +We have either one of the following cases: +(i) Suppose that ρ2 − γρ + β > 0. In this case, we deduce from (11) that +e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X ≤ C; +e2ρt∥x(t) − ξ∥2 +X ≤ +C +ρ2 − γρ + β . +Passing to the upper limit as t → +∞ yields +lim sup +t→+∞ e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X < +∞; +lim sup +t→+∞ e2ρt∥x(t) − ξ∥2 +X < +∞. +Moreover, in view of the basic inequality +∥ ˙x(t)∥2 +X ≤ 2∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + 2ρ2∥x(t) − ξ∥2 +X +and the fact that +∥ ˙λ(t)∥2 +Y = ∥A(x(t) − ξ)∥2 +Y ≤ ∥A∥2∥x(t) − ξ∥2 +X, +we obtain +lim sup +t→+∞ e2ρt�∥ ˙x(t)∥2 +X + ∥ ˙λ(t)∥2 +Y +� < +∞. +The remaining estimate is now readily deduced as in Corollary 3.4. +17 + +(ii) Suppose now that ρ2 − γρ + β = 0. In this case, we observe from (11) that +e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X ≤ C. +Passing to the upper limit as t → +∞ entails +lim sup +t→+∞ e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X < +∞. +Moreover, given the fact that +eρt∥x(t) − ξ∥X ≤ ∥x(0) − ξ∥X + +� t +0 +eρτ∥ ˙x(τ) + ρ(x(τ) − ξ)∥X dτ, +we deduce +eρt∥x(t) − ξ∥X ≤ +√ +Ct + ∥x(0) − ξ∥X. +Taking the square and multiplying the resulting inequality by t−2 yields +t−2e2ρt∥x(t) − ξ∥2 +X ≤ C + 2 +√ +C∥x(0) − ξ∥Xt−1 + ∥x(0) − ξ∥2 +Xt−2. +This majorization being valid for any t > 0, we conclude +lim sup +t→+∞ t−2e2ρt∥x(t) − ξ∥2 +X < +∞. +The remaining estimates now follow at once. +The previous result complements the exponential decay rate estimates obtained by +Polyak [19] based on spectral arguments. We further note that the above decay rate es- +timates are comparable to the spectral bounds known for ‘saddle matrices’; cf. the sur- +vey paper by Benzi et al. [31, Section 3.4] and references therein. +Assuming, moreover, that ∇2f( · ) = α Id, we have the following refined exponential +decay rate estimates. +Corollary 4.2. Let ∇2f( · ) = α Id and suppose that A : X → Y is bounded from below +with constant β. Let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for any +(ξ, η) ∈ S × M, the following assertions hold: +(i) If α2 < 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�e−αt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�e−αt� as t → +∞; +∥x(t) − ξ∥2 +X = O +�e−αt� as t → +∞; +(ii) If α2 = 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�t2e−αt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�t2e−αt� as t → +∞; +∥x(t) − ξ∥2 +X = O +�t2e−αt� as t → +∞; +18 + +(iii) If α2 > 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�e−(α−δ)t� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�e−(α−δ)t� as t → +∞; +∥x(t) − ξ∥2 +X = O +�e−(α−δ)t� as t → +∞, +where δ = +� +α2 − 4β. +Proof. (i)–(ii) This is an immediate consequence of Theorem 4.1(i)–(ii). +(iii) Suppose that α2 > 4β and let ρ = (α + δ)/2, where δ = +� +α2 − 4β, so that +ρ2 − αρ + β = 0. From (11) and the fact that ∇2f( · ) = α Id, we observe that there +exists C ≥ 0 such that for any t ≥ 0, +e(α+δ)t∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X ≤ C + 2δ +� t +0 e(α+δ)τ∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 +X dτ. +Applying Gronwall’s inequality yields +e(α+δ)t∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X ≤ Ce2δt. +Using this inequality together with the fact that +e(α+δ)t/2∥x(t) − ξ∥X ≤ ∥x(0) − ξ∥X + +� t +0 e(α+δ)τ/2∥ ˙x(τ) + ρ(x(τ) − ξ)∥X dτ, +we obtain +e(α+δ)t/2∥x(t) − ξ∥X ≤ +√ +C +δ eδt + ∥x(0) − ξ∥X − +√ +C +δ . +Consequently, +e(α−δ)t∥x(t) − ξ∥2 +X ≤ C +δ2 + 2 +√ +C +δ +� +∥x(0) − ξ∥X − +√ +C +δ +� +e−δt ++ +� +∥x(0) − ξ∥X − +√ +C +δ +�2 +e−2δt. +Passing to the upper limit as t → +∞ yields the desired estimate. The remaining as- +sertions are now easily obtained. +Remark 4.3. The previous result essentially recovers the optimal decay rate estimates +known for the classical damped harmonic oscillator. Indeed, in the case when ∇2f( · ) = +α Id, we observe in view of an immediate differentiation that the solutions of (AH) fur- +ther obey the second-order dynamics +¨x + α ˙x + ∇∥A(x − ¯x)∥2 +Y /2 = 0X, +where ¯x denotes the unique element in S. The above second-order differential system +was first introduced, from a more general optimization perspective, by Polyak [32] and is +known to inherit remarkable minimizing properties; see, e.g., Alvarez [33] and Attouch +et al. [34] for a general exposition. +19 + +4.2. ‘Dual exponential estimates’ +Let us now complement the previous discussion with decay rate estimates on the solu- +tions of (AH) under the assumption that A∗ : Y → X is bounded from below, i.e., +∃β > 0 ∀y ∈ Y +∥A∗y∥2 +X ≥ β∥y∥2 +Y . +Proposition 4.4. Let ∇2f( · ) = α Id and suppose that A∗ : Y → X is bounded from +below with constant β. Let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for +(ξ, η) ∈ S × M, the following assertions hold: +(i) If α2 < 4β, then it holds that +∥λ(t) − η∥2 +Y = O +�e−αt� as t → +∞; +∥ ˙λ(t)∥2 +Y = O +�e−αt� as t → +∞; +(ii) If α2 = 4β, then it holds that +∥λ(t) − η∥2 +Y = O +�t2e−αt� as t → +∞; +∥ ˙λ(t)∥2 +Y = O +�t2e−αt� as t → +∞; +(iii) If α2 > 4β, then it holds that +∥λ(t) − η∥2 +Y = O +�e−(α−δ)t� as t → +∞; +∥ ˙λ(t)∥2 +Y = O +�e−(α−δ)t� as t → +∞, +where δ = +� +α2 − 4β. +Proof. Let (ξ, η) be the unique element in S × M and let ρ > 0. Using similar deriva- +tions as in Theorem 4.1, we have for any t ≥ 0 +d +dt +�∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + ρ2∥λ(t) − η∥2 +Y + ∥A∗(λ(t) − η)∥2 +X +�/2 ++ ρ +�∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + ρ2∥λ(t) − η∥2 +Y + ∥A∗(λ(t) − η)∥2 +X +� ++ ⟨A(∇f(x(t)) − ∇f(ξ)), ˙λ(t) + ρ(λ(t) − η)⟩Y +− 2ρ∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y = 0. +From ∇2f( · ) = α Id together with (AH) and Aξ − b = 0Y , we obtain +d +dt +�∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + (ρ2 − αρ)∥λ(t) − η∥2 +Y + ∥A∗(λ(t) − η)∥2 +X +�/2 ++ ρ +�∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + (ρ2 − αρ)∥λ(t) − η∥2 +Y + ∥A∗(λ(t) − η)∥2 +X +� ++ (α − 2ρ)∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y = 0. +An immediate integration over [0, t] shows that there exists C ≥ 0 such that +∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + (ρ2 − αρ)∥λ(t) − η∥2 +Y + ∥A∗(λ(t) − η)∥2 +X ++ 2(α − 2ρ) +� t +0 +∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 +Y dτ = Ce−2ρt. +(12) +20 + +Since A∗ is bounded from below with constant β, we infer +∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + (ρ2 − αρ + β)∥λ(t) − η∥2 +Y ++ 2(α − 2ρ) +� t +0 +∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 +Y dτ ≤ Ce−2ρt. +The desired estimates are now readily deduced. +Remark 4.5. The above result again retrieves the well-known decay rate estimates for +the classical damped harmonic oscillator. As in the previous case (and, in fact, dual to +our observation in Remark 4.3), we note that whenever ∇2f( · ) = α Id, the solutions of +(AH) further obey the second-order dynamics +¨λ + α ˙λ + ∇∥A∗(λ − ¯λ)∥2 +X/2 = 0Y +with ¯λ denoting the unique element in M. We leave the details to the reader. +In order to obtain asymptotic estimates on the primal-dual gap function in the case +when A∗ is bounded from below, we utilize the following relation between the primal +and dual variables. +Lemma 4.6. Let ∇2f( · ) = α Id, let A∗ : Y → X be bounded from below with constant +β, and let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for (ξ, η) ∈ S × M, +there exists C ≥ 0 such that for any t ≥ 0, +β∥(x(t), λ(t)) − (ξ, η)∥2 − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X ≤ Ce−2αt. +Proof. Let (ξ, η) be the unique element in S × M. Using (AH) together with the fact +that T(ξ, η) = (0X, 0Y ), we have for any t ≥ 0, +d +dt +�β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X +�/2 ++ β⟨∇f(x(t)) − ∇f(ξ), x(t) − ξ⟩X − ⟨∇f(x(t)) − ∇f(ξ), A∗A(x(t) − ξ)⟩X = 0. +In view of ∇2f( · ) = α Id, the above equality reads +d +dt +�β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X +�/2 ++ α +�β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X +� ++ α +�∥A∗(λ(t) − η)∥2 +X − β∥λ(t) − η∥2 +Y +� = 0. +Since A∗ is bounded from below with constant β, we obtain +d +dt +�β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X +�/2 ++ α +�β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X +� ≤ 0. +An immediate integration over [0, t] then shows that there exists C ≥ 0 such that +β∥x(t) − ξ∥2 +X + β∥λ(t) − η∥2 +Y − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X ≤ Ce−2αt. +21 + +Combining the above results finally gives the following asymptotic estimates. +Corollary 4.7. Let ∇2f( · ) = α Id and suppose that A∗ : Y → X is bounded from +below with constant β. Let (x, λ) : [0, +∞) → X × Y be a solution of (AH). Then, for +(ξ, η) ∈ S × M, the following assertions hold: +(i) If α2 < 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�e−αt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�e−αt� as t → +∞; +∥(x(t), λ(t)) − (ξ, η)∥2 = O +�e−αt� as t → +∞; +(ii) If α2 = 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�t2e−αt� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�t2e−αt� as t → +∞; +∥(x(t), λ(t)) − (ξ, η)∥2 = O +�t2e−αt� as t → +∞; +(iii) If α2 > 4β, then it holds that +L(x(t), η) − L(ξ, λ(t)) = O +�e−(α−δ)t� as t → +∞; +∥( ˙x(t), ˙λ(t))∥2 = O +�e−(α−δ)t� as t → +∞; +∥(x(t), λ(t)) − (ξ, η)∥2 = O +�e−(α−δ)t� as t → +∞, +where δ = +� +α2 − 4β. +Proof. Let (ξ, η) be the unique element in S × M and let ρ > 0. Since ∇2f( · ) = α Id, +by combining (10) with (12) and subsequently using (AH) together with the fact that +Aξ − b = 0Y , there exists C ≥ 0 such that for any t ≥ 0, +∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y + ∥A(x(t) − ξ)∥2 +Y ++ (ρ2 − αρ) +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� + ∥A∗(λ(t) − η)∥2 +X ++ 2(α − 2ρ) +� t +0 e−2ρ(t−τ)κ(τ) dτ ≤ Ce−2ρt, +where κ(τ) = ∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 +X + ∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 +Y . Since A∗ is bounded from +below with constant β, we observe from Lemma 4.6 that there exists ˜C ≥ 0 such that +β +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� − ∥A(x(t) − ξ)∥2 +Y − ∥A∗(λ(t) − η)∥2 +X ≤ ˜Ce−2αt. +In view of the above inequalities, we obtain +∥ ˙x(t) + ρ(x(t) − ξ)∥2 +X + ∥ ˙λ(t) + ρ(λ(t) − η)∥2 +Y ++ (ρ2 − αρ + β) +�∥x(t) − ξ∥2 +X + ∥λ(t) − η∥2 +Y +� ++ 2(α − 2ρ) +� t +0 e−2ρ(t−τ)κ(τ) dτ ≤ Ce−2ρt + ˜Ce−2αt. +The desired estimates are now easily derived. +22 + +5. Structured convex minimization +In this section, we aim to extend some of our previous results on the Arrow–Hurwicz +differential system (AH) to the more general case of solving structured convex mini- +mization problems. Let X, Y and Z be real Hilbert spaces, and let X × Y × Z be +endowed with the product structure ⟨ · , · ⟩ = ⟨ · , · ⟩X +⟨ · , · ⟩Y +⟨ · , · ⟩Z and associated +norm ∥ · ∥. Consider the structured minimization problem +inf {f(x) + g(y) | Ax + By − c = 0Z} +(SP) +and suppose that the following assumptions are verified: +(A1)′ f : X → R and g : Y → R are convex and continuously differentiable; +(A2)′ ∇f : X → X and ∇g : Y → Y are Lipschitz continuous on bounded sets; +(A3)′ A : X → Z and B : Y → Z are linear and continuous, and c ∈ Z. +We associate with (SP) the Lagrangian +L : X × Y × Z −→ R +(x, y, λ) �−→ f(x) + g(y) + ⟨λ, Ax + By − c⟩Z, +which, given the above assumptions, is a convex function with respect to (x, y) ∈ X ×Y +and a concave (in fact, affine) function with respect to λ ∈ Z. We denote by S×U ×M ⊂ +X × Y × Z the (possibly empty) set of saddle points of the Lagrangian L. +Consider now the generalized Arrow–Hurwicz differential system + + + + + + + +˙x + ∇f(x) + A∗λ = 0X +˙y + ∇g(y) + B∗λ = 0Y +˙λ + c − Ax − By = 0Z +(GAH) +with initial data (x0, y0, λ0) ∈ X × Y × Z and observe that (GAH) admits, for any ini- +tial data, a unique (classical) solution (x, y, λ) : [0, +∞) → X × Y × Z. Recall that the +zeros of the maximally monotone operator +T : X × Y × Z −→ X × Y × Z +(x, y, λ) �−→ (∇f(x) + A∗λ, ∇g(y) + B∗λ, c − Ax − By) +are nothing but the saddle points of the Lagrangian L. +The following discussion suggests that our results on the (AH) differential system +directly convey to the (GAH) evolution system for solving the structured convex mini- +mization problem (SP). +Theorem 5.1. Let S × U × M be non-empty and let (x, y, λ) : [0, +∞) → X × Y × Z +be a solution of (GAH). Then, for any (ξ, ψ, η) ∈ S × U × M, +(i) limt→+∞∥(x(t), y(t), λ(t)) − (ξ, ψ, η)∥ exists; +(ii) limt→+∞∥( ˙x(t), ˙y(t), ˙λ(t))∥ exists; +(iii) it holds that +� ∞ +0 +L(x(τ), y(τ), η) − L(ξ, ψ, λ(τ)) dτ < +∞. +23 + +Let the Cesàro average (σ, τ, ω) : (0, +∞) → X × Y × Z of a solution of (GAH) be +defined by +(σ(t), τ(t), ω(t)) = 1 +t +� t +0 +(x(τ), y(τ), λ(τ)) dτ. +We have the following asymptotic estimate on the primal-dual gap function. +Proposition 5.2. Let S×U ×M be non-empty and let (σ, τ, ω) : (0, +∞) → X ×Y ×Z +be the Cesàro average of a solution of (GAH). Then, for any (ξ, ψ, η) ∈ S × U × M, it +holds that +L(σ(t), τ(t), η) − L(ξ, ψ, ω(t)) = O +�1 +t +� +as t → +∞. +Moreover, there exists (¯x, ¯y, ¯λ) ∈ S × U × M such that (σ(t), τ(t), ω(t)) ⇀ (¯x, ¯y, ¯λ) +weakly in X × Y × Z as t → +∞. +Let us now investigate the asymptotic properties of the solutions of (GAH) under +the more stringent assumption that +f + g : X × Y −→ R +(x, y) �−→ f(x) + g(y) +is α-strongly convex (or, equivalently, ∇(f + g) : X × Y → X × Y is α-strongly mo- +notone). In this case, the following asymptotic properties are verified. +Theorem 5.3. Let ∇(f +g) : X ×Y → X ×Y be α-strongly monotone, let S ×U ×M +be non-empty, and let (x, y, λ) : [0, +∞) → X × Y × Z be a solution of (GAH) with +initial data (x0, y0, λ0) ∈ X × Y × Z. Then, for any (ξ, ψ, η) ∈ S × U × M, it holds that +L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = o +� 1 +√ +t +� +as t → +∞; +∥( ˙x(t), ˙y(t), ˙λ(t))∥ = o +� 1 +√ +t +� +as t → +∞. +Moreover, (x(t), y(t), λ(t)) converges weakly, as t → +∞, to projS×U×M(x0, y0, λ0) ∈ +S × U × M. +Assuming, moreover, that (A B)∗ : Z → X × Y is bounded from below, i.e., +∃β > 0 ∀z ∈ Z +∥(A B)∗z∥2 +X×Y ≥ β∥z∥2 +Z, +we have the following refined asymptotic estimates. +Corollary 5.4. Under the hypotheses of Theorem 5.3, let (A B)∗ : Z → X × Y be +bounded from below. Then, for (ξ, ψ, η) ∈ S × U × M, it holds that +L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = o +�1 +t +� +as t → +∞; +∥(x(t), y(t), λ(t)) − (ξ, ψ, η)∥ = o +� 1 +√ +t +� +as t → +∞. +24 + +Consequently, (x(t), y(t), λ(t)) converges strongly, as t → +∞, to the unique element +in S × U × M. +Remark 5.5. The structured convex minimization problem (SP) has recently been ap- +proached by Attouch et al. [35] and Boţ and Nguyen [36] using the second-order non- +autonomous differential system + + + + + + + + + + + + + +¨x + ν +t ˙x + ∇xLµ(x, y, λ + θt ˙λ) = 0X +¨y + ν +t ˙y + ∇yLµ(x, y, λ + θt ˙λ) = 0Y +¨λ + ν +t +˙λ − ∇λLµ(x + θt ˙x, y + θt ˙y, λ) = 0Z +(AAH) +with ν ≥ 3, µ ≥ 0, θ ∈ [1/(ν − 1), 1/2], and initial data (x0, y0, λ0), (v0, w0, µ0) ∈ +X ×Y ×Z. As a decisive feature, the (AAH) dynamics are governed by ‘asymptotically +vanishing damping coefficients’ which relate the above system to Nesterov’s accelerated +gradient method (see Nesterov [37], Su et al. [38]), and additional ‘exploration terms’ +within the partial gradients of the augmented Lagrangian Lµ associated with (SP). This +particular structure allows for remarkably fast mini-maximizing properties with respect +to the Lagrangian L given the sole convexity hypothesis on the objective function of +(SP). In particular, the (classical) solutions (x, y, λ) : [t0, +∞) → X × Y × Z of (AAH) +with t0 > 0 evolve, for any (ξ, ψ, η) ∈ S × U × M, according to the asymptotic estimate +(see Attouch et al. [35], Boţ and Nguyen [36]) +L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = O +� 1 +t2 +� +as t → +∞. +If, in addition, ν > 3 and θ ∈ (1/(ν − 1), 1/2], then the solutions of (AAH) further re- +main bounded on [t0, +∞) and it holds that +∥( ˙x(t), ˙y(t), ˙λ(t))∥ = O +�1 +t +� +as t → +∞. +Assuming, moreover, that f +g : X ×Y → R is strongly convex, then, for any (ξ, ψ, η) ∈ +S × U × M, the following estimate is verified (see Attouch et al. [35]): +∥(x(t), y(t)) − (ξ, ψ)∥X×Y = O +�1 +t +� +as t → +∞. +The latter may be particularized to an exponential estimate by further introducing tem- +poral scaling factors in (AAH); cf. Attouch et al. [35, Remark 5.2]. +In view of the above discussion, we observe that the second-order differential system +(AAH) clearly outperforms the first-order differential system (AH) in the case of a con- +vex objective function. However, this may not be the case, as we shall see next, whenever +the objective function is strongly convex. +6. Numerical experiments +In this section, we perform numerical experiments on the Arrow–Hurwicz differential +system (AH) to support our theoretical findings. In particular, we consider two simple +25 + +0 +5 +10 +15 +20 +25 +30 +35 +40 +10-14 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +10-4 +10-3 +10-2 +10-1 +100 +101 +0 +5 +10 +15 +20 +25 +30 +35 +40 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +-1 +-0.5 +0 +0.5 +1 +1.5 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Figure 1. +Graphical view on the evolution of the primal-dual gap function L(x(t), η)−L(ξ, λ(t)), the squared +velocity ∥( ˙x(t), ˙λ(t))∥2, the squared error ∥x(t) − ξ∥2 +X, and the trajectories of the solution components x(t) = +(x1(t), x2(t)) of (AH) and (AAH). +but representative (strongly) convex minimization problems in two dimensions. +Example 6.1. Let X, Y = R2 and consider the quadratic function f : R2 → R defined +by f(x1, x2) = (x2 +1 − x1x2 + x2 +2)/2. Clearly, f is α-strongly convex with α = 1/2. +Moreover, ∇2f( · ) is γ-bounded with γ = 3/2. Further, let A(x1, x2) = (x1, x2), b = +(1, 1), and observe that A is bounded from below with constant β = 1. The unique +minimizer of f subject to the linear constraints corresponds to ξ = (1, 1); with associated +Lagrange multiplier η = (−1/2, −1/2). The evolution of the primal-dual gap function +L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, the squared error ∥x(t) − +ξ∥2 +X, and the trajectory of the solution component x(t) = (x1(t), x2(t)) of (AH) with +initial data x0 = (−1, 1) and λ0 = (1, 1) is depicted in Figure 1. For comparison, the +corresponding quantities of the (AAH) differential system are displayed with damping +parameter ν = 3, exploration coefficient θ = 1/2, and augmentation parameter µ = 1/2. +The initial data of the (AAH) differential system is set accordingly to x0 = (−1, 1), +λ0 = (1, 1), v0 = (0, 0), and µ0 = (1, 1). +Analyzing Figure 1, we observe that the solutions (x(t), λ(t)) of (AH) converge, as +t → +∞, towards the unique mini-maximizer (ξ, η) of the convex minimization problem +(P) and its associated Lagrange dual (D); cf. Proposition 3.5. Moreover, according to +Theorem 4.1(i), we find that the primal-dual gap function L(x(t), η) − L(ξ, λ(t)), the +squared velocity ∥( ˙x(t), ˙λ(t))∥2 and the squared error ∥x(t) − ξ∥2 +X obey the exponential +estimate O +�e−αt� as t → +∞. Compared to the (AAH) dynamics for which the quantity +∥( ˙x(t), ˙λ(t))∥2 evolves according to the estimate O +�1/t2� as t → +∞ (even though the +damping parameter is chosen to be ν = 3), we find that the solutions of (AH) indeed +26 + +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +Figure 2. +Exponential decay properties of the primal-dual gap function L(x(t), η) − L(ξ, λ(t)), the squared +velocity ∥( ˙x(t), ˙λ(t))∥2, and the squared error ∥(x(t), λ(t)) − (ξ, η)∥2 of the solutions (x(t), λ(t)) of (AH) for +distinct values of α. +admit a faster and less oscillatory decay. It is interesting to note that, in this example, +the primal-dual gap function L(x(t), η) − L(ξ, λ(t)) and the squared error ∥x(t) − ξ∥2 +X +for (AAH) appear to obey the estimate O +�1/t4� rather than O +�1/t2� as t → +∞. +Example 6.2. Let X, Y = R2 and consider the parameterized quadratic function f : +R2 → R defined by f(x1, x2) = α(x2 +1 + x2 +2)/2 with α > 0. Further, let A(x1, x2) = +√ +2(x1 + x2)/2 and β = +√ +2/2 so that A∗ is bounded from below with constant β = 1. +The unique minimizer of f subject to the linear constraints is denoted by ξ; with cor- +responding Lagrange multiplier η. Figure 2 illustrates the decay properties of the primal- +dual gap function L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, and the +squared error ∥(x(t), λ(t))−(ξ, η)∥2 of the solutions (x(t), λ(t)) of (AH) for the distinct +values α = 1, α = 2, and α = 3. The initial data is set to x0 = (−1, 1) and λ0 = 1. +Figure 2 suggests that the solutions (x(t), λ(t)) of (AH) converge, as t → +∞, at an +exponential rate towards the unique mini-maximizer (ξ, η) of the convex minimization +problem (P) and its associated Lagrange dual (D). Indeed, the decay properties of the +solutions of (AH) may be categorized as predicted by Corollary 4.7: In case (i), we +have α2 < 4β with the rate estimate O +�e−αt� as t → +∞. We refer to this case as the +‘under-damped case’ as the solutions of (AH) admit a significant oscillatory behavior. +In case (ii), we have α2 = 4β with the rate estimate O +�t2e−αt� as t → +∞. This +case refers to the ‘critically-damped case’ for which we observe the fastest possible +convergence of the solutions of (AH). Finally, in case (iii), we have α2 > 4β with the +rate estimate O +�e−(α−δ)t� as t → +∞, where δ = +� +α2 − 4β. In this case, referred to as +the ‘over-damped case’, the decay of the solutions of (AH) is considerably degraded. +27 + +Acknowledgment +The author expresses his gratitude to the two anonymous reviewers whose comments +and suggestions led to a significant improvement of this manuscript. +Disclosure statement +No potential conflict of interest was reported by the author. +Funding +Research supported by the German Research Foundation (DFG). +References +[1] Ekeland I, Témam R. Convex analysis and variational problems. Philadelphia: Society +for Industrial and Applied Mathematics; 1999. Classics in applied mathematics. +[2] Hiriart-Urruty JB, Lemaréchal C. Convex analysis and minimization algorithms I. New +York: Springer; 1993. Grundlehren der mathematischen Wissenschaften 305. +[3] Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert +spaces. New York: Springer; 2017. CMS Books in Mathematics. +[4] Arrow KJ, Hurwicz L. 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J Mach Learn Res. 2016;17:1–43. +29 + diff --git a/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/load_file.txt b/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e5119243c8a961fde2a22f109cfe2b17b1ee232 --- /dev/null +++ b/bNAyT4oBgHgl3EQfXPcY/content/tmp_files/load_file.txt @@ -0,0 +1,802 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf,len=801 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='00177v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='OC] 31 Dec 2022 On the Arrow–Hurwicz differential system for linearly constrained convex minimization Simon K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Niederländer Institute for Systems Theory and Automatic Control, University of Stuttgart, Stuttgart, Germany ARTICLE HISTORY Compiled January 3, 2023 Abstract In a real Hilbert space setting, we reconsider the classical Arrow–Hurwicz differential system in view of solving linearly constrained convex minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We in- vestigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Our convergence analysis mainly relies on a ‘Lagrangian identity’ which naturally extends on the well-known descent property of the classical continuous steepest descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In addition, we present asymptotic estimates on the decay of the solutions and the primal-dual gap function measured in terms of the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' These estimates are further refined to the ones of the classical damped harmonic oscillator provided that second-order information on the objective function of the convex minimization problem is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, we show that our results directly translate to the case of solving structured convex minimiza- tion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Numerical experiments further illustrate our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' KEYWORDS Arrow–Hurwicz differential system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Lyapunov analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' asymptotic properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' exponential stabilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' convex minimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' saddle-point problem 2010 MATHEMATICS SUBJECT CLASSIFICATIONS 37N40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 46N10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 49M30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 65K05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 90C25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Introduction Let X and Y be real Hilbert spaces endowed with inner products ⟨ · , · ⟩X, ⟨ · , · ⟩Y and induced norms ∥ · ∥X, ∥ · ∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consider the minimization problem inf {f(x) | Ax − b = 0Y }, (P) where f : X → R is a convex and continuously differentiable function, A : X → Y a linear and continuous operator, and b ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We associate with (P) the Lagrangian L : X × Y −→ R (x, λ) �−→ f(x) + ⟨λ, Ax − b⟩Y CONTACT Simon K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Niederländer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Email: niederlaender@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='de which, by construction, is a convex-concave and continuously differentiable bifunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' A pair (¯x, ¯λ) ∈ X × Y is a saddle point of the Lagrangian L if L(¯x, λ) ≤ L(¯x, ¯λ) ≤ L(x, ¯λ), ∀(x, λ) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' It is well known that (¯x, ¯λ) ∈ X ×Y is a saddle point of L if and only if ¯x is a minimizer of (P), ¯λ is a maximizer of the Lagrange dual to (P), that is sup {−f ∗(−A∗λ) − ⟨λ, b⟩Y | λ ∈ Y }, (D) and the optimal values of (P) and (D) coincide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Ekeland and Témam [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Here, f ∗ : X → R ∪ {+∞} denotes the Fenchel conjugate of f defined by f ∗(u) = sup {⟨u, x⟩X − f(x) | x ∈ X}, and A∗ : Y → X refers to the adjoint operator of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Equivalently, (¯x, ¯λ) ∈ X ×Y is a saddle point of L if and only if (¯x, ¯λ) solves the system of primal-dual optimality conditions � ∇f(x) + A∗λ = 0X Ax − b = 0Y with ∇f denoting the gradient of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Throughout the text, we denote by S ×M ⊂ X ×Y the (possibly empty) set of saddle points of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We recall that a saddle point of L exists whenever (P) admits a minimizer and, for instance, the constraint qualification b ∈ sri A(X) is verified1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Here, for a convex set C ⊂ Y , we denote by sri C = {x ∈ C | � µ>0 µ(C − x) is a closed linear subspace of Y } its strong relative interior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Bauschke and Combettes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We further recall that (P) admits a minimizer whenever its feasible set is non-empty and, for instance, f is coercive, that is, lim∥x∥X→+∞ f(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' On the other hand, if the feasible set of (P) is non-empty and f is strongly convex, then (P) admits a unique minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this work, we reconsider the classical Arrow–Hurwicz differential system � ˙x + ∇f(x) + A∗λ = 0X ˙λ + b − Ax = 0Y (AH) relative to the convex minimization problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The (AH) differential system was in essence originated by Arrow and Hurwicz [4] (see also Kose [5], Arrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [6]) and is known to be intimately related to the mini-maximization of the Lagrangian L associ- ated with (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Indeed, given the above system of primal-dual optimality conditions, we immediately observe that the zeros of the operator T : X × Y −→ X × Y (x, λ) �−→ (∇xL(x, λ), −∇λL(x, λ)), 1We remark that, in the finite-dimensional case, the condition amounts to b ∈ A(X) which is commonly re- ferred to as Slater assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Hiriart-Urruty and Lemaréchal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 2 that is, the ‘generator’ of the (AH) differential system, are precisely the saddle points of the Lagrangian L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', (¯x, ¯λ) ∈ S × M ⇐⇒ T(¯x, ¯λ) = (0X, 0Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, the operator T is maximally monotone on X × Y as it is both monotone and continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Minty [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Therefore, S × M can be interpreted as the set of zeros of the maximally monotone operator T and, as such, it is a closed and convex subset of X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The latter may also be deduced more elementary from the convexity-concavity properties of the ‘saddle function’ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Rockafellar [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Preliminary facts As emphasized by Rockafellar [9], the general theory for semi-groups of contractions generated by maximally monotone operators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Crandall and Pazy [10], Brézis [11]) applies to the Arrow–Hurwicz differential system (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' These results, dating back to the works of Kato [12] and K¯omura [13] (see also Browder [14]), imply that the Cauchy problem associated with (AH) is well posed and that its (classical) solutions (x, λ), (y, η) : [0, +∞) → X × Y verify the ‘non-expansiveness property’ d dt �∥x(t) − y(t)∥2 X + ∥λ(t) − η(t)∥2 Y � ≤ 0, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' If, in addition, the set S × M is non-empty, then the solutions (x(t), λ(t)) of (AH) remain bounded and, in fact, weakly converge in average, as t → +∞, towards a saddle point of the Lagrangian L (see Baillon and Brézis [15]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', there exists (¯x, ¯λ) ∈ S × M such that 1 t � t 0 (x(τ), λ(τ)) dτ ⇀ (¯x, ¯λ) as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The (asymptotic) stability properties of the solutions of (AH) (in the sense of Lya- punov) were further investigated by Venets [16] (see also Flåm and Ben-Israel [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' These results suggest that the solutions of (AH) tend towards a saddle point of L giv- en that, for any (x, λ) ∈ X × Y with x /∈ S, it holds that L(¯x, λ) ≤ L(¯x, ¯λ) < L(x, ¯λ), ∀(¯x, ¯λ) ∈ S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The above condition is, of course, trivially satisfied whenever f is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In the respective works, the authors further noted that the solutions (x, λ) : [0, +∞) → X ×Y of (AH) obey the ‘Lagrangian identity’ d dtL(x(t), λ(t)) + ∥ ˙x(t)∥2 X = ∥ ˙λ(t)∥2 Y , ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (1) The identity, however, was not pursued any further due to its indefinite character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We remark that, in the unconstrained case of (P), the above identity reduces to the well- known ‘descent property’ d dtf(x(t)) + ∥ ˙x(t)∥2 X = 0, ∀t ≥ 0 3 associated with the classical continuous steepest descent method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Brézis [11], Aubin and Cellina [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, the exponential decay properties of the solutions of (AH) were investigated by Polyak [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using spectral arguments, the work provides conditions for which the solutions (x(t), λ(t)) of (AH) converge at an exponential rate, as t → +∞, towards a saddle point (¯x, ¯λ) of L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', for which there exists ρ > 0 such that ∥x(t) − ¯x∥2 X + ∥λ(t) − ¯λ∥2 Y = O �e−ρt� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The decay rate estimates are, however, not derived in an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this work, our objective is to recover, unify and extend some of the previous results on the classical Arrow–Hurwicz differential system (AH) in view of solving the linearly constrained convex minimization problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using tools from monotone operator theory, we focus our attention on the convergence properties of the solutions of (AH) and further aim to characterize their limit within the set of saddle points of the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We also intend to make a contribution to the issue of finding (explicit) decay rate esti- mates on the solutions of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Presentation of the results The mini-maximizing properties of the solutions (x, λ) : [0, +∞) → X × Y of (AH) with respect to the convex minimization problem (P) and its associated Lagrange dual (D) are conveniently measured in terms of the ‘primal-dual gap function’ t �−→ L(x(t), · ) − L( · , λ(t)) relative to the set S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Whenever the function f is convex, we observe that the solutions (x(t), λ(t)) of (AH) may fail to converge as t → +∞ even though the set of saddle points of L is comprised of a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' As a consequence, it is natural to first study the average behavior of a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using the notion of the Cesàro average (σ, ω) : (0, +∞) → X × Y of a solution of (AH), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', (σ(t), ω(t)) = 1 t � t 0 (x(τ), λ(τ)) dτ, we find that the solutions of (AH) obey in average, for any (ξ, η) ∈ S ×M, the estimate L(σ(t), η) − L(ξ, ω(t)) = O �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, the Cesàro average (σ(t), ω(t)) of a solution of (AH) weakly converges, as t → +∞, towards a saddle point of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This result is in line with the work by Nemirovski and Yudin [20] on the classical Arrow–Hurwicz method and may also be deduced more elementary by the results of Baillon and Brézis [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Whenever f is strongly convex, we obtain more stringent mini-maximizing properties of the solutions of (AH) relative to the primal-dual gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' More precisely, we show that the solutions (x, λ) : [0, +∞) → X×Y of (AH) evolve, for any (ξ, η) ∈ S×M, according to the estimate L(x(t), η) − L(ξ, λ(t)) = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 4 Moreover, the solutions (x(t), λ(t)) of (AH) are proven to converge weakly, as t → +∞, towards an element of the set of saddle points of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In particular, we characterize the weak limit of a solution of (AH) as the orthogonal projection of its initial data (x0, λ0) ∈ X × Y onto the (closed and convex) set S × M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', (x(t), λ(t)) ⇀ projS×M(x0, λ0) as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' If, in addition, the linear operator A∗ is bounded from below, we observe that the so- lutions of (AH) obey, for (ξ, η) ∈ S × M, the refined estimate L(x(t), η) − L(ξ, λ(t)) = o �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, it is proven that the solutions (x(t), λ(t)) of (AH) strongly converge, as t → +∞, towards the unique saddle point of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, we show that the solutions of (AH) decay asymptotically at an exponential rate provided that f is twice continuously differentiable, satisfying ⟨∇2f(x)(x − y), x − y⟩X ≤ 2Df(y, x), ∀x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Here, Df denotes the Bregman distance associated with f, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Bregman [21], and ∇2f refers to the Hessian of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In particular, we show that under the above condition there exists ρ > 0 such that the solutions (x, λ) : [0, +∞) → X × Y of (AH) verify, for any (ξ, η) ∈ S × M, either one of the following exponential estimates: L(x(t), η) − L(ξ, λ(t)) = O �e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' L(x(t), η) − L(ξ, λ(t)) = O(t2e−2ρt) as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This result complements the decay rate estimates obtained earlier by Polyak [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Organization We begin our discussion by reviewing some basic properties of the solutions of (AH) in the case of a convex objective function of the minimization problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In Section 3, we then investigate their asymptotic properties under the more stringent assumption of a strongly convex objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In Section 4, we show that the solutions of (AH) decay at an exponential rate provided that second-order information on the objective function is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In Section 5, we highlight that the results on the Arrow–Hurwicz differential system (AH) may directly be conveyed to the case of solving structured minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, Section 6 is devoted to numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Basic properties Let X ×Y be equipped with the Hilbertian product structure ⟨ · , · ⟩ = ⟨ · , · ⟩X +⟨ · , · ⟩Y and induced norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Throughout the text, we assume that (A1) f : X → R is convex and continuously differentiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A2) ∇f : X → X is Lipschitz continuous on bounded sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A3) A : X → Y is linear and continuous, and b ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 5 Consider the Arrow–Hurwicz differential system � ˙x + ∇f(x) + A∗λ = 0X ˙λ + b − Ax = 0Y (AH) with initial data (x0, λ0) ∈ X × Y , and recall that (x, λ) : [0, +∞) → X × Y is a (classical) solution of (AH) if (x, λ) ∈ C1([0, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' X × Y ) and (x, λ) satisfies (AH) on [0, +∞) with (x(0), λ(0)) = (x0, λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The following result is an immediate consequence of the monotonicity of the operator T : X × Y −→ X × Y (x, λ) �−→ (∇f(x) + A∗λ, b − Ax) associated with the (AH) differential system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Brézis [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1], Aubin and Cel- lina [18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The existence and uniqueness of the (classical) solutions of (AH) thereby follow at once from the Cauchy–Lipschitz theorem2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Haraux [22, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' For any (x0, λ0) ∈ X × Y , there exists a unique solution (x, λ) : [0, +∞) → X × Y of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, (i) t �→ ∥( ˙x(t), ˙λ(t))∥ is non-increasing and ∥( ˙x(t), ˙λ(t))∥ ≤ ∥T(x0, λ0)∥, ∀t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) limt→+∞∥( ˙x(t), ˙λ(t))∥ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We note that the assertions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1 essentially remain valid even under the assumption that f : X → R∪{+∞} is a proper convex lower semi-continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, the (AH) dynamics generalize to the evolution system � ˙x + ∂f(x) + A∗λ ∋ 0X ˙λ + b − Ax = 0Y with ∂f denoting the convex subdifferential of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The existence and uniqueness of the (strong) solutions of the above differential system are then deduced from the general theory for semi-groups of contractions generated by maximally monotone operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Brézis [11] (see also Pazy [23], Peypouquet and Sorin [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In the following, let S × M denote the set of saddle points of the Lagrangian L : X × Y −→ R (x, λ) �−→ f(x) + ⟨λ, Ax − b⟩Y associated with the convex minimization problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using the convexity of f, we immediately observe that, for any (x, λ), (y, η) ∈ X × Y , it holds that ⟨T(x, λ), (x, λ) − (y, η)⟩ ≥ L(x, η) − L(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (2) 2Given the above assumptions, it is easy to verify that (x, λ) �→ T(x, λ) is Lipschitz continuous on the bounded subsets of X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 6 Anchoring the above inequality to the set S ×M yields the following integrability result for the primal-dual gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let S × M be non-empty and let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, (i) limt→+∞∥(x(t), λ(t)) − (ξ, η)∥ exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) it holds that � ∞ 0 L(x(τ), η) − L(ξ, λ(τ)) dτ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (i) Let (ξ, η) ∈ S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using (AH) together with (2), we have for any t ≥ 0, d dt �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y �/2 + L(x(t), η) − L(ξ, λ(t)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (3) Since (ξ, η) belongs to S ×M, it follows that L(x(t), η)−L(ξ, η) ≥ 0 as well as L(ξ, η)− L(ξ, λ(t)) ≥ 0 and thus, L(x(t), η) − L(ξ, λ(t)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consequently, t �→ ∥(x(t), λ(t)) − (ξ, η)∥ is non-increasing and bounded from below on [0, +∞) so that lim t→+∞∥(x(t), λ(t)) − (ξ, η)∥ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) Integration of (3) over [0, t] yields ∥x(t) − ξ∥2 X/2 + ∥λ(t) − η∥2 Y /2 + � t 0 L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ ∥x(0) − ξ∥2 X/2 + ∥λ(0) − η∥2 Y /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (4) Taking into account that ∥(x(t), λ(t)) − (ξ, η)∥2 ≥ 0, we obtain � t 0 L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ ∥x(0) − ξ∥2 X/2 + ∥λ(0) − η∥2 Y /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This majorization being valid for any t ≥ 0, taking the supremum gives � ∞ 0 L(x(τ), η) − L(ξ, λ(τ)) dτ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3(i) asserts that the solutions of (AH) remain bounded whenever the set S × M is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Conversely, it can be shown that the set S × M is non-empty whenever (AH) admits a bounded solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Pazy [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let us next focus on the asymptotic properties of the solutions of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' To this end, define the Cesàro average (σ, ω) : (0, +∞) → X × Y of a solution of (AH) by (σ(t), ω(t)) = 1 t � t 0 (x(τ), λ(τ)) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 7 The following result asserts the weak convergence of the Cesàro average of a solution of (AH) and further provides an estimate on the decay of the primal-dual gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let S × M be non-empty and let (σ, ω) : (0, +∞) → X × Y be the Cesàro average of a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, it holds that L(σ(t), η) − L(ξ, ω(t)) = O �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, there exists (¯x, ¯λ) ∈ S × M such that (σ(t), ω(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) ∈ S × M and recall from (4) that for any t ≥ 0, ∥x(t) − ξ∥2 X/2 + ∥λ(t) − η∥2 Y /2 + � t 0 L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ ∥x(0) − ξ∥2 X/2 + ∥λ(0) − η∥2 Y /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Dividing the above inequality by t > 0 and taking into account that ∥(x(t), λ(t)) − (ξ, η)∥2 ≥ 0, we obtain 1 t � t 0 L(x(τ), η) − L(ξ, λ(τ)) dτ ≤ 1 2t �∥x(0) − ξ∥2 X + ∥λ(0) − η∥2 Y �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' By Jensen’s inequality, as L( · , η) and −L(ξ, · ) are both convex, it follows L(σ(t), η) − L(ξ, ω(t)) ≤ 1 2t �∥x(0) − ξ∥2 X + ∥λ(0) − η∥2 Y � and thus, lim sup t→+∞ t �L(σ(t), η) − L(ξ, ω(t)) � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The weak convergence of (σ(t), ω(t)) as t → +∞ is an immediate consequence of the Opial–Passty lemma applied to the set S × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Passty [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We note that the weak ergodic convergence of the solutions of (AH) may also be deduced from the general theory for semi-groups of contractions generated by maximally monotone operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Baillon and Brézis [15] (see also Brézis [26] for a localization result of the weak limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The strongly monotone case In this section, we investigate the asymptotic properties of the solutions of (AH) under the more stringent assumptions that (A4) ∇f : X → X is α-strongly monotone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃α > 0 ∀x, y ∈ X ⟨∇f(x) − ∇f(y), x − y⟩X ≥ α∥x − y∥2 X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A5) S × M ⊂ X × Y is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 8 We recall that the latter assumption is verified whenever (P) admits a minimizer and, for instance, the constraint qualification b ∈ sri A(X) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Bauschke and Combettes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In turn, the former assumption implies that S is reduced to a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Weak convergence Let us begin with a result on the weak convergence of the solutions of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since f is α- strongly convex (∇f being α-strongly monotone), we have for any (x, λ), (y, η) ∈ X ×Y , ⟨T(x, λ), (x, λ) − (y, η)⟩ ≥ L(x, η) − L(y, λ) + α∥x − y∥2 X/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (5) Utilizing this inequality relative to the set S × M gives the following asymptotic prop- erties of the solutions of (AH) and the primal-dual gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇f : X → X be α-strongly monotone, let S × M be non-empty, and let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, it holds that L(x(t), η) − L(ξ, λ(t)) = o � 1 √ t � as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥ = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, there exists (¯x, ¯λ) ∈ S × M such that (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) ∈ S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using the ‘Lagrangian identity’ (1), we have for any t ≥ 0, d dt �L(ξ, η) − L(x(t), λ(t)) � + ∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y = 2∥ ˙x(t)∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Integration over [0, t] yields L(ξ, η) − L(x(t), λ(t)) + � t 0 ∥ ˙x(τ)∥2 X + ∥ ˙λ(τ)∥2 Y dτ = 2 � t 0 ∥ ˙x(τ)∥2 X dτ + L(ξ, η) − L(x(0), λ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since ∇f is α-strongly monotone, we readily deduce from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1(i) that ∥ ˙x(t)∥2 X/2 + ∥ ˙λ(t)∥2 Y /2 + α � t 0 ∥ ˙x(τ)∥2 X dτ ≤ ∥ ˙x(0)∥2 X/2 + ∥ ˙λ(0)∥2 Y /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (6) Using this inequality together with the above equation, we obtain L(ξ, η) − L(x(t), λ(t)) + ∥ ˙x(t)∥2 X/α + ∥ ˙λ(t)∥2 Y /α + � t 0 ∥ ˙x(τ)∥2 X + ∥ ˙λ(τ)∥2 Y dτ ≤ C, 9 where C = L(ξ, η)−L(x(0), λ(0))+∥ ˙x(0)∥2 X/α +∥ ˙λ(0)∥2 Y /α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, from (AH) and the fact that T(ξ, η) = (0X, 0Y ), we infer L(ξ, η) − L(x(t), λ(t)) = f(ξ) − f(x(t)) + ⟨ ˙x(t) + ∇f(x(t)), x(t) − ξ⟩X = d dt∥x(t) − ξ∥2 X/2 + Df(ξ, x(t)) (7) with Df denoting the Bregman distance associated with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since f is α-strongly convex, we have Df(ξ, x(t)) ≥ α∥x(t) − ξ∥2 X/2 and thus, L(ξ, η) − L(x(t), λ(t)) ≥ d dt∥x(t) − ξ∥2 X/2 + α∥x(t) − ξ∥2 X/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using this inequality together with the fact that α∥x(t) − ξ∥2 X/2 − α∥x(0) − ξ∥2 X/2 = α � t 0 d dτ ∥x(τ) − ξ∥2 X/2 dτ, we obtain L(ξ, η) − L(x(t), λ(t)) ≥ d dt∥x(t) − ξ∥2 X/2 + α � t 0 d dτ ∥x(τ) − ξ∥2 X/2 dτ + α∥x(0) − ξ∥2 X/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of the above derivations, we deduce d dt∥x(t) − ξ∥2 X/2 + α � t 0 d dτ ∥x(τ) − ξ∥2 X/2 dτ + ∥ ˙x(t)∥2 X/α + ∥ ˙λ(t)∥2 Y /α + α � t 0 ∥ ˙x(τ)∥2 X/α + ∥ ˙λ(τ)∥2 Y /α dτ ≤ C − α∥x(0) − ξ∥2 X/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Multiplying the above inequality by eαt and integrating over [0, t] yields α∥x(t) − ξ∥2 X/2 + � t 0 ∥ ˙x(τ)∥2 X + ∥ ˙λ(τ)∥2 Y dτ ≤ ˜C for some sufficiently large constant ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Taking into account that ∥x(t) − ξ∥2 X ≥ 0 and subsequently passing to the limit as t → +∞ gives � ∞ 0 ∥ ˙x(τ)∥2 X + ∥ ˙λ(τ)∥2 Y dτ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, since t �→ ∥( ˙x(t), ˙λ(t))∥2 is non-increasing on [0, +∞), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1(i), we have for any t ≥ 0, � t t/2∥ ˙x(τ)∥2 X + ∥ ˙λ(τ)∥2 Y dτ ≥ t �∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y �/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Observing that ∥( ˙x, ˙λ)∥2 belongs to L1([0, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' R) entails lim t→+∞ t �∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 10 Finally, from inequality (5), we have for any t ≥ 0, L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ ⟨T(x(t), λ(t)), (x(t), λ(t)) − (ξ, η)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of (AH) and the Cauchy–Schwarz inequality, we get L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ ∥(x(t), λ(t)) − (ξ, η)∥∥( ˙x(t), ˙λ(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (8) Using that (x, λ) remains bounded on [0, +∞), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3(i), there exists ˆC ≥ 0 such that L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ ˆC∥( ˙x(t), ˙λ(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Multiplying the above inequality by √ t gives √ t �L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 � ≤ ˆC √ t∥( ˙x(t), ˙λ(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Observing that limt→+∞ √ t∥( ˙x(t), ˙λ(t))∥ = 0, passing to the limit as t → +∞ yields lim t→+∞ √ t �L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The weak convergence of (x(t), λ(t)) as t → +∞ is an immediate consequence of the Opial lemma applied to the set S × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Opial [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We note that the weak convergence of the solutions of (AH) may also be deduced from the graph closedness property of the maximally monotone operator T with respect to the weak-strong topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Bauschke and Combettes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Yet another tool to establish the weak convergence of the solutions of (AH) is the concept of demipositivity, first developed by Bruck [28] for monotone operators, and later extended by Chbani and Riahi [29] to monotone bifunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' However, the maximally monotone operator T associated with the Lagrangian L of the convex minimization problem (P) need, in general, not be demipositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' To further localize the weak limit of a solution of (AH), recall that the set S × M (if non-empty) is of the form {¯x} × M, where ¯x ∈ X is the unique minimizer of (P) and M ⊂ Y refers to the closed affine subspace of Lagrange multipliers, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', M = {¯λ ∈ Y | ∇f(¯x) + A∗¯λ = 0X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The following result characterizes the weak limit of a solution of (AH) as the orthogonal projection of its initial data onto the (closed and convex) set S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Under the hypotheses of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, let (¯x, ¯λ) ∈ S × M be such that (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, (¯x, ¯λ) = projS×M(x0, λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (¯x, ¯λ) ∈ S × M be such that (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞ and let (ξ, η) ∈ S × M be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using (AH) and the fact that T(ξ, η) = (0X, 0Y ), we have for any t ≥ 0, ⟨(¯x, ¯λ) − (ξ, η), ( ˙x(t), ˙λ(t))⟩ + ⟨T(x(t), λ(t)) − T(ξ, η), (¯x, ¯λ) − (ξ, η)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 11 Observing that ⟨T(x(t), λ(t)) − T(ξ, η), (¯x, ¯λ) − (ξ, η)⟩ = ⟨∇f(x(t)) − ∇f(ξ), ¯x − ξ⟩X + ⟨∇f(¯x) − ∇f(ξ), x(t) − ξ⟩X, and noticing that the right-hand side of the above equation vanishes (as S is reduced to a singleton), it follows that ⟨(¯x, ¯λ) − (ξ, η), ( ˙x(t), ˙λ(t))⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Integration over [0, t] yields ⟨(¯x, ¯λ) − (ξ, η), (x(t), λ(t)) − (x(0), λ(0))⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since (x(t), λ(t)) ⇀ (¯x, ¯λ) weakly in X × Y as t → +∞, we infer ⟨(¯x, ¯λ) − (ξ, η), (¯x, ¯λ) − (x(0), λ(0))⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The above equality being true for any (ξ, η) ∈ S × M, we conclude by virtue of the pro- jection theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Bauschke and Combettes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, as an immediate consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, we have the following refined asymptotic estimates whenever A : X → Y is bounded from below3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃β > 0 ∀x ∈ X ∥Ax∥2 Y ≥ β∥x∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Under the hypotheses of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, let A : X → Y be bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, it holds that L(x(t), η) − L(ξ, λ(t)) = o �1 t � as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥X = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) ∈ S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since A is bounded from below, there exists β > 0 such that for any t ≥ 0, ∥ ˙x(t)∥2 X + β∥x(t) − ξ∥2 X ≤ ∥ ˙x(t)∥2 X + ∥A(x(t) − ξ)∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using (AH) together with the fact that Aξ − b = 0Y , we obtain ∥ ˙x(t)∥2 X + β∥x(t) − ξ∥2 X ≤ ∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Multiplying the above inequality by t and using limt→+∞ t∥( ˙x(t), ˙λ(t))∥2 = 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, we infer lim t→+∞ t �∥ ˙x(t)∥2 X + β∥x(t) − ξ∥2 X � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 3We recall that A : X → Y is bounded from below if and only if it is injective with closed range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Brézis [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 12 Moreover, from (5) together with T(ξ, η) = (0X, 0Y ), we observe that for any t ≥ 0, L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ ⟨T(x(t), λ(t)) − T(ξ, η), (x(t), λ(t)) − (ξ, η)⟩ = ⟨∇f(x(t)) − ∇f(ξ), x(t) − ξ⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using that ∇f is Lipschitz continuous on bounded sets, there exists γ ≥ 0 such that L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ γ∥x(t) − ξ∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Multiplying this inequality by t and using limt→+∞ t∥x(t) − ξ∥2 X = 0, we conclude lim t→+∞ t �L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Strong convergence Let us now complement the previous discussion with a result on the strong convergence of the solutions of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' To this end, we assume that A∗ : Y → X is bounded from below4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃β > 0 ∀y ∈ Y ∥A∗y∥2 X ≥ β∥y∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This clearly implies that the set S×M is reduced to {(¯x, ¯λ)}, where ¯x ∈ X is the unique minimizer of (P) and ¯λ ∈ Y refers to the corresponding Lagrange multiplier given by ¯λ = −(AA∗)−1A∇f(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇f : X → X be α-strongly monotone, let A∗ : Y → X be bounded from below, and let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for (ξ, η) ∈ S × M, it holds that L(x(t), η) − L(ξ, λ(t)) = o �1 t � as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥(x(t), λ(t)) − (ξ, η)∥ = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consequently, (x(t), λ(t)) converges strongly, as t → +∞, to the unique element in S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) be the unique element in S ×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using (AH) together with T(ξ, η) = (0X, 0Y ) and the fact that ∇f is α-strongly monotone, we have for any t ≥ 0, d dt �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y �/2 + α �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � ≤ α∥λ(t) − η∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (9) Since A∗ is bounded from below, there exists β > 0 such that β∥λ(t) − η∥2 Y ≤ ∥A∗(λ(t) − η)∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 4We note that A∗ : Y → X is bounded from below if and only if A is surjective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Brézis [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 13 Using again (AH) together with ∇f(ξ) + A∗η = 0X, we get β∥λ(t) − η∥2 Y ≤ ∥ ˙x(t) + ∇f(x(t)) − ∇f(ξ)∥2 X and thus, β∥λ(t) − η∥2 Y /2 ≤ ∥ ˙x(t)∥2 X + ∥∇f(x(t)) − ∇f(ξ)∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since (x, λ) remains bounded on [0, +∞), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3(i), and owing to the fact that ∇f is Lipschitz continuous on bounded sets, there further exists γ ≥ 0 such that β∥λ(t) − η∥2 Y /2 ≤ ∥ ˙x(t)∥2 X + γ2∥x(t) − ξ∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of the above derivations, we obtain d dt �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y �/2 + α �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � ≤ 2α∥ ˙x(t)∥2 X/β + 2αγ2∥x(t) − ξ∥2 X/β, which, by applying (9) again, reads (β/2 + γ2) d dt �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � + αβ �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � ≤ 2α∥ ˙x(t)∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Integration over [0, t] yields (β/2 + γ2) �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � + αβ � t 0 ∥x(τ) − ξ∥2 X + ∥λ(τ) − η∥2 Y dτ ≤ 2α � t 0 ∥ ˙x(τ)∥2 X dτ + (β/2 + γ2) �∥x(0) − ξ∥2 X + ∥λ(0) − η∥2 Y �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Combining the above inequality with (6) gives ∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y + (β/2 + γ2) �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � + αβ � t 0 ∥x(τ) − ξ∥2 X + ∥λ(τ) − η∥2 Y dτ ≤ C, where C = ∥ ˙x(0)∥2 X + ∥ ˙λ(0)∥2 Y + (β/2 + γ2) �∥x(0) − ξ∥2 X + ∥λ(0) − η∥2 Y �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Taking into account that ∥( ˙x(t), ˙λ(t))∥2 ≥ 0 and ∥(x(t), λ(t)) − (ξ, η)∥2 ≥ 0, and subsequently passing to the limit as t → +∞ yields � ∞ 0 ∥x(τ) − ξ∥2 X + ∥λ(τ) − η∥2 Y dτ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since t �→ ∥(x(t), λ(t))−(ξ, η)∥2 is non-increasing on [0, +∞), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3(i), we have for any t ≥ 0, � t t/2 ∥x(τ) − ξ∥2 X + ∥λ(τ) − η∥2 Y dτ ≥ t �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y �/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 14 Noticing that ∥(x, λ) − (ξ, η)∥2 belongs to L1([0, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' R), we classically deduce lim t→+∞ t �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, recall from (8) that for any t ≥ 0, L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 ≤ ∥(x(t), λ(t)) − (ξ, η)∥∥( ˙x(t), ˙λ(t))∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Multiplying the above inequality by t and using limt→+∞ √ t∥( ˙x(t), ˙λ(t))∥ = 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, together with limt→+∞ √ t∥(x(t), λ(t)) − (ξ, η)∥ = 0, we infer lim t→+∞ t �L(x(t), η) − L(ξ, λ(t)) + α∥x(t) − ξ∥2 X/2 � = 0, concluding the desired estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Exponential decay rate estimates In this section, we provide decay rate estimates of exponential type on the solutions of (AH) under the additional assumption that f is twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In particular, we presuppose that (A6) f : X → R satisfies condition (C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', 2Df(y, x) − ⟨∇2f(x)(x − y), x − y⟩X ≥ 0, ∀x, y ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A7) ∇2f( · ) : X → X is γ-bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃γ > 0 ∀x, y ∈ X ⟨∇2f(x)y, y⟩X ≤ γ∥y∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Here, Df denotes again the Bregman distance associated with f, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Bregman [21], and ∇2f refers to the Hessian of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We remark that condition (C) is verified whenever f is minorized by its second-order Taylor approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ‘Primal exponential estimates’ Let us first establish exponential decay rate estimates on the solutions of (AH) in the case when A : X → Y is bounded from below, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃β > 0 ∀x ∈ X ∥Ax∥2 Y ≥ β∥x∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇2f( · ) : X → X be α-elliptic and γ-bounded, and suppose that A : X → Y is bounded from below with constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let f : X → R satisfy condition (C) and set ρ = � α/2, if γ2 ≤ 4β, min{α, γ − � γ2 − 4β}/2, if γ2 > 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, the following assertions hold: 15 (i) If ρ2 − γρ + β > 0, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥2 X = O �e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) If ρ2 − γρ + β = 0, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �t2e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �t2e−2ρt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥2 X = O �t2e−2ρt� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) ∈ S × M and let ρ > 0 to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using again the ‘Lagrangian identity’ (1) together with (AH), we have for any t ≥ 0, d dt �∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y �/2 + ⟨∇2f(x(t)) ˙x(t), ˙x(t)⟩X + ρ d dt �L(ξ, η) − L(x(t), λ(t)) � + ρ �∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y � = 2ρ∥ ˙x(t)∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, from equation (7), we obtain d dt �L(ξ, η) − L(x(t), λ(t)) � = d dt⟨x(t) − ξ, ˙x(x)⟩X + ⟨∇2f(x(t)) ˙x(t), x(t) − ξ⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Combining the above expressions yields d dt �∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X + ∥ ˙λ(t)∥2 Y �/2 + ⟨(∇2f(x(t)) − 2ρ Id) ˙x(t), ˙x(t) + ρ(x(t) − ξ)⟩X + ρ �∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Developing the term in the second line gives d dt �∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 Y �/2 + ρ �∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 Y � + ⟨(∇2f(x(t)) − 2ρ Id)( ˙x(t) + ρ(x(t) − ξ)), ˙x(t) + ρ(x(t) − ξ)⟩X + ρ2�2Df(ξ, x(t)) − ⟨∇2f(x(t))(x(t) − ξ), x(t) − ξ⟩X � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since ∇2f( · ) is α-elliptic and f satisfies condition (C), we obtain d dt �∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 Y �/2 + ρ �∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 Y � + (α − 2ρ)∥ ˙x(t) + ρ(x(t) − ξ)∥2 X ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 16 An immediate integration over [0, t] shows that there exists C ≥ 0 such that ∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ρ2∥x(t) − ξ∥2 X − 2ρDf(ξ, x(t)) + ∥ ˙λ(t)∥2 Y + 2(α − 2ρ) � t 0 e−2ρ(t−τ)∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 X dτ ≤ Ce−2ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (10) Using that ∇2f( · ) is γ-bounded and that A is bounded from below with constant β, we have both Df(ξ, x(t)) ≤ γ∥x(t) − ξ∥2 X/2 and ∥A(x(t) − ξ)∥2 Y ≥ β∥x(t) − ξ∥2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of (AH) and Aξ − b = 0Y , we infer ∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + (ρ2 − γρ + β)∥x(t) − ξ∥2 X + 2(α − 2ρ) � t 0 e−2ρ(t−τ)∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 X dτ ≤ Ce−2ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (11) Let us now determine the largest value for ρ ∈ (0, α/2] such that ρ2 − γρ + β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Clearly, if γ2 ≤ 4β, then ρ2 − γρ + β ≥ 0 holds for any ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' On the other hand, if γ2 > 4β, then ρ2−γρ+β ≥ 0 is attained whenever ρ ≤ γ/2− � γ2 − 4β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consequently, we may take ρ = � α/2, if γ2 ≤ 4β, min{α, γ − � γ2 − 4β}/2, if γ2 > 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We have either one of the following cases: (i) Suppose that ρ2 − γρ + β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, we deduce from (11) that e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 X ≤ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' e2ρt∥x(t) − ξ∥2 X ≤ C ρ2 − γρ + β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Passing to the upper limit as t → +∞ yields lim sup t→+∞ e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 X < +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' lim sup t→+∞ e2ρt∥x(t) − ξ∥2 X < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, in view of the basic inequality ∥ ˙x(t)∥2 X ≤ 2∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + 2ρ2∥x(t) − ξ∥2 X and the fact that ∥ ˙λ(t)∥2 Y = ∥A(x(t) − ξ)∥2 Y ≤ ∥A∥2∥x(t) − ξ∥2 X, we obtain lim sup t→+∞ e2ρt�∥ ˙x(t)∥2 X + ∥ ˙λ(t)∥2 Y � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The remaining estimate is now readily deduced as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 17 (ii) Suppose now that ρ2 − γρ + β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, we observe from (11) that e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 X ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Passing to the upper limit as t → +∞ entails lim sup t→+∞ e2ρt∥ ˙x(t) + ρ(x(t) − ξ)∥2 X < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, given the fact that eρt∥x(t) − ξ∥X ≤ ∥x(0) − ξ∥X + � t 0 eρτ∥ ˙x(τ) + ρ(x(τ) − ξ)∥X dτ, we deduce eρt∥x(t) − ξ∥X ≤ √ Ct + ∥x(0) − ξ∥X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Taking the square and multiplying the resulting inequality by t−2 yields t−2e2ρt∥x(t) − ξ∥2 X ≤ C + 2 √ C∥x(0) − ξ∥Xt−1 + ∥x(0) − ξ∥2 Xt−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This majorization being valid for any t > 0, we conclude lim sup t→+∞ t−2e2ρt∥x(t) − ξ∥2 X < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The remaining estimates now follow at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The previous result complements the exponential decay rate estimates obtained by Polyak [19] based on spectral arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We further note that the above decay rate es- timates are comparable to the spectral bounds known for ‘saddle matrices’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' the sur- vey paper by Benzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [31, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Assuming, moreover, that ∇2f( · ) = α Id, we have the following refined exponential decay rate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇2f( · ) = α Id and suppose that A : X → Y is bounded from below with constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, η) ∈ S × M, the following assertions hold: (i) If α2 < 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥2 X = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) If α2 = 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥2 X = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 18 (iii) If α2 > 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �e−(α−δ)t� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �e−(α−δ)t� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥x(t) − ξ∥2 X = O �e−(α−δ)t� as t → +∞, where δ = � α2 − 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (i)–(ii) This is an immediate consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1(i)–(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (iii) Suppose that α2 > 4β and let ρ = (α + δ)/2, where δ = � α2 − 4β, so that ρ2 − αρ + β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' From (11) and the fact that ∇2f( · ) = α Id, we observe that there exists C ≥ 0 such that for any t ≥ 0, e(α+δ)t∥ ˙x(t) + ρ(x(t) − ξ)∥2 X ≤ C + 2δ � t 0 e(α+δ)τ∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 X dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Applying Gronwall’s inequality yields e(α+δ)t∥ ˙x(t) + ρ(x(t) − ξ)∥2 X ≤ Ce2δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using this inequality together with the fact that e(α+δ)t/2∥x(t) − ξ∥X ≤ ∥x(0) − ξ∥X + � t 0 e(α+δ)τ/2∥ ˙x(τ) + ρ(x(τ) − ξ)∥X dτ, we obtain e(α+δ)t/2∥x(t) − ξ∥X ≤ √ C δ eδt + ∥x(0) − ξ∥X − √ C δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consequently, e(α−δ)t∥x(t) − ξ∥2 X ≤ C δ2 + 2 √ C δ � ∥x(0) − ξ∥X − √ C δ � e−δt + � ∥x(0) − ξ∥X − √ C δ �2 e−2δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Passing to the upper limit as t → +∞ yields the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The remaining as- sertions are now easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The previous result essentially recovers the optimal decay rate estimates known for the classical damped harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Indeed, in the case when ∇2f( · ) = α Id, we observe in view of an immediate differentiation that the solutions of (AH) fur- ther obey the second-order dynamics ¨x + α ˙x + ∇∥A(x − ¯x)∥2 Y /2 = 0X, where ¯x denotes the unique element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The above second-order differential system was first introduced, from a more general optimization perspective, by Polyak [32] and is known to inherit remarkable minimizing properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', Alvarez [33] and Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [34] for a general exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ‘Dual exponential estimates’ Let us now complement the previous discussion with decay rate estimates on the solu- tions of (AH) under the assumption that A∗ : Y → X is bounded from below, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃β > 0 ∀y ∈ Y ∥A∗y∥2 X ≥ β∥y∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇2f( · ) = α Id and suppose that A∗ : Y → X is bounded from below with constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for (ξ, η) ∈ S × M, the following assertions hold: (i) If α2 < 4β, then it holds that ∥λ(t) − η∥2 Y = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥ ˙λ(t)∥2 Y = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) If α2 = 4β, then it holds that ∥λ(t) − η∥2 Y = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥ ˙λ(t)∥2 Y = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (iii) If α2 > 4β, then it holds that ∥λ(t) − η∥2 Y = O �e−(α−δ)t� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥ ˙λ(t)∥2 Y = O �e−(α−δ)t� as t → +∞, where δ = � α2 − 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) be the unique element in S × M and let ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using similar deriva- tions as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1, we have for any t ≥ 0 d dt �∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + ρ2∥λ(t) − η∥2 Y + ∥A∗(λ(t) − η)∥2 X �/2 + ρ �∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + ρ2∥λ(t) − η∥2 Y + ∥A∗(λ(t) − η)∥2 X � + ⟨A(∇f(x(t)) − ∇f(ξ)), ˙λ(t) + ρ(λ(t) − η)⟩Y − 2ρ∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' From ∇2f( · ) = α Id together with (AH) and Aξ − b = 0Y , we obtain d dt �∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + (ρ2 − αρ)∥λ(t) − η∥2 Y + ∥A∗(λ(t) − η)∥2 X �/2 + ρ �∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + (ρ2 − αρ)∥λ(t) − η∥2 Y + ∥A∗(λ(t) − η)∥2 X � + (α − 2ρ)∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' An immediate integration over [0, t] shows that there exists C ≥ 0 such that ∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + (ρ2 − αρ)∥λ(t) − η∥2 Y + ∥A∗(λ(t) − η)∥2 X + 2(α − 2ρ) � t 0 ∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 Y dτ = Ce−2ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (12) 20 Since A∗ is bounded from below with constant β, we infer ∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + (ρ2 − αρ + β)∥λ(t) − η∥2 Y + 2(α − 2ρ) � t 0 ∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 Y dτ ≤ Ce−2ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The desired estimates are now readily deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The above result again retrieves the well-known decay rate estimates for the classical damped harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' As in the previous case (and, in fact, dual to our observation in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3), we note that whenever ∇2f( · ) = α Id, the solutions of (AH) further obey the second-order dynamics ¨λ + α ˙λ + ∇∥A∗(λ − ¯λ)∥2 X/2 = 0Y with ¯λ denoting the unique element in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In order to obtain asymptotic estimates on the primal-dual gap function in the case when A∗ is bounded from below, we utilize the following relation between the primal and dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇2f( · ) = α Id, let A∗ : Y → X be bounded from below with constant β, and let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for (ξ, η) ∈ S × M, there exists C ≥ 0 such that for any t ≥ 0, β∥(x(t), λ(t)) − (ξ, η)∥2 − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X ≤ Ce−2αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) be the unique element in S × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Using (AH) together with the fact that T(ξ, η) = (0X, 0Y ), we have for any t ≥ 0, d dt �β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X �/2 + β⟨∇f(x(t)) − ∇f(ξ), x(t) − ξ⟩X − ⟨∇f(x(t)) − ∇f(ξ), A∗A(x(t) − ξ)⟩X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of ∇2f( · ) = α Id, the above equality reads d dt �β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X �/2 + α �β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X � + α �∥A∗(λ(t) − η)∥2 X − β∥λ(t) − η∥2 Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since A∗ is bounded from below with constant β, we obtain d dt �β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X �/2 + α �β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' An immediate integration over [0, t] then shows that there exists C ≥ 0 such that β∥x(t) − ξ∥2 X + β∥λ(t) − η∥2 Y − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X ≤ Ce−2αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 21 Combining the above results finally gives the following asymptotic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇2f( · ) = α Id and suppose that A∗ : Y → X is bounded from below with constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (x, λ) : [0, +∞) → X × Y be a solution of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for (ξ, η) ∈ S × M, the following assertions hold: (i) If α2 < 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥(x(t), λ(t)) − (ξ, η)∥2 = O �e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) If α2 = 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥(x(t), λ(t)) − (ξ, η)∥2 = O �t2e−αt� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (iii) If α2 > 4β, then it holds that L(x(t), η) − L(ξ, λ(t)) = O �e−(α−δ)t� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙λ(t))∥2 = O �e−(α−δ)t� as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥(x(t), λ(t)) − (ξ, η)∥2 = O �e−(α−δ)t� as t → +∞, where δ = � α2 − 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let (ξ, η) be the unique element in S × M and let ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since ∇2f( · ) = α Id, by combining (10) with (12) and subsequently using (AH) together with the fact that Aξ − b = 0Y , there exists C ≥ 0 such that for any t ≥ 0, ∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + ∥A(x(t) − ξ)∥2 Y + (ρ2 − αρ) �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � + ∥A∗(λ(t) − η)∥2 X + 2(α − 2ρ) � t 0 e−2ρ(t−τ)κ(τ) dτ ≤ Ce−2ρt, where κ(τ) = ∥ ˙x(τ) + ρ(x(τ) − ξ)∥2 X + ∥ ˙λ(τ) + ρ(λ(τ) − η)∥2 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Since A∗ is bounded from below with constant β, we observe from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6 that there exists ˜C ≥ 0 such that β �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � − ∥A(x(t) − ξ)∥2 Y − ∥A∗(λ(t) − η)∥2 X ≤ ˜Ce−2αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of the above inequalities, we obtain ∥ ˙x(t) + ρ(x(t) − ξ)∥2 X + ∥ ˙λ(t) + ρ(λ(t) − η)∥2 Y + (ρ2 − αρ + β) �∥x(t) − ξ∥2 X + ∥λ(t) − η∥2 Y � + 2(α − 2ρ) � t 0 e−2ρ(t−τ)κ(τ) dτ ≤ Ce−2ρt + ˜Ce−2αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The desired estimates are now easily derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Structured convex minimization In this section, we aim to extend some of our previous results on the Arrow–Hurwicz differential system (AH) to the more general case of solving structured convex mini- mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let X, Y and Z be real Hilbert spaces, and let X × Y × Z be endowed with the product structure ⟨ · , · ⟩ = ⟨ · , · ⟩X +⟨ · , · ⟩Y +⟨ · , · ⟩Z and associated norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consider the structured minimization problem inf {f(x) + g(y) | Ax + By − c = 0Z} (SP) and suppose that the following assumptions are verified: (A1)′ f : X → R and g : Y → R are convex and continuously differentiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A2)′ ∇f : X → X and ∇g : Y → Y are Lipschitz continuous on bounded sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (A3)′ A : X → Z and B : Y → Z are linear and continuous, and c ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We associate with (SP) the Lagrangian L : X × Y × Z −→ R (x, y, λ) �−→ f(x) + g(y) + ⟨λ, Ax + By − c⟩Z, which, given the above assumptions, is a convex function with respect to (x, y) ∈ X ×Y and a concave (in fact, affine) function with respect to λ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We denote by S×U ×M ⊂ X × Y × Z the (possibly empty) set of saddle points of the Lagrangian L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Consider now the generalized Arrow–Hurwicz differential system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ˙x + ∇f(x) + A∗λ = 0X ˙y + ∇g(y) + B∗λ = 0Y ˙λ + c − Ax − By = 0Z (GAH) with initial data (x0, y0, λ0) ∈ X × Y × Z and observe that (GAH) admits, for any ini- tial data, a unique (classical) solution (x, y, λ) : [0, +∞) → X × Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Recall that the zeros of the maximally monotone operator T : X × Y × Z −→ X × Y × Z (x, y, λ) �−→ (∇f(x) + A∗λ, ∇g(y) + B∗λ, c − Ax − By) are nothing but the saddle points of the Lagrangian L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The following discussion suggests that our results on the (AH) differential system directly convey to the (GAH) evolution system for solving the structured convex mini- mization problem (SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let S × U × M be non-empty and let (x, y, λ) : [0, +∞) → X × Y × Z be a solution of (GAH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, ψ, η) ∈ S × U × M, (i) limt→+∞∥(x(t), y(t), λ(t)) − (ξ, ψ, η)∥ exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (ii) limt→+∞∥( ˙x(t), ˙y(t), ˙λ(t))∥ exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' (iii) it holds that � ∞ 0 L(x(τ), y(τ), η) − L(ξ, ψ, λ(τ)) dτ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 23 Let the Cesàro average (σ, τ, ω) : (0, +∞) → X × Y × Z of a solution of (GAH) be defined by (σ(t), τ(t), ω(t)) = 1 t � t 0 (x(τ), y(τ), λ(τ)) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We have the following asymptotic estimate on the primal-dual gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let S×U ×M be non-empty and let (σ, τ, ω) : (0, +∞) → X ×Y ×Z be the Cesàro average of a solution of (GAH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, ψ, η) ∈ S × U × M, it holds that L(σ(t), τ(t), η) − L(ξ, ψ, ω(t)) = O �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, there exists (¯x, ¯y, ¯λ) ∈ S × U × M such that (σ(t), τ(t), ω(t)) ⇀ (¯x, ¯y, ¯λ) weakly in X × Y × Z as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let us now investigate the asymptotic properties of the solutions of (GAH) under the more stringent assumption that f + g : X × Y −→ R (x, y) �−→ f(x) + g(y) is α-strongly convex (or, equivalently, ∇(f + g) : X × Y → X × Y is α-strongly mo- notone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, the following asymptotic properties are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let ∇(f +g) : X ×Y → X ×Y be α-strongly monotone, let S ×U ×M be non-empty, and let (x, y, λ) : [0, +∞) → X × Y × Z be a solution of (GAH) with initial data (x0, y0, λ0) ∈ X × Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for any (ξ, ψ, η) ∈ S × U × M, it holds that L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = o � 1 √ t � as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥( ˙x(t), ˙y(t), ˙λ(t))∥ = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, (x(t), y(t), λ(t)) converges weakly, as t → +∞, to projS×U×M(x0, y0, λ0) ∈ S × U × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Assuming, moreover, that (A B)∗ : Z → X × Y is bounded from below, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=', ∃β > 0 ∀z ∈ Z ∥(A B)∗z∥2 X×Y ≥ β∥z∥2 Z, we have the following refined asymptotic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Under the hypotheses of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='3, let (A B)∗ : Z → X × Y be bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Then, for (ξ, ψ, η) ∈ S × U × M, it holds that L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = o �1 t � as t → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' ∥(x(t), y(t), λ(t)) − (ξ, ψ, η)∥ = o � 1 √ t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 24 Consequently, (x(t), y(t), λ(t)) converges strongly, as t → +∞, to the unique element in S × U × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The structured convex minimization problem (SP) has recently been ap- proached by Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [35] and Boţ and Nguyen [36] using the second-order non- autonomous differential system \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¨x + ν t ˙x + ∇xLµ(x, y, λ + θt ˙λ) = 0X ¨y + ν t ˙y + ∇yLµ(x, y, λ + θt ˙λ) = 0Y ¨λ + ν t ˙λ − ∇λLµ(x + θt ˙x, y + θt ˙y, λ) = 0Z (AAH) with ν ≥ 3, µ ≥ 0, θ ∈ [1/(ν − 1), 1/2], and initial data (x0, y0, λ0), (v0, w0, µ0) ∈ X ×Y ×Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' As a decisive feature, the (AAH) dynamics are governed by ‘asymptotically vanishing damping coefficients’ which relate the above system to Nesterov’s accelerated gradient method (see Nesterov [37], Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [38]), and additional ‘exploration terms’ within the partial gradients of the augmented Lagrangian Lµ associated with (SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This particular structure allows for remarkably fast mini-maximizing properties with respect to the Lagrangian L given the sole convexity hypothesis on the objective function of (SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In particular, the (classical) solutions (x, y, λ) : [t0, +∞) → X × Y × Z of (AAH) with t0 > 0 evolve, for any (ξ, ψ, η) ∈ S × U × M, according to the asymptotic estimate (see Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [35], Boţ and Nguyen [36]) L(x(t), y(t), η) − L(ξ, ψ, λ(t)) = O � 1 t2 � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' If, in addition, ν > 3 and θ ∈ (1/(ν − 1), 1/2], then the solutions of (AAH) further re- main bounded on [t0, +∞) and it holds that ∥( ˙x(t), ˙y(t), ˙λ(t))∥ = O �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Assuming, moreover, that f +g : X ×Y → R is strongly convex, then, for any (ξ, ψ, η) ∈ S × U × M, the following estimate is verified (see Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [35]): ∥(x(t), y(t)) − (ξ, ψ)∥X×Y = O �1 t � as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The latter may be particularized to an exponential estimate by further introducing tem- poral scaling factors in (AAH);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' [35, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In view of the above discussion, we observe that the second-order differential system (AAH) clearly outperforms the first-order differential system (AH) in the case of a con- vex objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' However, this may not be the case, as we shall see next, whenever the objective function is strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Numerical experiments In this section, we perform numerical experiments on the Arrow–Hurwicz differential system (AH) to support our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In particular, we consider two simple 25 0 5 10 15 20 25 30 35 40 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 0 2 4 6 8 10 12 14 16 18 20 10-4 10-3 10-2 10-1 100 101 0 5 10 15 20 25 30 35 40 10-12 10-10 10-8 10-6 10-4 10-2 100 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Graphical view on the evolution of the primal-dual gap function L(x(t), η)−L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, the squared error ∥x(t) − ξ∥2 X, and the trajectories of the solution components x(t) = (x1(t), x2(t)) of (AH) and (AAH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' but representative (strongly) convex minimization problems in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let X, Y = R2 and consider the quadratic function f : R2 → R defined by f(x1, x2) = (x2 1 − x1x2 + x2 2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Clearly, f is α-strongly convex with α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, ∇2f( · ) is γ-bounded with γ = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Further, let A(x1, x2) = (x1, x2), b = (1, 1), and observe that A is bounded from below with constant β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The unique minimizer of f subject to the linear constraints corresponds to ξ = (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' with associated Lagrange multiplier η = (−1/2, −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The evolution of the primal-dual gap function L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, the squared error ∥x(t) − ξ∥2 X, and the trajectory of the solution component x(t) = (x1(t), x2(t)) of (AH) with initial data x0 = (−1, 1) and λ0 = (1, 1) is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' For comparison, the corresponding quantities of the (AAH) differential system are displayed with damping parameter ν = 3, exploration coefficient θ = 1/2, and augmentation parameter µ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The initial data of the (AAH) differential system is set accordingly to x0 = (−1, 1), λ0 = (1, 1), v0 = (0, 0), and µ0 = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Analyzing Figure 1, we observe that the solutions (x(t), λ(t)) of (AH) converge, as t → +∞, towards the unique mini-maximizer (ξ, η) of the convex minimization problem (P) and its associated Lagrange dual (D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Moreover, according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='1(i), we find that the primal-dual gap function L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2 and the squared error ∥x(t) − ξ∥2 X obey the exponential estimate O �e−αt� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Compared to the (AAH) dynamics for which the quantity ∥( ˙x(t), ˙λ(t))∥2 evolves according to the estimate O �1/t2� as t → +∞ (even though the damping parameter is chosen to be ν = 3), we find that the solutions of (AH) indeed 26 0 2 4 6 8 10 12 14 16 18 20 10-12 10-10 10-8 10-6 10-4 10-2 100 0 2 4 6 8 10 12 14 16 18 20 10-10 10-8 10-6 10-4 10-2 100 0 2 4 6 8 10 12 14 16 18 20 10-10 10-8 10-6 10-4 10-2 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='5 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='8 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Exponential decay properties of the primal-dual gap function L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, and the squared error ∥(x(t), λ(t)) − (ξ, η)∥2 of the solutions (x(t), λ(t)) of (AH) for distinct values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' admit a faster and less oscillatory decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' It is interesting to note that, in this example, the primal-dual gap function L(x(t), η) − L(ξ, λ(t)) and the squared error ∥x(t) − ξ∥2 X for (AAH) appear to obey the estimate O �1/t4� rather than O �1/t2� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Let X, Y = R2 and consider the parameterized quadratic function f : R2 → R defined by f(x1, x2) = α(x2 1 + x2 2)/2 with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Further, let A(x1, x2) = √ 2(x1 + x2)/2 and β = √ 2/2 so that A∗ is bounded from below with constant β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The unique minimizer of f subject to the linear constraints is denoted by ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' with cor- responding Lagrange multiplier η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Figure 2 illustrates the decay properties of the primal- dual gap function L(x(t), η) − L(ξ, λ(t)), the squared velocity ∥( ˙x(t), ˙λ(t))∥2, and the squared error ∥(x(t), λ(t))−(ξ, η)∥2 of the solutions (x(t), λ(t)) of (AH) for the distinct values α = 1, α = 2, and α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' The initial data is set to x0 = (−1, 1) and λ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Figure 2 suggests that the solutions (x(t), λ(t)) of (AH) converge, as t → +∞, at an exponential rate towards the unique mini-maximizer (ξ, η) of the convex minimization problem (P) and its associated Lagrange dual (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Indeed, the decay properties of the solutions of (AH) may be categorized as predicted by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content='7: In case (i), we have α2 < 4β with the rate estimate O �e−αt� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' We refer to this case as the ‘under-damped case’ as the solutions of (AH) admit a significant oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In case (ii), we have α2 = 4β with the rate estimate O �t2e−αt� as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' This case refers to the ‘critically-damped case’ for which we observe the fastest possible convergence of the solutions of (AH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Finally, in case (iii), we have α2 > 4β with the rate estimate O �e−(α−δ)t� as t → +∞, where δ = � α2 − 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' In this case, referred to as the ‘over-damped case’, the decay of the solutions of (AH) is considerably degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' 27 Acknowledgment The author expresses his gratitude to the two anonymous reviewers whose comments and suggestions led to a significant improvement of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Disclosure statement No potential conflict of interest was reported by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Funding Research supported by the German Research Foundation (DFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' References [1] Ekeland I, Témam R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Convex analysis and variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfXPcY/content/2301.00177v1.pdf'} +page_content=' Philadelphia: Society for Industrial and Applied Mathematics;' metadata={'source': 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We study tail risk dynamics in high-frequency financial markets and their connection with +trading activity and market uncertainty. We introduce a dynamic extreme value regression model ac- +commodating both stationary and local unit-root predictors to appropriately capture the time-varying +behaviour of the distribution of high-frequency extreme losses. To characterize trading activity and mar- +ket uncertainty, we consider several volatility and liquidity predictors, and propose a two-step adaptive +L1-regularized maximum likelihood estimator to select the most appropriate ones. We establish the oracle +property of the proposed estimator for selecting both stationary and local unit-root predictors, and show +its good finite sample properties in an extensive simulation study. Studying the high-frequency extreme +losses of nine large liquid U.S. stocks using 42 liquidity and volatility predictors, we find the severity of +extreme losses to be well predicted by low levels of price impact in period of high volatility of liquidity and +volatility. +Keywords: high-frequency financial data · peaks-over-threshold (POT) · time-varying generalized Pareto +distribution· L1-regularized maximum likelihood estimation · nonstationary variable selection +1 +Introduction +Measuring tail risk at high-frequency has become of utmost importance to market players and regulators (Weller, +2017). While much efforts have been devoted to the measurement of tail risk at low-frequency (Nieto and Ruiz, +2016), few attempts have been made to measure risk at high-frequency, see Giot (2005), Dionne et al. (2009) +and Chavez-Demoulin and Davison (2012). Moreover, although these models can be very accurate, they explain +the tail risk evolution in a “reduced form” manner, i.e., using autoregressive terms exploiting the persistence +of the time series. They thus fail to provide a deeper structural understanding of the factors driving tail risk. +As much as understanding the macroeconomic determinants of tail risk is a relevant problem at low-frequency +(Massacci, 2017), it is important to understand how market uncertainty and trading activity impacts tail risk +at high-frequency. +From a market microstructure perspective, though the intensification of high-frequency trading has improved +trading costs and liquidity (Hendershott et al., 2011), it is also suspected to be responsible for more frequent +extreme price movements over short periods of time (Brogaard et al., 2018). Such extreme fluctuations are +often the result of an aggressive directional market making activity initiated when the market is already under +stress. Brogaard et al. (2018) find that market wide extreme shocks are likely to trigger the risk controls of +high-frequency liquidity providers that thus withdraw from the market to reduce their risk exposure. Similarly, +Kirilenko et al. (2017) find that during the market turbulence induced by the 2010 Flash Crash, many high- +frequency liquidity providers withdrew from the market, thus exacerbating the price fall. Studying how market +uncertainty and trading activity affect extreme losses can thus provide a deeper understanding of the evolution +of tail risk at high-frequency, and this paper proposes appropriate econometric techniques to do so. +We consider a dynamic extreme value regression framework (Chavez-Demoulin et al., 2016; Massacci, 2017; +Schwaab et al., 2021) where the distribution of extreme losses is assumed to be well approximated by a general- +ized Pareto distribution (GPD) with time-varying parameters driven by exogenous preditors and autoregressive +⋆ Corresponding author: ir.li.sun@gmail.com +arXiv:2301.01362v1 [econ.EM] 3 Jan 2023 + +terms. To assess the impact of market uncertainty and trading activity on extreme losses, we consider several +volatility predictors, proxing for market uncertainty, and liquidity predictors, characterizing trading activity. +Despite extreme value regression techniques have been widely applied in finance (Chavez-Demoulin et al., 2016; +Hambuckers et al., 2018; Bee et al., 2019), our investigation presents new challenges: (i) as the financial litera- +ture proposes several volatility and liquidity measures, we face a variable selection problem aimed at identifying +predictors capturing the most relevant aspects of trading activity affecting extremes as well as improving the +predictive accuracy of tail risk; (ii) volatility and liquidity measures observed at high-frequency exhibit strong +persistence and seasonalities, thus violating the classical stationary assumptions required for inference with the +maximum likelihood estimator (MLE). To overcome these issues, we develop a two-step adaptive L1-regularized +maximum likelihood estimator (ALMLE) that allows performing variable selection with both stationary and +local unit-root predictors (Lee et al., 2022), and establish its oracle property. +We investigate the impact of 42 liquidity and volatility indicators on the distribution of high-frequency +extreme losses of nine large liquid U.S. stocks observed from 2006 to 2014. We find that the severity of tail +risk, as measured by the shape parameter of the GPD, is well predicted by low price impact (Goyenko et al., +2009) during periods of high volatility of volatility and high volatility of liquidity. This finding is coherent with +the evidence in Brogaard et al. (2018) that market markers liquidity supply is outstripped by liquidity demand +after large uncertainty shocks, and their rush to leave the market to lower their risk exposures amplify extreme +price movements. Our two-step ALMLE is necessary to reveal this pattern as the standard MLE finds almost +all predictors to be significant. To validate our estimating strategy, we provide an out-of-sample VaR forecast +analysis and find that the estimated model performs well in the out-of-sample. +The remainder of the paper is organized as follows: Section 2 presents the time-varying GPD model accom- +modating stationary and local unit-root predictors as well as autoregressive components; Section 3 presents the +MLE and shows its asymptotic non-normality when local unit-root predictors are included in the model; Section 4 +introduces the two-step ALMLE and prove the oracle property of this estimator in selecting both stationary +and local unit-root predictors; Section 5 provides an extensive simulation study comparing the performance of +the two-step ALMLE to those of the MLE, showing the superiority of the former in finite samples. Section 6 +discusses the results of the empirical study whereas Section 7 concludes. Additional results and mathematical +proofs are relegated to the Appendix. +2 +Extreme value regression +We denote the logarithmic loss and return time series of a financial asset by {lt}T +t=1 and {rt}T +t=1, respectively, +with lt = −rt, and denote zt a vector of exogenous predictors observed at time t. +Assumption M.1. {lt}T +t=1 and {zt}T +t=1 are on a complete probability space (Ω, F, P). At each time t ∈ +{1, 2, . . . , T}, we have an information set Ft−1 available which is the σ-algebra generated by {zt−1, lt−1, zt−2, +lt−2, . . .}. +Let assume {lt}T +t=1 is independent and identically distributed (i.i.d.) with a cumulative distribution function +(c.d.f.) F(·). Probabilistic results from extreme value theory show that if there exist real sequences aT > 0 +and βT such that limT →∞ F T (aT x + βT ) converges to a non-degenerate distribution G(·), then F(·) belongs +to the max-domain of attraction of G(·), i.e. F ∈ D(G), and G(·) must be the generalized extreme value (GEV) +distribution (see Theorem 3.1.1. of Coles (2001)). +Let {yt}T +t=1 be a censored sequence of excess losses above a high threshold u, such that the excess loss +yt = lt − u, if lt > u, and yt = 0 otherwise. Define the conditional distribution of excess losses, +F|u(y) := P {lt − u ≤ y|lt > u} = P {yt ≤ y|yt > 0} , +0 < yt ≤ LF − u, +2 + +with LF := sup{x : F(x) < 1} the right end point of F(·). Pickands (1975) and Balkema and De Haan (1974) +show that if F(·) ∈ D(G) then the limiting distribution of F|u(y) is a GPD, i.e. +lim +u→+∞ +sup +0 0, +GPD(y; k, σ) = 1 − +� +1 + k y +σ +�−1/k +. +(2) +Eq. (1) suggests that F|u(y) with u large enough can be approximated by a GPD(·; k, σu), where the scale +parameter σu depends on u. The peaks-over-threshold (POT) approach assumes this relationship holds exactly +above a fixed threshold u and uses the exceedances of such threshold to estimate the GPD parameters σ and k +(see section 4.3 of Coles (2001)). +2.1 +Time-varying peaks-over-threshold (POT) approach +The classical POT approach assumes that {lt} is i.i.d. However, financial data typically exhibit dependence +features such as time-varying heteroscedasticity and extremal clustering that violate this assumption. To capture +these aspects, we adopt a dynamic POT approach. Let {yt}T +t=1 be a censored sequence of excess losses over a +threshold time series {ut}T +t=1, we model the excess loss distribution conditional on the information set Ft−1, +Ft|ut(y) := P{yt ≤ y|yt > 0, Ft−1}, using a GPD with time-varying parameters kt and σt. See, e.g., Chavez- +Demoulin et al. (2014); Massacci (2017); Bee et al. (2019). +Consider the vector-valued time series of p ∈ N explanatory variables {zt := [z1,t, . . . , zp,t]′}T +t=1. Given the +information set Ft−1, we consider the following specification for {(kt, σt)}, +log +� +kt +0.5 − kt +� += β1,0 + +p +� +j=1 +β1,j zj,t−1, +(3) +log(σt) = β2,0 + +p +� +j=1 +β2,j zj,t−1 + β2,p+1 log(σt−1). +(4) +We impose that 0 < kt < 0.5 and σt > 0 (see Hosking and Wallis (1987)) to ensure a finite conditional variance +of yt and numerical stability in the estimation. As the scale parameter σt can be associated with the variance +of the underlying distribution Ft(·), we accommodate an autoregressive term in log(σt) in the spirit of GARCH +models (Engle, 2001). We allow for both stationary and unit-root explanatory variables in (3) and (4), such +that persistent predictors can be accommodated. +3 +Maximum likelihood estimation +Let β := [β1,0, β1,1, . . . , β1,p, β2,0, β2,1, . . . , β2,p+1]′ denote the vector of the model coefficients in (3)-(4), and +define the coefficient space Θ of β as a subspace of R2p+2 × (−1, 1) accomodating all permissible coefficient +vectors β. We present the MLE of the model coefficients in (3)-(4) and show it is consistent but asymptotically +non-normal when local unit-root explanatory variables are included in the model. +3.1 +Maximum likelihood estimator +Assumption M.2. We assume that for a given {ut}, the conditional c.d.f. Ft(·) of lt given Ft−1 exists for +t ∈ {1, 2, . . . , T} and yt := lt − ut > 0 follows a time-varying GPD, i.e. +Ft|ut(yt) = GPD(yt; kt, σt) = 1 − +� +1 + kt +yt +σt +�− 1 +kt +, +(5) +3 + +where {kt} and {σt} are specified by (3)-(4) with the true coefficient vector βo ∈ Θ ⊂ R2p+2×(−1, 1). Moreover, +{ut} returns a constant unconditional exceedance rate, i.e., P{yt > 0} = τ for all t ∈ {1, . . . , T} with a constant +τ close to zero. +Assumption M.3. Among the explanatory variables in Model (3)-(4), we assume that {zi,t, i = 1, . . . , p0} ∈ +I(0) and {zj,t, j = p0 + 1, . . . , p} ∈ I(1) with ϵj,t := zj,t − zj,t−1 and {ϵj,t, j = p0 + 1, . . . , p} ∈ I(0). We denote +by I(0) and I(1) the set of stationary and unit-root predictors, respectively. +Under Assumption M.2, the conditional probability density function (p.d.f.) of yt|{yt > 0, Ft−1} is +ft(yt) = 1 +σt +� +1 + kt +yt +σt +�− 1 +kt −1 +, +(6) +and the log-likelihood function L(·) of {yt|yt > 0, Ft−1} can be defined as (Schwaab et al., 2021), +L(β; {yt}, {zt−1}) = +T +� +t=1 +1{yt > 0} log(ft(yt)) += +T +� +t=1 +1{yt > 0} +� +− log(σt) − +� 1 +kt ++ 1 +� +log(1 + kt +yt +σt +) +� +, +(7) +where +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +kt(β) = 0.5 +� +�1 + exp +� +�−(β1,0 + +p +� +j=1 +β1,j zj,t−1) +� +� +� +� +−1 +, +σt(β) = exp +� +�β2,0 + +p +� +j=1 +β2,j zj,t−1 + β2,p+1 log(σt−1) +� +� , +(8) +for t = 1, 2, . . . , T, with 1{·} the indicator function taking value one if the input is true and zero otherwise. +We consider standardized predictors {z∗ +t } in the estimation to get stochastically bounded variables, i.e., for +each t ∈ {1, . . . , T}, we standardize zt as follows, +z∗ +t := [ z1,t, . . . , zp0,t, zp0+1,t +√ +T +, . . . , zp,t +√ +T +]′. +(9) +Replacing {zt} with {z∗ +t } into the likelihood function in (7) and maximizing we obtain +�βmle = arg max +β∈Θ +L(β; {yt}, {z∗ +t−1}), +(10) +where Θ ⊂ R2p+2 × (−1, 1). We denote the corresponding vector of true coefficients βo∗. +Remark. Assumption M.2 assumes a constant unconditional probability for the exceedance 1{yt > 0} for +t ∈ {1, . . . , T}, which is more general than assuming a constant conditional probability for 1{yt > 0|Ft−1}. This +causes us no extra burden to obtain the limiting behaviour of the MLE because 1{yt > 0} is bounded and not +a function of the model coefficients. Assumption M.3 allows for both stationary and unit-root predictors among +zt. +3.2 +Asymptotic properties of the MLE +Smith (1985) establishes the asymptotic properties of the MLE of a GPD with constant k and σ in an i.i.d. +setting. We extend Smith (1985) establishing the consistency and limiting distribution of the MLE of the dynamic +GPD with stationary and unit-root predictors in (10). In what follows, we list the assumptions required to derive +the asymptotic behaviour of the MLE, and establish the consistency and limiting distribution of �β. +4 + +Assumption M.4. We assume that, +� +� +� +� +� +� +� +βo∗ +s,i := βo +s,i = O(1), +for +i = 0, 1, . . . , p0, and s = 1, 2; +βo∗ +s,j := +√ +Tβo +s,j = O(1), +for +j = p0 + 1, . . . , p0 + p and s = 1, 2; +βo∗ +2,p+1 := βo +2,p+1 ∈ (−1, 1), +(11) +and βo∗ := [βo∗ +1,0, . . . , βo∗ +1,p, βo∗ +2,0, . . . , βo∗ +2,p+1]′ ∈ R2p+2 × (−1, 1). +Assumption M.5. {ϵt := [z1,t, . . . , zp0,t, ϵp0+1,t, . . . , ϵp,t]′}T +t=1 is assumed i.i.d. (0, Σ(0)) with mean 0 and posi- +tive definite covariance matrix Σ(0). With +� +z∗ +t := [ z1,t, . . . , zp0,t, zp0+1,t +√ +T +, . . . , zp,t +√ +T ]′� +, we assume that as T → ∞, +we have that +(1) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +√ +T +T +� +t=1 +z∗ +i,t = Op(1), +1 +T +T +� +t=1 +(z∗ +i,t)2 = Op(1), +i = 1, 2, . . . , p0; +1 +T +T +� +t=1 +z∗ +j,t = Op(1), +1 +T +T +� +t=1 +(z∗ +j,t)2 = Op(1), +j = (p0 + 1), . . . , p; +1 +T +T +� +t=1 +z∗ +t z∗′ +t +is positive definite in probability one; +and there exists a positive definite matrix Σ := [Σi,j]i,j=1,...,p such that +(2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +lim +T →∞ +1 +T +T +� +t=1 +z∗ +i,t = 0, +lim +T →∞ +1 +T +T +� +t=1 +(z∗ +i,t)2 = 1, +1 +√ +T +t0 +� +t=1 +z∗ +i,t +D∼ Wi(t0 +T ), +i = 1, 2, . . . , p0; +1 +T +T +� +t=1 +z∗ +j,t +D∼ +� 1 +0 +Σ1/2 +j,j Wj(t) dt, +1 +T +T +� +t=1 +(z∗ +j,t)2 D∼ +� 1 +0 +Σj,jW 2 +j (t) dt, +j = (p0 + 1), . . . , p; +1 +T +T +� +t=1 +z∗ +t z∗′ +t +D∼ +� 1 +0 +� +Σ1/2 Wz∗(t) +� � +Σ1/2 Wz∗(t) +�′ +dt, +1 +T +T +� +t=1 +z∗ +t z∗′ +t +is positive definite in probability one; +where 1 ≤ t0 ≤ T, and Wi(·), Wj(·) are independent Brownian motions. And denote ‘ +D∼ ’ for convergence in +distribution and Wz∗(t) := [ dW1(t) +√ +dt , . . . , dWp0(t) +√ +dt +, Wp0+1(t), . . . , Wp(t)]′. +Assumption M.6. Θ is a compact subspace of R2p+2 × (−1, 1) containing the true coefficient vector βo∗ such +that HL(·) is positive definite in Θ almost surely. +Assumption M.7. Given Assumptions M.3 and M.5, we further assume that as T → ∞, it holds that +1 +√ +T +T +� +t=1 +ψt(βo∗) +D∼ Sψ, +(12) +where Sψ is a non-degenerate distribution: +Sψ = Σ1/2 +ψ +� 1 +0 +� +������� +dWgk(t) +Wz∗(t)dWgk(t) +dWgσ(t) +Wz∗(t)dWgσ(t) +dWar(t) dWgσ(t) +� +������� +(13) +where Σψ is a positive definite (2p + 3) × (2p + 3) matrix and {Wgk(t)}, {Wgσ(t)}, {War(t)} are Brownian +motions. +5 + +Assumption M.8. +HL(βo∗;{lt},{z∗ +t }) +T +at βo∗ is assumed to weakly converge to a stochastic integral ΩH, i.e., +HL(βo∗; {lt}, {z∗ +t } +T +D∼ ΩH, +when T → ∞, +(14) +where Ω−1 +H +exists upon the limiting behaviours of z∗ +t in Assumption M.5. +Remark. Assumption M.4 imposes the orders of magnitude of βo to ensure that the unit-root explanatory +variables zp0+1,t, . . . , zp,t have coefficients of local-to-zero rate being 1 +2, see Phillips and Lee (2013) and Lee +et al. (2022). Assumption M.5 ensures that the partial sums of {z∗ +t } and {z∗ +t z∗′ +t } converge at specific rates. +Assumption M.5 is also used by Saikkonen (1993, 1995); Lee et al. (2022), and was shown to hold for time +series with a moderate degree of temporal dependence and heteroscedasticity of {ϵt}. See, e.g.,Theorem 18.2 +of Billingsley (2013), Phillips and Durlauf (1986); Phillips (1991). Assumption M.6 restricts the permissible +parameter space Θ for the ML estimation, especially maintaining the positive definiteness of HL(β) in analogy +to Assumption 9 of Smith (1985) for settling the uniqueness of the estimator. Assumption M.7 assumes the +limiting distribution of the likelihood gradient function at βo∗, see Lemma A.1.1 of Lee (2016). Assumption M.8 +assumes the existence of the limiting distribution of the likelihood Hessian matrix at βo∗, see Lemma A.1.2 of +Lee (2016). +Theorem 1 (MLE consistency). +Under Assumptions M.1, M.2, M.3, M.5(1), M.4 and M.6, and for any ϵ > 0, +lim +T →∞ P +���� �βmle − βo∗��� > ϵ +� += 0, +(15) +Proof. See Appendix A.1. +Theorem 2 (MLE asymptotics). +Under Assumptions M.1 to M.8, we have +√ +T +� +�βmle − βo∗� D∼ Ω−1 +H Sψ, +as T → ∞. +(16) +Proof. See Appendix A.1. +4 +Adaptive L1-regularized maximum likelihood estimation +Variable selection facilitates interpretation of a regression model and solves the trade-off issue between bias and +efficiency so as to achieve predictive accuracy, see James et al. (2013). Although variable selection performed +via inferential tests based on the asymptotic normality of the MLE might seem a viable solution, it is not +appropriate in our setting because of the following three issues: (i) the inability to control type I error for +multiple predictor selection; (ii) severe size distortion for selecting unit-root predictors because of the non-normal +limiting distribution; (iii) low power in selecting predictors for the shape parameter due to high standard errors +of coefficients, see simulations in Section 5. +To circumvent these issues, we adopt L1-regularized MLE for automatic variable selection (Tibshirani, 1996). +Due to the constraining nature of L1-regularization, this estimator sets some coefficients exactly to zero so as +to perform variable selection. Zou (2006) explore the advantages of using weighted L1-regularization on model +coefficients and proposed the adaptive LASSO. With proper adaptive weights, the adaptive LASSO exhibits +the oracle property, which produces an asymptotic efficient estimator of variable selection consistency as if the +true underlying model were given from the outset. Medeiros and Mendes (2016) prove the oracle property for +the adaptive LASSO in high-dimensional time series with non-Gaussian and heteroscedastic errors as well as +with highly correlated regressors. Kock (2016) show that the adaptive LASSO is oracle efficient in stationary +and non-stationary autoregressions. Lee et al. (2022) prove the oracle property of the adaptive LASSO with +6 + +stationary and local unit-root predictors, and propose a novel post-selection adaptive LASSO for selecting +mixed-root predictors i.e. stationary, local unit root, and cointegrated predictors. +Drawing on this literature, we extend the adaptive LASSO to the MLE in (10) to estimate and select sta- +tionary and local unit-root predictors in (3)-(4). A general form of adaptive L1-regularized maximum likelihood +estimator (ALMLE) can be drawn directly from Zou (2006) and is formulated as follows: +�βal = arg min +β∈Θ∗ +− L(β; {yt}, {z∗ +t }) + λk,T +p +� +i=1 +wk,i|β1,i| + λσ,T +p+1 +� +j=1 +wσ,j|β2,j|, +(17) +where L(β; {yt}, {z∗ +t }) is the log-likelihood function specified in (7); λk,T , λσ,T > 0 are tuning parameters; and +wk,i, wσ,j are adaptive weights for penalizing coefficients differently. We consider two tuning parameters instead +of one to be less restrictive on tuning parameter selection, thereby stabilizing the variable selection for both the +shape and scale models in (3)-(4). To set the tuning parameters, we start off with large enough values of λk,T and +λσ,T such that no predictors are selected by �βal, and denote these two values as λk,T,max and λσ,T,max, respec- +tively. We then search for the optimal tuning parameters using an information criterion (IC) over equally-spaced +grids of nλk and nλσ nodes1 defined on the intervals [λk,T,max, 10−6] and [λσ,T,max, 10−6]. Formally, the grids +for the shape and scale parameters are defined as Sλk,T := {exp(log(λk,T,max) − j log(λk,T,max)−log(10−6) +nλk −1 +), j = +0, 1, . . . , (nλk − 1)} and Sλσ,T := {exp(log(λσ,T,max) − j log(λσ,T,max)−log(10−6) +nλσ −1 +), j = 0, 1, . . . , (nλσ − 1)}, respec- +tively. We consider different information criteria, namely the Bayesian Information Criterion (BIC), the Han- +nan–Quinn information criterion (HQ) and the Akaike Information Criterion (AIC), and thus select the optimal +tuning parameters (�λk,T , �λσ,T ) according to the following rules, +AIC: +(�λk,T , �λσ,T ) = +arg min +λk,T ∈Sλk,T , λσ,T ∈Sλσ,T +− 2 log(L( �βal(λk,T , λσ,T ); {yt}, {z∗ +t })) + 2 +�� +i=1,...,p 1{�βal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βal +2,j ̸= 0} +� +(18) +HQ: +(�λk,T , �λσ,T ) = +arg min +λk,T ∈Sλk,T , λσ,T ∈Sλσ,T +− 2 log(L( �βal(λk,T , λσ,T ); {yt}, {z∗ +t })) + 2 log(log(T)) +�� +i=1,...,p 1{�βal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βal +2,j ̸= 0} +� +(19) +BIC: +(�λk,T , �λσ,T ) = +arg min +λk,T ∈Sλk,T , λσ,T ∈Sλσ,T +− 2 log(L( �βal(λk,T , λσ,T ); {yt}, {z∗ +t })) + log(T) +�� +i=1,...,p 1{�βal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βal +2,j ̸= 0} +� +(20) +The sequential strong rules of Tibshirani et al. (2012) is typically employed for computing LASSO-type +problems. However, when kt presents persistent dynamics the sequential strong rules for �βal fails to screen +among truly active and inactive predictors due to estimation bias when the tuning parameters are not small +enough, and, as a byproduct, favors the boundary solution kt = 0.5. To reach variable selection consistency, it is +necessary to enforce the optimizer to stay away from the boundary of the parameter space. Theorem 3 illustrates +the restriction on the permissible coefficient space Θ in order to achieve the model selection consistency of �βal, +i.e., +lim +T →∞ P +� +Aal +T = A +� += 1, +(21) +where Aal +T := Aal +k,T ∪ Aal +σ,T with Aal +k,T := +� +(1, i) : i ≥ 1, �βal +1,i ̸= 0 +� +and Aal +σ,T := +� +(2, j) : j ≥ 1, �βal +2,j ̸= 0 +� +, and +A := Ak ∪ Aσ with Ak := +� +(1, i) : i ≥ 1, βo∗ +1,i ̸= 0 +� +and Aσ := +� +(2, j) : j ≥ 1, βo∗ +2,j ̸= 0 +� +. +Theorem 3. Under the assumptions in Theorem 2, if there is no �βal(λk,T , λσ,T ) with λk,T , λσ,T ∈ O(T +1 +2 ) such +that +det +� +∂2L(β) +∂[β′ +Ak, β′ +Aσ]′∂[β′ +Ak, β′ +Aσ] +�����β= � +βal(λk,T ,λσ,T ) +� +̸= 0, +(22) +1 We use nλk = 50 and nλσ = 30 across this paper unless stated otherwise. We also have tried nλk = 100 and nλσ = 100 +to check the sufficiency of nλk = 50 and nλσ = 30, and found that differences in the results are small. +7 + +then limT →∞ P +� +Aal +T = A +� +̸= 1, where det(·) is the matrix determinant operator; wk,i and wσ,j are set using +the MLE in section 3 such that +√ +T( +1 +wk,i − β∗o +1,i) = Op(1) and +√ +T( +1 +wσ,j − β∗o +2,j) = Op(1), for i = 1, . . . , p, +j = 1, . . . , p + 1. +Proof. See Appendix A.3. +Theorem 3 shows that if not all the truly active predictors are able to enter the regression model with +λk,T , λσ,T ∈ O(T +1 +2 ), then truly inactive predictors start to be selected for compensating for the missing ones +since λk,T , λσ,T ∈ O(T +1 +2 ) and thereby fail �βal in the variable selection. The necessary condition in Theorem 3 +tends to be broken when the underlying {kt(βo∗)} involves local unit-root predictors. To solve this issue we +propose a two-step ALMLE and prove its oracle property. +4.1 +Two-Step ALMLE +From the previous discussion, we know that ALMLE can be improved if we ensure the estimation to stay away +from {kt(·) = 0.5} for every λk,T . Therefore, we propose a two-step ALMLE, denoted as �βtal, to avoid the local +minimizer issue of �βal by selecting predictors for the shape at the first step and running the ALMLE in (17) at +the second step with the selected �λk,T in the first step. Specifically, the two-step ALMLE �βtal is obtained using +the following procedure: +Step 1: Select the optimal tuning parameter �λk,T ∈ Sλk,T using an IC as follows, +AIC: +�λk,T = +arg min +λk,T ∈Sλk,T +− 2 log(L( �βk,al(λk,T ); {yt}, {z∗ +t })) + 2 +� +i=1,...,p +1{�βk,al +1,i ̸= 0} +HQ: +�λk,T = +arg min +λk,T ∈Sλk,T +− 2 log(L( �βk,al(λk,T ); {yt}, {z∗ +t })) + 2 log(log(T)) +� +i=1,...,p +1{�βk,al +1,i ̸= 0} +BIC: +�λk,T = +arg min +λk,T ∈Sλk,T +− 2 log(L( �βk,al(λk,T ); {yt}, {z∗ +t })) + log(T) +� +i=1,...,p +1{�βk,al +1,i ̸= 0} , +where �βk,al(λk,T ) := [�βk,al +1,0 , . . . , �βk,al +1,p , �βk,al +2,0 , 0, . . . , 0]′ restricts �βk,al +2,1 , . . . , �βk,al +2,p+1 to zero and define +� +�βk,al +1,0 , . . . , �βk,al +1,p , �βk,al +2,0 +� += +arg min +β1,0,...,β1,p,β2,0 +− L(β; {yt}, {z∗ +t }) + λk,T +p +� +i=1 +�wk,i|β1,i|. +(23) +Step 2: Select the optimal tuning parameter �λσ,T ∈ Sλσ,T using the IC and �λk,T from Step 1 as follows, +AIC: +�λσ,T = +arg min +λσ,T ∈Sλσ,T +− 2 log(L( �βtal(λσ,T ); {yt}, {z∗ +t })) + 2 +�� +i=1,...,p 1{�βtal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βtal +2,j ̸= 0} +� +HQ: +�λσ,T = +arg min +λσ,T ∈Sλσ,T +− 2 log(L( �βtal(λσ,T ); {yt}, {z∗ +t })) + 2 log(log(T)) +�� +i=1,...,p 1{�βtal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βtal +2,j ̸= 0} +� +BIC: +�λσ,T = +arg min +λσ,T ∈Sλσ,T +− 2 log(L( �βal(λσ,T ); {yt}, {z∗ +t })) + log(T) +�� +i=1,...,p 1{�βtal +1,i ̸= 0} + � +j=1,...,(p+1) 1{�βtal +2,j ̸= 0} +� +, +where �βtal(λσ,T ) := [ �βtal′ +1· , �βtal′ +2· ]′ = [�βtal +1,0, . . . , �βtal +1,p, �βtal +2,0, . . . , �βtal +2,(p+1)]′, with �βtal +1,i = �βal +1,i = 0, ∀(1, i) ̸∈ Ak,al +T +and +�� +�βtal +1,i +� +(1,i)∈{(1,0)}∪Ak,al +T +, �βtal′ +2· +� += +arg min +{β1,i|(1,i)∈{(1,0)}∪Ak,al +T +},β2· +− L(β; {yt}, {z∗ +t }) + �λk,T +p +� +i=1 +�wk,i|β1,i| + λσ,T +p+1 +� +j=1 +�wσ,j|β2,j|, +(24) +where Ak,al +T +:= +� +(1, i) : i ≥ 1, �βk,al +1,i +̸= 0 +� +. +The final two-step ALMLE �βtal is obtained using the optimal tuning parameters �λk,T and �λσ,T . +8 + +We use two MLEs to set up �wk,i and �wσ,j as the two-step ALMLE involves two different likelihood functions +in each step. Specifically, we set +� +� +� +� +� +� +� +� +� +�wk,i = +1 +�βk,mle +1,i +1 +�βmle +1,i +, i = 1, . . . , p, +�wσ,j = +1 +�βmle +2,j +, j = 1, . . . , p + 1. +(25) +where �βmle := [�βmle +1,0 , . . . , �βmle +1,p , �βmle +2,0 , . . . , �βmle +2,p+1] is the full-model MLE (10) and �βk,mle := [�βk,mle +1,0 +, . . . , �βk,mle +1,p +, �βk,mle +2,0 +, +0, . . . , 0] is the partial-model MLE defined below +�βk,mle = +arg min +{β∈Θ|β2,j=0,j=1,...,p+1} +− L(β; {yt}, {z∗ +t }). +(26) +In this way, we choose �wk,i and �wσ,j such that truly active predictors are ensured to be selected efficiently +with Sλk,T and Sλσ,T before the truly inactive ones in both Step 1 and Step 2. Therefore, we achieve the oracle +property of �βtal as shown in Theorem 4. +Assumption L1. There exist λk,T = O(T +1 +2 −γ1) and λσ,T = O(T +1 +2 −γ2) with 0 < γ1 < 1 +2 and 0 < γ2 < 1 +2. +Assumption L2. We assume that there exists βk,o := [βk,o +1,0, βk,o +1,1, . . . , βk,o +1,p, βk,o +2,0, 0, . . . , 0]′ ∈ {β ∈ R2p+3|β2,j = +0, j = 1, . . . , p + 1} such that for any ϵ > 0 +lim +T →∞ P +�����βk,mle +1,i +− βk,o +1,i +��� > ϵ +� += 0, +i = 1, . . . , p; +(27) +and βk,o +1,i ̸= 0 for any (1, i) ∈ Ak. +Theorem 4 (Oracle Property of �βtal). +Under Assumptions L1, L2 and the assumptions in Theorem 2, we have that +(a) Model selection consistency: +lim +T →∞ P +� +Atal +T += A +� += 1, +(28) +where Atal +T +:= Atal +k,T ∪ Atal +σ,T with Atal +k,T := +� +(1, i) : �βtal +1,i ̸= 0, i = 1, . . . , p. +� +and Atal +σ,T := {(2, j) : �βtal +2,j ̸= 0, +j = 1, . . . , p + 1.}. +(a) Limiting distribution of �βtal: +√ +T +� +�βtal +A − βo∗ +A +� D∼ Ω−1 +HA SψA, +√ +T +� +�βtal +Ac − βo∗ +Ac +� +→ 0, +(29) +as T → ∞, where SψA and ΩHA are defined in Assumption M.7 and M.8 under the model specification with +only the truly active predictors involved and ordered according to A. +Proof. See Appendix A.3. +The superiority of the proposed two-step ALMLE to the ALMLE (17) is not just in the oracle property +when local unit-root predictors are included in the regression model but also in the computing cost. The +ALMLE (17) is computed over a two-dimensional tuning parameter grid in order to select an optimal pair of +(λk,T , λσ,T ) ∈ Sλk,T × Sλσ,T , while the two-step ALMLE is computed over two separate one-dimensional tuning +parameter grids in order to select the optimal λk,T ∈ Sλk,T first and λσ,T ∈ Sλσ,T after. +5 +Simulation study +We assess the finite sample properties of �βmle and �βtal from the perspectives of their biases, mean square errors +(MSEs) and model selection using four data generating processes (DGPs). These four DGPs are designed to +9 + +reflect the characteristics of the high-frequency financial data used in Section 6. First, DGPs are heteroscedastic +and the conditional exceedance rates can change over time. Second, DGPs involve predictors which are func- +tions of lagged loss rates characterizing the serial dependence structure in {(kt, σt)}. Third, we consider either +stationary or local unit-root predictors or both. +We simulate {lt} from the following conditional distribution, +lt = +� +� +� +� +� +� +� +� +� +F −1 +t( 1 +kt )(τt), +if +τt ≤ Ft( 1 +kt )(u) +u + F −1 +GPD(kt,σt) +�τt − Ft( 1 +kt )(u) +1 − τt +� +, +if +τt > Ft( 1 +kt )(u), +(30) +where {τt} is i.i.d. standard uniform distributed, Ft(ν)(·) and F −1 +t(ν)(·) denote the distribution and quantile +functions of a Student’s t distribution with ν degrees of freedom. The processes of {kt} and {σt} are specified +according to the following specifications: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +log +� +kt +0.5 − kt +� += β10 + β11 log(|lt−1| + 1 − rm) + +14 +� +j=1 +β1,1+j zj,t−1 , +log(σt) = β20 + β21 log(σt−1) + β22 log(|lt−1| + 1 − rm) + +14 +� +j=1 +β2,2+j zj,t−1 , +zi,t = φi zi,t−1 + ϵi,t−1, +i = 1, 2, . . . , 14, +φ := [φ1, . . . , φ14], +{ϵt := [ϵ1,t, . . . , ϵ14,t]′} +i.i.d. +∼ N(0, I14×14) , +βo +1· := [βo +1,0, βo +1,1, . . . , βo +1,15]′ , +βo +2· := [βo +2,0, βo +2,1, . . . , βo +2,16]′ , +βo := [βo′ +1·, βo′ +2·]′ . +(31) +We set u = F −1 +t(3)(0.8) and rm = 0.05, but use different φ and βo to obtain different degrees of serial dependence. +DGP 1. There are five truly active stationary predictors for both {kt} and {σt}, namely log(|lt−1| + 1 − rm), +z1,t−1, . . . , z4,t−1. Among truly inactive predictors z5,t−1, . . . , z14,t−1, two of them are local unit-root, i.e. +z13,t−1 and z14,t−1, and the others are stationary. +� +� +� +� +� +� +� +φ = [0, 0, 0, 0, 0 . . . , 0, 1, 1], +βo +1· = [−1, 0.3, −0.4, 0.2, 0.6, 0.6, 0, . . . , 0]′ , +βo +2· = [−1, 0, 0.7, 0.4, 0.3, 0.5, 0.6, 0, . . . , 0]′ , +(32) +DGP 2. As DGP 1 but with the difference that βo +2,1 is changed to nonzero, and hence log(σt−1) is now truly +active. We set βo +2,1 = 0.7 and keep the true values of the other coefficients unchanged. +� +� +� +� +� +� +� +φ = [0, 0, 0, 0, 0 . . . , 0, 1, 1], +βo +1· = [−1, 0.3, −0.4, 0.2, 0.6, 0.6, 0, . . . , 0]′ , +βo +2· = [−1, 0.7, 0.7, 0.4, 0.3, 0.5, 0.6, 0, . . . , 0]′ , +(33) +DGP 3. As DGP 1 but with the difference that φ4 = 1 and (βo +1,5, βo +2,6) = ( 0.6 +√ +T , 0.6 +√ +T ). +� +� +� +� +� +� +� +� +� +� +� +� +� +φ = [0, 0, 0, 1, 0 . . . , 0, 1, 1], +βo +1· = [−1, 0.3, −0.4, 0.2, 0.6, 0.6 +√ +T +, 0, . . . , 0]′ , +βo +2· = [−1, 0, 0.7, 0.4, 0.3, 0.5, 0.6 +√ +T +, 0, . . . , 0]′ , +(34) +10 + +DGP 4. As DGP 3 but with the difference that log(σt−1) is truly active. We set βo +2,1 = 0.7 and and keep the +true values of the other coefficients unchanged. +� +� +� +� +� +� +� +� +� +� +� +� +� +φ = [0, 0, 0, 1, 0 . . . , 0, 1, 1], +βo +1· = [−1, 0.3, −0.4, 0.2, 0.6, 0.6 +√ +T +, 0, . . . , 0]′ , +βo +2· = [−1, 0.7, 0.7, 0.4, 0.3, 0.5, 0.6 +√ +T +, 0, . . . , 0]′ , +(35) +In each simulation, we obtain a sample {lt}T +t=1 of T observations, and extract the excess time series +{yt = max(lt − u, 0)} using the true threshold u. We standardize the predictors using their empirical stan- +dard deviations. We then fit the full model specification (31) to {yt} using standardized predictors, estimating +the model parameter by �βmle and �βtal. Bias and mean squared error (MSE) are then computed as +Bias = +1 +#βo +� +� +� +i=1,2;j=0 +����βi,j − βo +i,j +��� + +� +i=1,2;j≥1 +si,j +����βi,j �si,j − βo +i,j +��� +� +� +(36) +MSE = +1 +#βo +� +� +� +i=1,2;j=0 +� +�βi,j − βo +i,j +�2 ++ +� +i=1,2;j≥1 +s2 +i,j +� +�βi,j �si,j − βo +i,j +�2 +� +� +(37) +where #βo denotes the number of parameters in βo, �si,j denotes the empirical standard deviation of the (i, j)-th +predictor, and si,j = 1 for I(0) predictors and si,j = +√ +T for I(1) predictors. +Table 1 presents the average absolute bias and average MSE of the coefficient estimates obtained over +100 replications. These results show that �βmle and �βtal have decreasing biases and MSEs when T increases, +coherently with the theoretical results presented in Sections 3.2 and 4. Moreover �βtal under BIC always has the +lowest bias and MSE across the DGPs, supporting the use of �βtal with BIC in the empirical section. Boxplots +for the bias in Figure 1 support these conclusions. +Table 2 presents the variable selection results for both �βtal and �βmle. Note that for the latter, we perform +variable selection based on the significance of the t-statistics associated to the candidate predictors. To measure +the ability to select the correct predictors, we assess the average selection rates of truly active and inactive +predictors for both the shape and scale parameters. Moreover, we compute the correct classification rate (CCR) +of each estimator, i.e. the proportion of selected truly active and unselected truly inactive predictors on the +total candidate predictors. Results in Table 2 show that variable selection improves as T increases for each +estimator. For �βmle the average selection rates of truly inactive stationary predictors approach the significance +level α = 0.05, whereas the average selection rates of truly inactive local unit-root predictors are much higher +than α = 0.05, for both k and σ, and regardless of the DGP. These results are coherent with the asymptotic +results derived in Section 3.2, and echo the size distortion concerns of using t-tests to select non-stationary +predictors discussed in Section 4. Remarkably, the average selection rates of truly active predictors for �βmle are +much lower than those for �βtal. Moreover, we see that the power of t-tests performed with �βmle +1· +is lower than +the one for �βmle +2· +due to the uncertainty in the estimation of �βmle +1· . Finally, Table 2 shows that �βtal with BIC +always has the highest CCR and produces the most accurate selection regardless the DGP, supporting the use +of �βtal with BIC for the empirical application. +6 +Empirical Study +We study the high-frequency excess loss distributions of nine large liquid U.S. stocks: American Express (AXP), +Boeing (BA), General Electric (GE), Home Depot (HD), IBM, Johnson and Johnson (JNJ), JPMorgan Chase +(JPM), Coca-Cola (KO), and ExxonMobil (XOM). Our data covers all transactions observed from January +2006 to December 2014. Market uncertainty and liquidity being elusive concepts, we study their impact on the +excess loss distribution using as predictors several high-frequency volatility and liquidity indicators, and select +11 + +(a) +(b) +(c) +(d) +Fig. 1: Boxplots of bias (36) obtained from 100 replications with �βmle and �βtal using the optimal tuning pa- +rameters selected by AIC, HQ and BIC criteria. +12 + +DGP3 +0.35 +E +2-StepALMLE+AIC +2-StepALMLE+HQ +2-Step ALMLE + BIC +MLI ++ ++ ++ +0.3 +T +average absolute bias in each simulation +25 +.2 +-- +++→ +#+ ++++ +0.05 +-白 +T=25000 T=50000 T=100000 +T=25000T=50000 T=100000 +T=25000 T=50000 T=100000 +T=25000 T=50000 T=100000DGP 4 +0.35 +E +2-StepALMLE+AIC +HQ +BIC +MLI ++ ++ +2-StepALMLE +2-StepALMLE +0.3 +十 +bias in each simulation ++ +.25 +1 +0.2 ++ +absolute +1.15 ++ ++ +average +0.1 +- ++ ++ ++ +0.05 +T=25000T=50000T=100000 +T=25000T=50000T=100000 +T=25000T=50000T=100000 +T=25000T=50000T=100000DGP 5 +MLE +2-Step ALMLE +AIC +OH +BIC ++ +2-Step ALMLE + +2-StepALMLE+ +0.5 +0.4 +T +- +- ++ +0.3 +- ++ ++ ++ +#+ ++# +++ +0.2 +--- +丰 ++ ++ ++ ++ +- +-++ ++ ++ +0.1 +- +1 +一 +L +X +0F +T=25000 T=50000T=100000 +T=25000 T=50000 T=100000 +T=25000 T=50000T=100000 +T=25000 +T=50000T=100000DGP 6 +MLE ++AIC +HQ +BIC ++ +2-StepALMLE+ +2-Step ALMLE +2-StepALMLE +1.8 +absolutebiasineachsimulation +1.6 +1.4 +1.2 +0.8 +averagea +0.6 +0.4 ++ +0.2 +白 +X +丰 +丰 +山 +T=25000T=50000 T=100000 +T=25000T=50000T=100000 +T=25000T=50000T=100000 +T=25000T=50000 T=100000Table 1: Average absolute bias (36) and MSE (37) over 100 replications obtained with �βmle and �βtal using the +optimal tuning parameters selected by AIC, HQ and BIC criteria. +Bias +MSE +DGPs Estimators +T +25,000 +50,000 +100,000 25,000 +50,000 +100,000 +DGP 1 +�βmle +0.031 +0.015 +0.008 +0.105 +0.031 +0.016 +�βtal + AIC +0.014 +0.007 +0.004 +0.036 +0.014 +0.007 +�βtal + HQ +0.010 +0.006 +0.004 +0.030 +0.012 +0.006 +�βtal + BIC +0.010 +0.005 +0.004 +0.026 +0.011 +0.006 +DGP 2 +�βmle +0.027 +0.017 +0.010 +0.091 +0.031 +0.014 +�βtal + AIC +0.015 +0.011 +0.006 +0.047 +0.020 +0.009 +�βtal + HQ +0.012 +0.009 +0.004 +0.034 +0.012 +0.007 +�βtal + BIC +0.009 +0.007 +0.003 +0.030 +0.011 +0.005 +DGP 3 +�βmle +0.047 +0.023 +0.009 +0.197 +0.051 +0.023 +�βtal + AIC +0.021 +0.015 +0.005 +0.082 +0.024 +0.013 +�βtal + HQ +0.018 +0.013 +0.003 +0.067 +0.021 +0.011 +�βtal + BIC +0.017 +0.012 +0.003 +0.060 +0.020 +0.010 +DGP 4 +�βmle +0.029 +0.025 +0.019 +0.128 +0.165 +0.263 +�βtal + AIC +0.018 +0.017 +0.006 +0.071 +0.099 +0.013 +�βtal + HQ +0.015 +0.014 +0.006 +0.065 +0.093 +0.012 +�βtal + BIC +0.015 +0.014 +0.005 +0.057 +0.091 +0.011 +the most appropriate ones with the two-step ALMLE developed in Section 4. We perform an in-sample analysis +providing an economic interpretation for the impact of the selected predictors on the excess loss distribution, +and an out-of-sample VaR forecast analysis to assess the goodness of fit of the predicted excess loss distribution. +6.1 +Variables description +The raw intraday data of the studied stocks contain transaction timestamps in milliseconds, transaction prices +per share, and transaction volume in shares for each trade. We cleaned the raw data according to standard +procedures in Brownlees and Gallo (2006) and Barndorff-Nielsen et al. (2009). Since transaction data are +irregularly-spaced, we need to define an equally-spaced grid at a fixed frequency to analyse losses with our +model. We choose to analyse losses at the five minute frequency. Let Pt,i be the transaction price of the i-th +trade in the t-th five minute interval, and let Vt,i be the corresponding quantity of traded shares, with 0 ≤ i ≤ nt +where nt is the number of trades in the t-th five minute interval and 0 < t ≤ T. We define 5-min prices, Pt, +as the median transaction price in the t-th five minute interval, and compute 5-min losses as the negative t-th +return, Rt := log(Pt) − log(Pt−1). To obtain the time series of excess losses we consider a dynamic threshold +accounting for the time-varying behavior of losses at high-frequency. Specifically, the threshold ut at time t is +defined as the 90%-quantile of the losses observed over the period (t − 1, t − h), with h > 1 the moving window +size. We consider 12 possible values of h ranging from one week to twelve weeks. +Liquidity refers to the ability to trade large volume of a financial instrument with low price impact, cost +and postponement. As liquidity can be decomposed into different dimensions (Harris et al., 1990), we consider +several liquidity indicators as possible predictors. Similarly, to characterize market uncertainty we consider +several indicators for the observed dispersion of transaction prices. Moreover, to disentangle the impact of +trading activity at different frequencies, we build our set of candidate predictors considering both information +within the t-th five minute interval and across neighbourhoods of the t-th five minute interval. Let Pt,BU := +[Pt,1, . . . , Pt,nt]′ and Rt,BU := [Rt,1, . . . , Rt,nt]′ be the vectors of traded prices and trade returns observed +within the t-th five minute interval, with Rt,i := log(Pt,i) − log(Pt,i−1). Let Tw be a neighborhood size, and +define Pt,Tw := [Pt, . . . , Pt−Tw+1]′ the vector of 5-min prices within a neighborhood of size Tw and Rt,Tw := +13 + +Table 2: Average selection rate across 100 replication for truly active (t.p.) and truly inactive (f.p.) stationary (I(0)) and local unit-root (I(1)) predictors in the shape +(k) and scale (σ) parameters, correct classification rate (CCR), and selection rates of log(σt−1). +DGPs +T +estimators selection criteria t.p.(k) of I(0)s +f.p.(k) of I(0)s +t.p.(k) of I(1)s +f.p.(k) of I(1)s +t.p.(σ) of I(0)s +f.p.(σ) of I(0)s +t.p.(σ) of I(1)s +f.p.(σ) of I(1)s +CCR +log(σt−1) +DGP 3 +25,000 +�βmle +t test (α = 0.05) +0.546 +0.063 +− +0.110 +1.000 +0.110 +− +0.215 +0.858 +0.060 +�βtal +AIC +0.998 +0.160 +− +0.465 +1.000 +0.156 +− +0.220 +0.869 +0.150 +HQ +0.996 +0.123 +− +0.420 +1.000 +0.024 +− +0.060 +0.930 +0.020 +BIC +0.984 +0.076 +− +0.350 +1.000 +0.006 +− +0.010 +0.953 +0.000 +50,000 +�βmle +t test (α = 0.05) +0.718 +0.050 +− +0.085 +1.000 +0.112 +− +0.185 +0.892 +0.050 +�βtal +AIC +0.998 +0.100 +− +0.450 +1.000 +0.147 +− +0.160 +0.892 +0.110 +HQ +0.998 +0.075 +− +0.410 +1.000 +0.032 +− +0.040 +0.942 +0.020 +BIC +0.998 +0.053 +− +0.330 +1.000 +0.004 +− +0.015 +0.963 +0.000 +100,000 +�βmle +t test (α = 0.05) +0.788 +0.045 +− +0.130 +1.000 +0.109 +− +0.220 +0.900 +0.040 +�βtal +AIC +0.988 +0.045 +− +0.435 +1.000 +0.154 +− +0.200 +0.901 +0.160 +HQ +0.988 +0.034 +− +0.385 +1.000 +0.030 +− +0.040 +0.953 +0.020 +BIC +0.988 +0.024 +− +0.345 +1.000 +0.001 +− +0.010 +0.969 +0.000 +DGP 4 +25,000 +�βmle +t test (α = 0.05) +0.588 +0.071 +− +0.100 +1.000 +0.101 +− +0.195 +0.870 +1.000 +�βtal +AIC +0.982 +0.414 +− +0.665 +1.000 +0.176 +− +0.155 +0.792 +1.000 +HQ +0.966 +0.213 +− +0.525 +1.000 +0.044 +− +0.040 +0.892 +1.000 +BIC +0.916 +0.086 +− +0.450 +1.000 +0.004 +− +0.005 +0.934 +1.000 +50,000 +�βmle +t test (α = 0.05) +0.718 +0.055 +− +0.110 +1.000 +0.085 +− +0.170 +0.900 +1.000 +�βtal +AIC +0.996 +0.310 +− +0.630 +1.000 +0.145 +− +0.140 +0.832 +1.000 +HQ +0.992 +0.178 +− +0.535 +1.000 +0.038 +− +0.025 +0.907 +1.000 +BIC +0.982 +0.109 +− +0.500 +1.000 +0.001 +− +0.005 +0.936 +1.000 +100,000 +�βmle +t test (α = 0.05) +0.828 +0.039 +− +0.090 +1.000 +0.094 +− +0.190 +0.920 +1.000 +�βtal +AIC +1.000 +0.248 +− +0.560 +1.000 +0.134 +− +0.100 +0.859 +1.000 +HQ +1.000 +0.111 +− +0.435 +1.000 +0.036 +− +0.040 +0.931 +1.000 +BIC +0.998 +0.060 +− +0.345 +1.000 +0.000 +− +0.000 +0.962 +1.000 +DGP 5 +25,000 +�βmle +t test (α = 0.05) +0.458 +0.065 +0.280 +0.140 +1.000 +0.120 +1.000 +0.225 +0.832 +0.080 +�βtal +AIC +0.998 +0.161 +0.930 +0.410 +1.000 +0.152 +1.000 +0.175 +0.874 +0.190 +HQ +0.998 +0.098 +0.900 +0.365 +1.000 +0.039 +1.000 +0.045 +0.934 +0.050 +BIC +0.998 +0.083 +0.890 +0.350 +1.000 +0.006 +1.000 +0.005 +0.950 +0.000 +50,000 +�βmle +t test (α = 0.05) +0.620 +0.058 +0.390 +0.135 +1.000 +0.111 +1.000 +0.215 +0.862 +0.060 +�βtal +AIC +0.993 +0.085 +0.990 +0.355 +1.000 +0.151 +1.000 +0.175 +0.899 +0.150 +HQ +0.993 +0.058 +0.980 +0.285 +1.000 +0.036 +1.000 +0.020 +0.954 +0.030 +BIC +0.993 +0.044 +0.970 +0.260 +1.000 +0.003 +1.000 +0.005 +0.969 +0.000 +100,000 +�βmle +t test (α = 0.05) +0.755 +0.036 +0.510 +0.095 +1.000 +0.092 +1.000 +0.180 +0.899 +0.040 +�βtal +AIC +0.993 +0.033 +1.000 +0.300 +1.000 +0.124 +1.000 +0.170 +0.924 +0.140 +HQ +0.993 +0.025 +1.000 +0.220 +1.000 +0.028 +1.000 +0.040 +0.968 +0.010 +BIC +0.993 +0.020 +1.000 +0.195 +1.000 +0.000 +1.000 +0.005 +0.981 +0.000 +DGP 6 +25,000 +�βmle +t test (α = 0.05) +0.480 +0.058 +0.290 +0.095 +1.000 +0.091 +1.000 +0.215 +0.852 +1.000 +�βtal +AIC +0.890 +0.304 +0.970 +0.640 +1.000 +0.148 +1.000 +0.135 +0.818 +1.000 +HQ +0.873 +0.219 +0.970 +0.535 +1.000 +0.039 +1.000 +0.020 +0.880 +1.000 +BIC +0.828 +0.144 +0.950 +0.445 +1.000 +0.004 +1.000 +0.000 +0.909 +1.000 +50,000 +�βmle +t test (α = 0.05) +0.683 +0.070 +0.370 +0.095 +1.000 +0.098 +1.000 +0.195 +0.877 +1.000 +�βtal +AIC +0.940 +0.241 +0.990 +0.600 +1.000 +0.175 +1.000 +0.180 +0.834 +1.000 +HQ +0.930 +0.140 +0.990 +0.500 +1.000 +0.034 +1.000 +0.045 +0.911 +1.000 +BIC +0.903 +0.076 +0.990 +0.425 +1.000 +0.011 +1.000 +0.015 +0.936 +1.000 +100,000 +�βmle +t test (α = 0.05) +0.813 +0.048 +0.450 +0.070 +1.000 +0.069 +1.000 +0.155 +0.914 +1.000 +�βtal +AIC +0.918 +0.095 +0.990 +0.510 +1.000 +0.125 +1.000 +0.090 +0.894 +1.000 +HQ +0.918 +0.036 +0.990 +0.385 +1.000 +0.033 +1.000 +0.025 +0.945 +1.000 +BIC +0.913 +0.023 +0.990 +0.340 +1.000 +0.001 +1.000 +0.005 +0.960 +1.000 +14 + +log(Pt,Tw) − log(Pt−1,Tw) the corresponding vector of returns. Let durt,i denote the execution duration of the +i-th transaction in the t-th five minute interval, i.e. the time difference between the order executed time and +order placed time. Table 3 lists the liquidity predictors we consider in the analysis. They are classified according +to their frequency, i.e. within or across the five minute interval, and by their nature of price impact or spread +proxies (Goyenko et al., 2009) or volatility of liquidity measures. Table 4 lists the volatility predictors we consider +in the analysis and are classified according to the frequency at which they are computed, i.e., within or across +the five minute interval. +Table 3: Liquidity measures proxying for price impact (PI), spread (S) and volatility of liquidity (Vol) computed +using information within the five minute interval (W), across (A) 5-min observations in a neighbourhood of size +Tw, or as a ratio (R) between the two frequencies. △ denotes the first difference operator for vectors; cov(,) +and cor(,) denote the covariance and the correlation between two input variables, respectively; var() denotes +the variance of the input variable or vector. +Frequency +Proxy +Liquidity Predictors +Formula +W +PI +Transaction Volume +TVt = �nt +i=1 Pt,i Vt,i +W +PI +Transaction Quantity +TQt = �nt +i=1 Vt,i +W +Vol +Micro Transaction Volume Volatility +MTVVt = +� +1 +nt +�nt +j=1 +� +Pt,j Vt,j − 1 +nt +�nt +i=1 Pt,i Vt,i +�2 +W +Vol +Micro Volatility of Trading Quantity in shares +MTQVt = +� +1 +nt +�nt +j=1 +� +Vt,j − 1 +nt +�nt +i=1 Vt,i +�2 +W +PI +Amihud Illiquidity Measure +AMt = +1 +nt +�nt +i=1 +|Rt,i| +Pt,i Vt,i +W +PI +Extended Amihud Measures (Goyenko et al., 2009) +EAMt = max(Pt,BU)−min(Pt,BU) +TVt +W +PI +Transaction Duration +durt = +�nt +i=1 durt,i +nt +A +S +Roll (Roll, 1984) +Rollt = cov (△Pt,Tw, △Pt−1.Tw) +A +S +Modified Roll +RollModt = cov(△Pt,Tw ,△Pt−1.Tw ) +P m +t +A +S +Negative Roll +Roll− +t = Rollt 1{Rollt < 0} +A +S +Negative Modified Roll +RollMod− +t = RollModt 1{RollModt < 0} +A +S +Return Autocorrelation (Grossman and Miller, 1988) RACt = cor (Rt,Tw, Rt−1.Tw) +A +PI +Amihud Illiquidity Measure +AMIt = +1 +Tw +�Tw−1 +j=0 +|Rt−j| +TVt−j +A +Vol +Transaction Volume Volatility +TVVt = +� +1 +Tw +�Tw−1 +j=0 +� +TVt−j − TVt,Tw +�2 +A +Vol +Relative Transaction Volume Volatility +RTVVt = +TVVt +1 +Tw +�Tw−1 +j=0 +TVt−j +A +Vol +Trading Quantity Volatility +TQVt = +� +1 +Tw +�Tw−1 +j=0 +� +TQt−j − +1 +Tw +�Tw−1 +j=0 +TQt−j +�2 +A +Vol +Relative Trading Quantity Volatility +RTQVt = +TQVt +TQt,Tw +R +S +Variance Ratio (Hasbrouck and Schwartz, 1988) +VRt = Tw·var(log(Pt,BU)−log(Pt)) +nt var(Rt,Tw ) +6.2 +In-sample estimates +We divide each time series into an in-sample period covering the first 90% of the observations and an out-of- +sample period spanning the last 10% of the sample. We model the excess losses {yt} of each stock with the +time-varying GPD regression model in (3)-(4), using the variables defined in Tables 3 and 4, with Tw ∈ {2, 6, 12}, +as possible predictors in both scale and shape parameters. Coefficient estimates obtained with the two-step +ALMLE are presented in Tables 5 and 6 for the shape and scale parameters, respectively. +Results for the shape parameter in Table 5 show that estimated coefficients have almost always the same +sign across the stocks. As to liquidity predictors, we find that price impact proxies are selected for almost +all the stocks, suggesting that they better capture liquidity effects on extreme losses. In particular, TV and +TQ display positive coefficients while AM and EAM display negative coefficients, entailing that larger extreme +15 + +Table 4: Volatility measures computed using information within the five minute interval (W), across (A) 5-min +observations in a neighbourhood of size Tw, or as a ratio (R) between the two frequencies. +Frequency +Volatility Predictors +Formulas +W +Micro Noise Return Volatility +MNRVt = +� +1 +nt +�nt +i=1 (log(Pt,i) − log(Pt))2 +W +Micro Realized Volatility +MRVt = +��nt +i=1 (log(Pt,i) − log(Pt,i−1))2 +A +Realized Volatility +RVt,Tw = +��Tw−1 +j=0 +R2 +t−j +R +MNRV2RV +MNRV2RV = MNRVt +RVt +R +MRV2RV +MRV2RV = MRVt +RVt +losses are associated with high levels of liquidity in the last five minutes. Although counter-intuitive at first, this +result is very interesting when read together with the other selected variables. As to the volatility of liquidity, +we notice that RTVV(Tw = 6) and RTQV(Tw = 6) are selected across most of the stocks and display large +and positive coefficients, indicating that extreme losses tend to be larger during periods of high volatility of +liquidity. Almost for every stock, we select the ratio MRV2RV(Tw = 12), essentially capturing the impact of the +volatility of volatility or jump risk on extreme losses, and associate a positive coefficient to it, conveying the idea +that extreme losses tend to be larger during periods of high uncertainty. Altogether these results are coherent +with the findings in Brogaard et al. (2018), i.e. that market markers amplify extreme price movements while +withdrawing from the market after large uncertainty shocks that caused their liquidity supply to be outstripped +by liquidity demand. +Table 6 shows that more variables are selected for the scale parameter but their pattern is less stable across +stocks. In general, we notice that the autoregressive component contributes to the dynamics, and that the +realized volatility predictor computed within the five-minute interval is always selected and displays positive +coefficient. This is coherent with the fact that the scale parameter captures the time-varying heteroscedasticity +in the data. +For comparison purposes, we report estimated regression coefficients for the shape and scale parameters +obtained with MLE in Tables 7-8. All of the estimated coefficients are nonzero and we cannot compute the +corresponding standard errors because the obtained Fisher information matrix of the MLE is not positive +definitive. This makes variable interpretation very difficult if not impossible. +6.3 +Out-of-sample VaR forecast +The coefficient estimates �β obtained on the in-sample period are used to compute a one-step ahead VaR +prediction in the out-of-sample period. Specifically, the VaR of each stock at a risk level α at time t given xt−1 +and �β is obtained as +� +VaRt(α) = �σt(xt−1, �β) +�kt(xt−1, �β) +�� +1 − α − Ft(�ut) +1 − Ft(�ut) +�−�kt(xt−1, � +β) +− 1 +� ++ �ut. +(38) +where Ft(�ut) is the probability of exceeding the threshold �ut and is fixed to 90%. The coverage rate of +{ � +VaRt(α)}Tis+Tos +t=Tis+1 for the out-of sample period is obtained as follows, +Coverage Rate = +�Tis+Tos +t=Tis+1 1{lt ≤ � +VaRt(α)} +Tos +. +(39) +Table 9 shows the coverage rate of { � +VaRt(α)}Tis+Tos +t=Tis+1 at the risk level α for various α ∈ [90%, 100%). We resort +to the Kolmogorov–Smirnov (K-S) test to test the goodness of fit of the predicted GPD over the out-of-sample +16 + +Table 5: Empirical estimates of the regression coefficients for the shape parameter obtained with the two-step +ALMLE. +Predictors +Stocks +AXP BA +GE +HD +IBM +JNJ +JPM KO +XOM +selection per stock +TV +1.105 0.676 +0.581 1.912 1.205 0.660 1.109 +2.014 +0.89 +TQ +1.314 1.101 0.501 0.874 0.813 1.130 0.756 +0.0004 +0.89 +AM +-0.345 +0.719 -0.037 +-0.522 -0.030 +0.56 +MTVV +0.067 +0.11 +EAM +-1.637 -1.760 +-0.913 -1.789 -1.681 -1.396 -1.650 +-0.201 +0.89 +MTQV +0.200 +0.166 +0.009 0.177 +0.44 +MRV +0.328 +-0.901 -0.367 -0.274 +0.44 +MNRV +-0.794 -0.242 -0.491 +0.020 +0.44 +dur +0.00 +AMI (Tw = 2) +0.00 +VR (Tw = 2) +0.00 +RV (Tw = 2) +0.00 +MNRV2RV (Tw = 2) +-0.084 +0.11 +MRV2RV (Tw = 2) +0.095 +0.11 +AMI (Tw = 6) +-0.003 +0.11 +Roll (Tw = 6) +0.00 +Roll− (Tw = 6) +0.00 +RollMod (Tw = 6) +0.00 +RollMod− (Tw = 6) +0.00 +TVV (Tw = 6) +0.00 +TQV (Tw = 6) +0.00 +RTVV (Tw = 6) +0.068 +0.298 +0.159 0.218 +0.264 +0.56 +RTQV (Tw = 6) +0.069 +0.300 +0.159 0.218 +0.270 +0.56 +RAC (Tw = 6) +0.00 +VR (Tw = 6) +0.00 +RV (Tw = 6) +-0.203 +0.11 +MNRV2RV (Tw = 6) +-0.415 +-0.185 +0.22 +MRV2RV (Tw = 6) +0.194 -0.010 +0.0003 +0.33 +AMI (Tw = 12) +0.524 +0.648 +0.22 +Roll (Tw = 12) +0.00 +Roll− (Tw = 12) +0.00 +RollMod (Tw = 12) +0.00 +RollMod− (Tw = 12) +0.00 +TVV (Tw = 12) +-0.851 +0.11 +TQV (Tw = 12) +0.00 +RTVV (Tw = 12) +-0.090 +0.11 +RTQV (Tw = 12) +-0.090 +0.11 +RAC (Tw = 12) +0.00 +VR (Tw = 12) +0.415 +0.244 +-0.00002 +0.33 +RV (Tw = 12) +-0.436 +0.11 +MNRV2RV (Tw = 12) +0.297 +0.11 +MRV2RV (Tw = 12) +0.994 0.608 0.287 0.554 0.548 0.487 0.544 +0.766 +0.89 +total number of selected variables +6 +7 +10 +8 +5 +7 +11 +17 +8 +17 + +Table 6: Empirical estimates of the regression coefficients for the scale parameter obtained with the two-step +ALMLE. +Predictors +Stocks +AXP BA +GE +HD +IBM +JNJ +JPM +KO +XOM selection per stock +TV +-0.004 +-0.005 +0.042 +-0.044 0.212 0.095 +0.67 +TQ +0.039 0.110 0.015 0.042 +0.047 +0.092 -0.102 -0.095 +0.89 +AM +0.024 0.050 +0.22 +MTVV +0.048 +-0.029 +-0.137 -0.260 +0.44 +EAM +-0.040 -0.012 +-0.043 +0.33 +MTQV +-0.123 0.016 +0.021 +0.135 0.168 -0.003 +0.67 +MRV +0.049 +0.032 +0.057 +0.33 +MNRV +0.049 0.139 0.056 0.055 0.067 0.079 +0.048 0.128 0.091 +1.00 +dur +0.00 +AMI (Tw = 2) +-0.018 +0.11 +VR (Tw = 2) +0.00 +RV (Tw = 2) +0.017 +-0.021 +0.22 +MNRV2RV (Tw = 2) +0.00 +MRV2RV (Tw = 2) +0.00 +AMI (Tw = 6) +-0.014 +-0.003 +0.22 +Roll (Tw = 6) +-0.054 +0.11 +Roll− (Tw = 6) +-0.035 +-0.086 +0.22 +RollMod (Tw = 6) +-0.061 +-0.048 +-0.050 +0.33 +RollMod− (Tw = 6) +0.077 +0.008 0.051 0.053 0.062 +0.104 +0.061 +0.78 +TVV (Tw = 6) +0.006 0.029 +0.22 +TQV (Tw = 6) +-0.015 -0.048 -0.006 +0.33 +RTVV (Tw = 6) +0.015 +0.11 +RTQV (Tw = 6) +0.0003 +0.11 +RAC (Tw = 6) +0.00 +VR (Tw = 6) +-0.003 +0.11 +RV (Tw = 6) +0.017 0.038 +0.22 +MNRV2RV (Tw = 6) +-0.002 -0.056 +-0.037 +-0.082 +0.44 +MRV2RV (Tw = 6) +-0.028 0.041 +0.044 +0.33 +AMI (Tw = 12) +0.039 +0.11 +Roll (Tw = 12) +-0.005 +0.027 +0.22 +Roll− (Tw = 12) +0.006 +0.11 +RollMod (Tw = 12) +-0.005 +-0.038 +-0.014 +0.33 +RollMod− (Tw = 12) +0.004 +0.006 +0.022 +0.33 +TVV (Tw = 12) +-0.036 +-0.009 +0.22 +TQV (Tw = 12) +-0.013 +-0.001 -0.010 +-0.011 +0.44 +RTVV (Tw = 12) +0.003 +0.11 +RTQV (Tw = 12) +0.046 +0.11 +RAC (Tw = 12) +0.00 +VR (Tw = 12) +-0.004 +-0.032 +0.22 +RV (Tw = 12) +0.027 +0.034 +0.22 +MNRV2RV (Tw = 12) +0.014 0.007 0.032 0.037 +0.061 +0.023 +0.103 +0.78 +MRV2RV (Tw = 12) +0.022 +0.021 -0.005 -0.00005 0.014 -0.016 +0.67 +log(σt−1) +0.849 0.631 0.758 0.724 0.832 0.778 +0.829 0.668 0.770 +1.00 +total number of selected variables +14 +10 +12 +12 +20 +11 +16 +7 +18 +18 + +Table 7: Empirical estimates of the regression coefficients for the scale parameter obtained with the MLE. +Predictors +Stocks +AXP BA +GE +HD +IBM +JNJ +JPM KO +XOM selection per stock +TV +0.396 0.264 0.116 0.264 0.563 0.631 0.239 0.602 0.239 +1.00 +TQ +0.469 0.313 0.173 0.326 0.457 0.619 0.259 0.519 0.216 +1.00 +AM +-0.083 -0.167 0.303 -0.042 -0.287 -0.394 -0.067 -0.221 -0.080 +1.00 +TVV +0.249 0.017 0.064 0.094 0.135 0.115 0.055 0.090 0.059 +1.00 +EAM +-0.621 -0.406 -0.062 -0.451 -0.525 -1.095 -0.236 -0.966 -0.350 +1.00 +TQV +0.195 0.020 0.076 0.100 0.151 0.112 0.053 0.087 0.057 +1.00 +MRV +0.062 0.001 0.224 -0.003 0.136 0.188 0.051 0.086 0.048 +1.00 +MNRV +-0.055 -0.081 0.072 -0.140 -0.021 0.028 0.009 -0.121 -0.035 +1.00 +dur +-0.025 0.000 -0.026 0.016 0.000 0.033 0.020 0.021 0.000 +1.00 +AMI (Tw = 2) +-0.130 -0.048 0.036 0.129 -0.386 -0.194 -0.008 0.098 -0.239 +1.00 +VR (Tw = 2) +0.000 0.060 0.000 0.013 0.000 0.034 -0.001 0.016 0.000 +0.56 +RV (Tw = 2) +-0.047 -0.048 0.036 -0.178 0.047 0.006 0.027 -0.005 -0.070 +1.00 +MNRV2RV (Tw = 2) +0.000 0.048 0.000 0.012 0.000 0.084 -0.043 -0.066 0.000 +0.78 +MRV2RV (Tw = 2) +0.000 0.035 0.000 0.010 0.000 0.007 -0.025 0.036 0.000 +0.78 +AMI (Tw = 6) +0.005 -0.154 0.107 -0.059 -0.401 -0.188 0.000 0.111 0.019 +1.00 +Roll (Tw = 6) +0.043 0.035 -0.010 0.018 0.032 -0.045 -0.004 0.001 0.001 +1.00 +Roll− (Tw = 6) +0.052 0.019 -0.016 0.045 0.011 -0.066 -0.011 0.001 0.010 +1.00 +RollMod (Tw = 6) +0.026 0.024 -0.022 -0.014 0.029 -0.051 -0.007 0.001 0.005 +1.00 +RollMod− (Tw = 6) +0.024 0.013 -0.031 0.011 0.008 -0.076 -0.016 0.001 0.006 +1.00 +TVV (Tw = 6) +0.015 0.026 -0.048 0.063 0.004 0.010 0.059 -0.021 0.023 +1.00 +TQV (Tw = 6) +0.062 0.038 -0.004 0.095 -0.015 0.006 0.070 -0.049 0.020 +1.00 +RTVV (Tw = 6) +0.132 0.081 0.080 0.182 0.080 0.152 0.093 0.168 0.147 +1.00 +RTQV (Tw = 6) +0.131 0.081 0.080 0.182 0.080 0.152 0.092 0.168 0.147 +1.00 +RAC (Tw = 6) +-0.042 -0.063 -0.029 -0.071 -0.020 -0.013 0.053 -0.078 -0.101 +1.00 +VR (Tw = 6) +0.103 0.032 -0.010 0.115 -0.018 -0.020 0.033 -0.052 0.043 +1.00 +RV (Tw = 6) +-0.116 -0.104 0.116 -0.176 -0.036 -0.099 -0.015 -0.114 -0.128 +1.00 +MNRV2RV (Tw = 6) +0.011 0.090 -0.151 0.058 0.009 0.044 -0.004 -0.087 -0.018 +1.00 +MRV2RV (Tw = 6) +0.097 0.165 0.031 0.188 0.231 0.107 0.068 0.142 0.132 +1.00 +AMI (Tw = 12) +-0.032 -0.133 0.162 -0.036 -0.259 -0.030 -0.021 0.152 -0.015 +1.00 +Roll (Tw = 12) +-0.019 0.005 -0.023 0.000 0.047 0.009 0.011 0.002 0.009 +1.00 +Roll− (Tw = 12) +0.074 0.059 -0.013 0.081 0.041 0.031 0.014 0.003 0.040 +1.00 +RollMod (Tw = 12) +-0.021 0.012 -0.026 -0.040 0.045 0.006 0.009 0.002 0.013 +1.00 +RollMod− (Tw = 12) +0.026 0.051 -0.029 0.020 0.039 0.018 0.006 0.002 0.031 +1.00 +TVV (Tw = 12) +-0.125 -0.049 -0.143 -0.051 -0.132 -0.141 -0.019 -0.285 -0.069 +1.00 +TQV (Tw = 12) +-0.059 -0.031 -0.096 -0.021 -0.159 -0.150 -0.008 -0.308 -0.070 +1.00 +RTVV (Tw = 12) +-0.008 0.035 0.010 -0.001 0.006 0.033 0.030 -0.028 0.056 +1.00 +RTQV (Tw = 12) +-0.008 0.036 0.010 -0.002 0.005 0.032 0.030 -0.027 0.056 +1.00 +RAC (Tw = 12) +-0.097 -0.007 -0.063 -0.124 0.036 0.132 0.019 -0.054 0.031 +1.00 +VR (Tw = 12) +0.087 0.029 0.011 0.046 0.031 0.043 0.062 -0.003 0.085 +1.00 +RV (Tw = 12) +-0.171 -0.167 0.028 -0.241 -0.118 -0.267 -0.074 -0.227 -0.258 +1.00 +MNRV2RV (Tw = 12) +0.034 0.138 -0.050 0.043 0.161 0.217 0.050 0.136 0.091 +1.00 +MRV2RV (Tw = 12) +0.163 0.228 0.170 0.190 0.376 0.309 0.131 0.376 0.257 +1.00 +total number of selected variables +39 +42 +39 +42 +41 +42 +42 +42 +41 +19 + +Table 8: Empirical estimates of the regression coefficients for the scale parameter obtained with the MLE. +Predictors +Stocks +AXP BA +GE +HD +IBM +JNJ +JPM KO +XOM selection per stock +TV +-0.033 -0.138 0.058 -0.064 0.114 -0.055 -0.075 0.314 0.253 +1.00 +TQ +0.094 0.325 -0.033 0.100 0.012 0.090 0.166 -0.190 -0.110 +1.00 +AM +-0.010 0.018 0.046 0.039 0.021 0.006 0.021 0.029 0.027 +1.00 +TVV +-0.045 0.157 -0.127 -0.018 0.051 -0.035 -0.224 -0.378 -0.043 +1.00 +EAM +-0.026 -0.039 -0.007 -0.055 -0.060 -0.018 -0.029 -0.008 -0.056 +1.00 +TQV +0.045 -0.312 0.142 0.011 -0.241 0.048 0.219 0.279 -0.148 +1.00 +MRV +0.028 0.024 0.021 0.018 -0.007 0.019 0.055 0.023 -0.015 +1.00 +MNRV +0.099 0.137 0.060 0.070 0.199 0.060 0.061 0.094 0.149 +1.00 +dur +-0.002 -0.031 0.000 -0.005 0.046 -0.007 -0.001 -0.005 0.003 +1.00 +AMI (Tw = 2) +0.030 0.009 -0.012 0.000 0.045 0.019 0.005 -0.045 0.043 +1.00 +VR (Tw = 2) +-0.005 -0.009 0.000 -0.007 0.014 0.001 0.020 -0.014 -0.007 +1.00 +RV (Tw = 2) +-0.037 -0.050 0.011 -0.006 -0.068 -0.007 -0.071 0.031 -0.015 +1.00 +MNRV2RV (Tw = 2) +0.010 0.001 0.001 -0.001 0.026 -0.010 -0.033 -0.015 -0.007 +1.00 +MRV2RV (Tw = 2) +-0.024 0.009 0.001 0.008 0.023 0.015 0.017 0.039 -0.007 +1.00 +AMI (Tw = 6) +0.009 -0.016 -0.009 -0.009 -0.047 -0.032 0.006 0.002 -0.106 +1.00 +Roll (Tw = 6) +0.069 0.014 -0.002 0.040 0.019 -0.087 0.020 0.050 -0.001 +1.00 +Roll− (Tw = 6) +-0.062 -0.013 -0.046 -0.090 -0.024 -0.022 -0.066 0.070 -0.009 +1.00 +RollMod (Tw = 6) +-0.114 0.022 -0.072 -0.027 -0.037 0.019 -0.058 0.040 -0.041 +1.00 +RollMod− (Tw = 6) +0.127 -0.016 0.139 0.096 0.049 0.113 0.125 0.058 0.068 +1.00 +TVV (Tw = 6) +0.016 0.045 -0.025 0.070 -0.033 0.055 -0.003 0.031 -0.011 +1.00 +TQV (Tw = 6) +-0.031 -0.041 0.024 -0.078 0.016 -0.074 -0.010 -0.027 -0.002 +1.00 +RTVV (Tw = 6) +-0.011 -0.010 -0.005 -0.004 0.014 0.006 -0.016 -0.003 -0.009 +1.00 +RTQV (Tw = 6) +-0.006 0.012 -0.005 0.004 0.013 0.008 0.005 -0.001 0.000 +1.00 +RAC (Tw = 6) +0.005 -0.010 0.008 -0.013 -0.012 -0.008 -0.004 -0.011 -0.002 +1.00 +VR (Tw = 6) +0.050 0.088 0.138 0.035 0.080 0.046 0.012 0.017 0.038 +1.00 +RV (Tw = 6) +0.020 -0.009 0.026 0.029 0.006 0.016 0.012 -0.064 0.031 +1.00 +MNRV2RV (Tw = 6) +-0.163 -0.190 -0.076 -0.080 -0.077 -0.114 -0.064 -0.112 -0.243 +1.00 +MRV2RV (Tw = 6) +0.081 0.054 0.014 0.010 -0.031 0.035 0.025 0.068 0.153 +1.00 +AMI (Tw = 12) +0.014 0.038 0.014 0.016 0.078 0.018 0.013 0.017 0.069 +1.00 +Roll (Tw = 12) +0.047 -0.052 0.097 0.041 -0.017 0.032 0.089 0.069 0.010 +1.00 +Roll− (Tw = 12) +-0.063 0.033 -0.071 -0.033 0.016 0.017 -0.114 0.063 -0.050 +1.00 +RollMod (Tw = 12) +-0.068 0.041 -0.077 -0.058 -0.023 -0.041 -0.094 0.063 -0.037 +1.00 +RollMod− (Tw = 12) +0.086 -0.024 0.047 0.051 0.043 -0.009 0.118 0.063 0.086 +1.00 +TVV (Tw = 12) +0.010 -0.092 0.006 -0.037 0.032 0.100 0.008 -0.029 0.001 +1.00 +TQV (Tw = 12) +-0.025 0.046 -0.015 0.031 -0.034 -0.095 -0.022 0.015 -0.011 +1.00 +RTVV (Tw = 12) +0.009 0.022 0.005 -0.001 0.007 -0.007 -0.002 0.003 0.013 +1.00 +RTQV (Tw = 12) +0.010 0.044 -0.001 0.008 -0.003 -0.003 0.016 0.011 0.012 +1.00 +RAC (Tw = 12) +0.000 0.016 -0.005 0.008 0.003 0.012 0.002 0.013 0.009 +1.00 +VR (Tw = 12) +-0.072 -0.205 -0.026 -0.029 -0.056 -0.063 -0.037 -0.022 -0.070 +1.00 +RV (Tw = 12) +0.038 0.041 0.001 0.024 0.066 0.054 0.022 0.097 0.042 +1.00 +MNRV2RV (Tw = 12) +0.183 0.149 0.083 0.073 -0.033 0.194 0.121 0.131 0.211 +1.00 +MRV2RV (Tw = 12) +-0.045 -0.019 -0.007 0.016 0.091 -0.060 -0.028 -0.071 -0.074 +1.00 +log(σt−1) +0.752 0.480 0.736 0.674 0.445 0.739 0.761 0.645 0.545 +1.00 +total number of selected variables +43 +43 +43 +43 +43 +43 +43 +43 +43 +20 + +period, i.e., we test whether { �F(yt|yt > 0) = GPD(yt; xt−1, �β)} follows a standard uniform distribution. The +p-values of the K-S tests in Table 9 indicate that we reject the regression model on three stocks out of nine at +the 1% significance level. +Table 9: Out-of-sample VaR Coverage Rates and p-values for the K-S Test. +Stock Names +VaR risk level +0.9 +0.91 +0.92 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 0.999 0.9999 K-S Test p-values +AXP +0.8999 0.9111 0.9224 0.9314 0.9414 0.9520 0.9631 0.9729 0.9827 0.9913 0.9986 0.9996 +0.208 +BA +0.9000 0.9098 0.9210 0.9309 0.9417 0.9522 0.9630 0.9738 0.9836 0.9930 0.9985 0.9996 +0.0431 +GE +0.8999 0.9171 0.9254 0.9376 0.9475 0.9622 0.9709 0.9811 0.9889 0.9943 0.9989 0.9999 +2.68 × 10−12 +HD +0.9001 0.9097 0.9217 0.9339 0.9453 0.9551 0.9652 0.9756 0.9840 0.9934 0.9990 0.9998 +0.0023 +IBM +0.9003 0.9094 0.9204 0.9312 0.9414 0.9509 0.9614 0.9702 0.9809 0.9905 0.9980 0.9993 +0.8954 +JNJ +0.9000 0.9083 0.9195 0.9284 0.9382 0.9480 0.9571 0.9682 0.9786 0.9888 0.9977 0.9995 +0.3322 +JPM +0.9000 0.9094 0.9176 0.9294 0.9385 0.9500 0.9612 0.9711 0.9815 0.9904 0.9982 0.9997 +0.3993 +KO +0.8996 0.9109 0.9180 0.9273 0.9380 0.9489 0.9594 0.9709 0.9813 0.9909 0.9980 0.9994 +0.3786 +XOM +0.8988 0.9056 0.9136 0.9218 0.9303 0.9396 0.9502 0.9630 0.9742 0.9863 0.9972 0.9992 +6.32 × 10−8 +7 +Conclusion +This paper proposes a novel extreme value regression framework to study the dynamics of high-frequency tail +risk. The proposed model allows for both stationary and local unit-root predictors to capture the persistence +of high-frequency extreme losses. We propose a two-step regularized approach to perform automatic variable +selection, and establish the oracle property of the corresponding ALMLE in selecting stationary and local +unit-root predictors. We use the proposed approach to investigate the predictive content of 42 liquidity and +volatility indicators on the distribution of extreme losses for nine large liquid U.S. stocks. Our variable selection +procedure reveals that the severity of tail risk is strongly associated to low price impact in periods of high +volatility of liquidity and volatility of volatility. These findings can contribute to timely alert high-frequency +traders of rising risk levels and facilitate improvements of their algorithmic trading practices for financial risk +management. 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Journal of the American Statistical Association, +101(476):1418–1429. +23 + +Appendices +A +Proofs +A.1 +MLE - Gradient and Hessian matrix of the loglikelihood function L(·) +The gradient function of L(β; {yt}, {zt−1}) is given by +∂L(β; {yt}, {zt−1}) +∂β += +T +� +t=1 +ψt(β), +(40) +ψt(β) := 1{yt > 0} +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Xt−1 +� +���������������� +gk,t(β) +... +gk,t(β) +gσ,t(β) +gσ,t(β) +... +gσ,t(β) +gσ,t(β) +� +���������������� ++ gσ,t(β) +t−1 +� +i=0 +� +���������������� +0 +... +0 +βi +2,p+1 +βi +2,p+1z1,t−1−i +... +βi +2,p+1zp,t−1−i +�p +j=1 β2,j i βi−1 +2,p+1 zj,t−1−i +� +���������������� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(41) +Xt := diag(xt), +xt := [1, z′ +t, 1, z′ +t, log(σt(β))]′, +gk,t(β):= +� +1 +kt(β) − 2 +� +log +� +1 + kt(β) +yt +σt(β) +� ++ +� +1 +kt(β) − 1 − 2kt(β) +� � +1 +1+kt(β) +yt +σt(β) − 1 +� +, +gσ,t(β) := +1 +kt(β) +� +1 − +1 +1 + kt(β) +yt +σt(β) +� +− +1 +1 + kt(β) +yt +σt(β) +, +gA,i,t := +� +0, . . . , 0, βi +2,p+1, βi +2,p+1z1,t−1−i, . . . , βi +2,p+1zp,t−1−i, �p +j=1 β2,j i βi−1 +2,p+1 zj,t−1−i +�′ +, +(42) +where diag(xt) denotes the square diagonal matrix with the elements of vector xt on the main diagonal. Using +the gradient information in (42), we give the Hessian matrix of L(β; {yt}, {zt−1}) below. +HL(β) := ∂2L(β; {yt}, {zt−1}) +∂β∂β′ += +T +� +t=1 +∂ψt(β) +∂β′ +∂ψt(β) +∂β′ += 1{yt > 0} +� +Xt−1Hg + +t−1 +� +i=0 +diag (gA,i,t) +� +I(2p+3)×1 +� ∂gσ,t(β) +∂β′ +� ++ gσ,t(β)HA +� +, +Hg := +� +����������� +∂gk,t(β) +∂β′ +... +∂gk,t(β) +∂β′ +∂gσ,t(β) +∂β′ +... +∂gσ,t(β) +∂β′ +� +����������� +, +HA := �t−1 +i=0 +� +���������������� +0 . . . 0 +0 +0 +. . . +0 +0 +0 . . . 0 +... +... +... +... +0 . . . 0 +0 +0 +. . . +0 +0 +0 . . . 0 +0 +0 +. . . +0 +i βi−1 +2,p+1 +0 . . . 0 +0 +0 +. . . +0 +i βi−1 +2,p+1z1,t−1−i +0 . . . 0 +... +... +... +... +0 . . . 0 +0 +0 +. . . +0 +i βi−1 +2,p+1zp,t−1−i +0 . . . 0 i βi−1 +2,p+1 i βi−1 +2,p+1 z1,t . . . i βi−1 +2,p+1 zp,t +�p +j=1 β2,j i (i − 1) βi−2 +2,p+1 zj,t−i−i +� +���������������� +(43) +∂gk,t(β) +∂β′ += +� +∂gk,t(β) +∂logit(kt) . . . +∂gk,t(β) +∂logit(kt) +∂gk,t(β) +∂ log(σt) . . . ∂gk,t(β) +∂ log(σt) +� +diag(Xt−1) +∂gσ,t(β) +∂β′ += +� +∂gσ,t(β) +∂logit(kt) . . . +∂gσ,t(β) +∂logit(kt) +∂gσ,t(β) +∂ log(σt) . . . ∂gσ,t(β) +∂ log(σt) +� +diag(Xt−1) +(44) +24 + +∂gk,t(β) +∂logit(kt) = (2 − 1 +kt +) log(1 + kt +yt +σt +) + (1 + kt +yt +σt +)−1((1 − 2 kt)2 − ( 1 +kt +− 2 kt)(1 − 2 kt)) ++ (1 + kt +yt +σt +)−2 yt +σt +(2k2 +t + kt − 1) +∂gk,t(β) +∂ log(σt) = kt +yt +σt +((1 − 1 +kt +)(1 + kt +yt +σt +)−1 + ( 1 +kt +− 1 − 2 kt)(1 + kt +yt +σt +)−2) +∂gσ,t(β) +∂logit(kt) = (1 + kt +yt +σt +)−2 yt +σt +(1 + kt)(1 − 2 kt) + ((1 + kt +yt +σt +)−1 − 1)(1 − 2 kt) 1 +kt +∂gσ,t(β) +∂ log(σt) = − yt +σt +(1 + kt +yt +σt +)−2(1 + kt), +(45) +where � denotes the Kronecker product operator, and we suppress the coefficients in kt := kt(β), σt := σt(β) +and the conditional variables in ψt(β) := ψ(β; Lt−1, Zt−1, Lt−2, Zt−2, . . .) for ease of the notations. We also +denote specifically that ko +t := kt(βo) and σo +t := σt(βo) to ease the notation. +A.2 +MLE - Proofs +Proposition 1. Under Assumptions M.1, M.2, M.3, M.5(1) and M.6, for any ε > 0 and any interior �β in +Θ, it holds that there exists an δ > 0 such that +sup +∥β− � +β∥<δ +���ψt(β) − ψt( �β) +��� < ε, +(46) +where ∥·∥ is the Euclidean norm. +Proof. This proposition claims that ψt(·) is uniformly continuous in β ∈ Θ. To prove this proposition, we show +that ψt(·) is continuous and tight in β ∈ Θ below. +First, viewing the formula of ψt(·) in Appendix A.1, we know that ψt(·) is a composition of continuous functions +in β and hence is continuous in β. Secondly, by Assumptions M.1, M.3, M.5(1) and M.6, we know that β and +zt−1 are bounded in probability and thereby kt(·) and σt(·) are bounded and their lower bounds are above +zero in probability. It follows that we get that gσ,t(·) and gA,i,t(·) are also bounded in probability. Additionally, +under Assumption M.2, we know that |yt| is bounded in probability. By Jensen’s inequality, we obtain that +gk,t(·) is bounded in probability. Therefore, we obtain that ψt(·) is bounded in probability in β ∈ Θ and thereby +accomplish this proof. +Proposition 2. Under Assumptions M.1, M.2, M.3, M.5(1) and M.6, it holds that {E[ψt(β)], β ∈ Θ} has +unique zero at β = βo∗ and for any ϵ > 0, +lim +T →∞ P +������ +1 +T +T +� +t=1 +ψt(βo∗) +����� > ϵ +� += 0, +(47) +Proof. First, the equality (47) claims that +1 +T +�T +t=1 ψt(βo∗) converges to zero in probability. Let us prove the +equality (47) using the conditions in Theorem 1 of Cs¨org˝o (1968). +Under Assumptions M.1 and M.2, we take the expectation with respect to the true conditional probability +function of yt and obtain that +E [ψt(βo∗)|Ft−1] = 0, +for t = 1, . . . , T. +(48) +We also have that +T +� +t=1 +1 +t2 E [ψt(βo∗)] ≤ +T +� +t=1 +1 +t2 M 12p+3 +< ∞, +(49) +25 + +where M ∈ R is a large finite number and 12p+3 denotes a vector of (2p + 3) ones; the second last inequality is +obtained by the tightness of ψt(·) in Proposition 1; the last inequality is obtained by �T +t=1 +1 +t2 < ∞. Therefore, +by applying Theorem 1 of Cs¨org˝o (1968) we conclude that 1 +T +�T +t=1 ψt(βo∗) converges to zero in probability. +Secondly, we are going to show the uniqueness of βo∗ in Θ such that E[ψt(βo∗)] = 0 by contradiction. Suppose +there is a β ∈ Θ and β ̸= βo∗ such that +E[ψt(β)] = 0. +(50) +On the other hand, by the mean value theorem it holds that +ψt(β) − ψt(βo∗) = HL(β) (β − βo∗) , +(51) +where β is between β and βo∗. Since HL(·) is positive definite in Θ almost surely by Assumption M.6 and +β ̸= βo∗, we obtain that ψt(β) ̸= ψt(βo∗) almost surely and contradict (50). Hence, we conclude the uniqueness +of βo∗ in Θ. +Proof of Theorem 1 +Proof. First, let us prove that �βmle is in Θ. By Proposition 2, we can get that for any ϵ ∈ R2p+3 and ϵ > 0 , +there exists TN such that for T > TN we have +����� +1 +T +T +� +t=1 +ψt(βo∗) +����� < ϵ. +(52) +Under Assumptions M.4 and M.6, we know that βo∗ is an interior point in Θ. Since βo∗ is an interior point in +Θ and 1 +T +�T +t=1 ψt(·) is uniformly continuous in Θ by Proposition 1, there exists δ > 0 and a ball B(βo∗, δ) := +{β ∈ Θ| ∥βo∗ − β∥ < δ} such that +����� +1 +T +T +� +t=1 +ψt(βo∗) − 1 +T +T +� +t=1 +ψt(β) +����� < 1 +2ϵ. +(53) +Moreover, there exist β1, β2 ∈ B(βo∗, δ) such that if +��� 1 +T +�T +t=1 ψt(βo∗) +��� ̸= 0, then +0 < 1 +T +T +� +t=1 +ψt(βo∗) − 1 +T +T +� +t=1 +ψt(β1) ≤ 1 +2 +����� +1 +T +T +� +t=1 +ψt(βo∗) +����� +(54) +−1 +2 +����� +1 +T +T +� +t=1 +ψt(βo∗) +����� ≤ 1 +T +T +� +t=1 +ψt(βo∗) − 1 +T +T +� +t=1 +ψt(β2) < 0 +(55) +which results in +1 +T +T +� +t=1 +ψt(βo∗) − 1 +2 +����� +1 +T +T +� +t=1 +ψt(βo∗) +����� ≤ 1 +T +T +� +t=1 +ψt(β1) < 1 +T +T +� +t=1 +ψt(βo∗) +(56) +1 +T +T +� +t=1 +ψt(βo∗) < 1 +T +T +� +t=1 +ψt(β2) ≤ 1 +T +T +� +t=1 +ψt(βo∗) + 1 +2 +����� +1 +T +T +� +t=1 +ψt(βo∗) +����� . +(57) +Hence we find +��� 1 +T +�T +t=1 ψt(β1) +��� < +��� 1 +T +�T +t=1 ψt(βo∗) +��� < ϵ or +��� 1 +T +�T +t=1 ψt(β2) +��� < +��� 1 +T +�T +t=1 ψt(βo∗) +��� < ϵ. +Continue this process, and we can find a sequence of points in Θ has decreasing values of +��� 1 +T +�T +t=1 ψt(·) +���. +There exists a subsequence of the resulted point sequence and the limit of the subsequence is �βmle in Θ with +��� 1 +T +�T +t=1 ψt( �βmle) +��� = 0. Since limT →∞ 1 +T +�T +t=1 ψt(βo∗) = 0, then we obtain limT →∞ �βmle = βo∗ and conclude +this proof. +26 + +Proof of Theorem 2 +Proof. Apply the Taylor expansion of �T +t=1 ψt( �βmle) at βo∗ and the mean value theorem, we obtain +1 +√ +T +T +� +t=1 +ψt( �βmle) = +1 +√ +T +T +� +t=1 +ψt(βo∗) + 1 +T HL(β) +√ +T +� +�βmle − βo∗� +, +(58) +where β is between �βmle and βo∗. Since �T +t=1 ψt( �βmle) = 0, the above expansion results in +� 1 +T HL(β) +�−1 +1 +√ +T +T +� +t=1 +ψt(βo∗) = +√ +T +� +�βmle − βo∗� +. +(59) +1 +T HL(·) is uniformly continuous in Θ, which can be proved analogously to the proof of Proposition 1 with knowing +y2 +t ∈ Op(1) thanks to 0 < kt(βo∗) < 0.5. By the continuous mapping theorem and knowing limT →∞ �βmle = βo∗ +from Theorem 1, we obtain that +lim +T →∞ +� 1 +T HL(β) +�−1 += +� 1 +T HL(βo∗) +�−1 +. +From Assumptions M.7 and M.8 we know that +� +� +� +� +� +� +� +� +� +1 +√ +T +T +� +t=1 +ψt(βo∗) +D∼ Sψ +1 +T HL(βo∗) +D∼ ΩH +(60) +and thus by Slutsky’s theorem, we obtain that +√ +T +� +βo∗ − �βmle� D∼ Ω−1 +H Sψ. +(61) +A.3 +ALMLE - Proofs +Proof of Theorem 3 +Proof. We prove this theorem by contradiction. In the proof below, we show that truly inactive predictors have +a non-zero probability to get selected since λk,T , λσ,T ∈ O(T +1 +2 ) if there is no �βal(λk,T , λσ,T ) with λk,T , λσ,T ∈ +O(T +1 +2 ) such that the condition (22) is met. +We set wk,i, wσ,j, e.g. using the MLE in section 3, such that +√ +T( +1 +wk,i − β∗o +1,i) = Op(1) and +√ +T( +1 +wσ,j − β∗o +2,j) = +Op(1), for i = 1, . . . , p, j = 1, . . . , p + 1. +Let us go over the tuning parameter grid {(λk,T , λσ,T ) : λk,T ∈ Sλk,T , λσ,T ∈ Sλσ,T , }. λk,T,max and λσ,T,max +are chosen large enough such that �βal(λk,T,max, λσ,T,max) = 0, i.e., no predictors get selected. By the Karush- +Kuhn-Tucker (KKT) optimality condition and with (λk,T,max, λσ,T,max), we have +� +� +� +� +� +� +� +� +� +� +� +����� +∂L +∂ �βal +1,i +����� ≤ λk,T,max wk,i, +(1, i) ∈ Ak ∪ Ac +k, +����� +∂L +∂ �βal +2,j +����� ≤ λσ,T,max wσ,j, +(2, j) ∈ Aσ ∪ Ac +σ, +(62) +where we denote Ak := +� +(1, i) : i ≥ 1, β∗o +1,i ̸= 0 +� +, Ac +k := +� +(1, i) : i ≥ 1, β∗o +1,i = 0 +� +, Aσ := +� +(2, j) : j ≥ 1, β∗o +2,j ̸= 0 +� +, +and Ac +σ := +� +(2, j) : j ≥ 1, β∗o +2,j = 0 +� +. We rewrite +∂L +∂ � +βal(λk,T,max,λσ,T,max) using the mean value theorem because +27 + +L(·; {yt}, {z∗ +t }) is twice continuously differentiable, and substitute this rewriting into the above KKT condition +and get +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +������ +∂L +∂β∗o +1,i +− +p +� +ig=0 +∂2L +∂ �βal +1,i∂ �βal′ +1,ig +β +(2 nλk nλσ ) +1,ig +− +p+1 +� +jg=0 +∂2L +∂ �βal +1,i∂ �βal′ +2,jg +β +(2 nλk nλσ ) +2,jg +������ +≤λk,T,max wk,i, +(1, i) ∈ Ak ∪ Ac +k, +������ +∂L +∂β∗o +2,j +− +p +� +ig=0 +∂2L +∂ �βal +2,j∂ �βal′ +1,ig +β +(2 nλk nλσ −1) +1,ig +− +p+1 +� +jg=0 +∂2L +∂ �βal +2,j∂ �βal′ +2,jg +β +(2 nλk nλσ −1) +2,jg +������ +≤λσ,T,max wσ,j, +(2, j) ∈ Aσ ∪ Ac +σ, +(63) +where β +(2 nλk nλσ ) +1,ig +, β +(2 nλk nλσ −1) +1,ig +∈ +� +β∗o +1,ig, �βal +1,ig(λk,T,max, λσ,T,max) +� +, and β +(2 nλk nλσ ) +1,jg +, β +(2 nλk nλσ −1) +1,jg +∈ [β∗o +2,jg, +�βal +2,jg(λk,T,max, λσ,T,max)]. In fact, we can find a pair of (λk,T,max, λσ,T,max) in O(T). +Lower the tuning parameters from (λk,T,max, λσ,T,max), and �βal is able to screen variables. Truly inactively +predictors always meet the inequality of the KKT condition with λk,T +√ +T → ∞ and λσ,T +√ +T → ∞, thereby not being +selected. Coming to λk,T , λσ,T ∈ O(T +1 +2 ), we get that λk,T wk,i = Op(T +1 +2 ) for (1, i) ∈ Ak; λk,T wk,i = Op(T) for +(1, i) ∈ Ac +k; λσ,T wσ,j = Op(T +1 +2 ) for (2, j) ∈ Aσ; λσ,T wσ,i = Op(T) for (2, i) ∈ Ac +σ. If with any λk,T ∈ O(T +1 +2 ) +and λσ,T ∈ O(T +1 +2 ), there is no �βal(λk,T , λσ,T ) such that the condition (22) is met, then there exists (1, ia) ∈ Ak +or (2, ja) ∈ Aσ such that +����� +∂L +∂ �βal +1,ia +����� ≤ λk,T wk,ia +(64) +or +����� +∂L +∂ �βal +2,ja +����� ≤ λσ,T wσ,ja +(65) +but �βal +1,ia = 0 or �βal +2,ja = 0 correspondingly for any λk,T , λσ,T ∈ O(T +1 +2 ). +Under the condition (22) is broken, if +���� +∂L +∂ �βal +1,ia +���� ≤ λk,T wk,ia with �βal +1,ia = 0 and any λk,T , λσ,T ∈ O(T +1 +2 ), we +get that λk,T wk,i = Op(T +1 +2 ) for (1, i) ∈ Ak and λk,T wk,i = Op(T) for (1, i) ∈ Ac +k. Then each truly inactively +predictor but correlated with the ia-th predictor of the shape model gains a non-zero probability to get selected +due to +P +������ +∂L +∂ �βal +1,i +����� ≥ λk,T wk,i, i ∈ Ac +k +����� +�βal +1,ia = 0, λk,T ∈ O(T +1 +2 ), λσ,T ∈ O(T +1 +2 ) +� +̸= 0. +(66) +The above probability is obtained easily by rewriting +���� +∂L +∂ �βal +1,i +���� using the mean value theorem as done in (63) and +obtaining that +���� +∂L +∂ �βal +1,i +���� ∈ Op(T). For each �βal +1,i ̸= 0, i ∈ Ac +k with λk,T ∈ O(T +1 +2 ), the bias of �βal +1,i is Op(1). Lowering +λk,T , it follows that �βal +1,i = Op(T −γ) meets the equality of the KKT condition with λk,T ∈ O(T +1 +2 −γ) and remains +selected for γ ∈ [0, 1 +2]. +The same reasoning applies if +���� +∂L +∂ �βal +2,ja +���� ≤ λσ,T wσ,ja with �βal +1,ja = 0 and any λk,T , λσ,T ∈ O(T +1 +2 ), we get that +each truly inactively predictor but correlated with the ja-th predictor of the scale model then gains a non-zero +probability to get selected, i.e., +P +������ +∂L +∂ �βal +2,j +����� ≥ λσ,T wσ,j, j ∈ Ac +σ +����� +�βal +2,ja = 0 λk,T ∈ O(T +1 +2 ), λσ,T ∈ O(T +1 +2 ) +� +̸= 0. +(67) +Therefore, the model selection consistency cannot be achieved if the condition (22) is not met, and we finish +the proof. +Proof of Theorem 4 +28 + +Proof. In this proof, we first prove the selection consistency of �βk,al on truly active predictors of the shape +model, i.e., +lim +T →∞ P +� +�βk,al +1,i +̸= 0, ∀(1, i) ∈ Ak +� += 1 +(68) +which is also equivalent to limT →∞ P +� +Ak,al +k,T ⊇ Ak +� += 1 with Ak,al +k,T := {(1, i) : �βk,al +1,i +̸= 0}. It follows that this +over-selection possibility for the shape model due to �βk,al is proved to be curbed to zero by �βtal, and the oracle +property of �βtal is obtained lastly. +Firstly, we rewrite the objective function to obtain �βk,al as follows: according to Assumption L2, there exists +βk,o, and +V (k)(ν(k)) = −L +� +βk,o + ν(k) +√ +T +; {yt}, {z∗ +t } +� ++ λk,T +p +� +i=1 +�wk,i +�����βk,o +1,i + +ν(k) +1,i +√ +T +����� , +(69) +where ν(k) := [ν(k) +1,0, . . . , ν(k) +1,p, ν(k) +2,0, 0, . . . , 0]′ ∈ Rp+2 ×0(p+1). We obtain that �ν(k) = +arg min +ν(k)∈Rp+2×0(p+1) +V (k)(ν(k))− +V (k)(0) such that equivalently +�βk,al = βk,o + �ν(k) +√ +T +. +(70) +Specifically, +V (k)(ν(k)) − V (k)(0) += − +� +L(βk,o + ν(k) +√ +T +; {yt}, {z∗ +t }) − L +� +βk,o; {yt}, {z∗ +t } +�� ++ λk,T +√ +T +p +� +i=1 +�wk,i +√ +T +������βk,o +1,i + +ν(k) +1,i +√ +T +����� − +���βk,o +1,i +��� +� +. +(71) +According to Assumption L2 and Theorem 2, we have that +√ +T(�βk,mle +1,i +− βk,o +1,i ) = Op(1) and +√ +T(�βmle +1,i − β∗o +1,i) = +Op(1). Considering that +lim +T →∞ +√ +T +������βk,o +1,i + +ν(k) +1,i +√ +T +����� − +���βk,o +1,i +��� +� += +� +� +� +ν(k) +1,i sgn(βk,o +1,i ), +if βk,o +1,i ̸= 0, +���ν(k) +1,i +��� , +if βk,o +1,i = 0, +(72) +and under Assumption L1 we obtain that +λk,T +√ +T +�wk,i +√ +T +������βk,o +1,i + +ν(k) +1,i +√ +T +����� − +���βk,o +1,i +��� +� += +� +� +� +� +� +� +� +� +� +Op(T −γ1) ν(k) +1,i sgn(βk,o +1,i ), +if β∗o +1,i ̸= 0, +Op(T +1 +2 −γ1) ν(k) +1,i sgn(βk,o +1,i ), +if β∗o +1,i = 0 but βk,o +1,i ̸= 0, +Op(T 1−γ1) +���ν(k) +1,i +��� , +if β∗o +1,i = 0 and βk,o +1,i = 0. +(73) +Also, we know that +− +� +L(βk,o + ν(k) +√ +T +; {yt}, {z∗ +t }) − L +� +βk,o; {yt}, {z∗ +t } +�� += Op(1) max{(ν(k) +1,i )2} +(74) +by Taylor’s expansion and the tightness of 1 +T HL(·) in Θ as shown in the proof of Theorem 2. Substitute Eq. (73) +back into Eq. (71), and by Slutsky’s theorem we obtain that +V (k)(ν(k) +1,i ) − V (k)(0) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Op(T), +if +ν(k) +1,i +√ +T += Op(1), ∀(1, i) ∈ Ak,al +k +and ν(k) +1,i = Op(1), ∀(1, i) /∈ Ak,al +k +; +Op(T 1−2γ), +if +ν(k) +1,i = Op(T +1 +2 −γ), ∀(1, i) ∈ Ak,al +k +with 0 < 2γ < γ1 and ν(k) +1,i = Op(1), ∀(1, i) /∈ Ak,al +k +; +Op(T 1−γ1), +if +ν(k) +1,i = Op(T +1 +2 −γ), ∀(1, i) ∈ Ak,al +k +with γ1 +2 ≤ γ ≤ 1 +2 and ν(k) +1,i = Op(1), ∀(1, i) /∈ Ak,al +k +; +Op(T 1−2γ), +if +ν(k) +1,i = Op(T +1 +2 −γ), ∀(1, i) ∈ Ak,al +k +with 0 < γ < γ1 and ν(k) +1,i = 0, ∀(1, i) /∈ Ak,al +k +; +Op(T 1−γ−γ1), +if +ν(k) +1,i = Op(T +1 +2 −γ), ∀(1, i) ∈ Ak,al +k +with γ1 ≤ γ ≤ 1 +2 and ν(k) +1,i = 0, ∀(1, i) /∈ Ak,al +k +, +(75) +29 + +where Ak,al +k +:= {(1, i) : βk,o +1,i ̸= 0}. Therefore, we have that +�ν(k) +1,i = Op(1), ∀(1, i) ∈ Ak,al +k +and �ν(k) +1,i = 0, ∀(1, i) /∈ Ak,al +k +, +(76) +minimizing V (k)(ν(k)) − V (k)(0), and hence that +lim +T →∞ P +� +�βk,al +1,i += βk,o +1,i + +�ν(k) +1,i +√ +T +̸= 0, ∀(1, i) ∈ Ak,al +k +� += P +� +βk,o +1,i ̸= 0, ∀(1, i) ∈ Ak,al +k +� += 1. +(77) +Thus we obtain limT →∞ P +� +�βk,al +1,i +̸= 0, ∀(1, i) ∈ Ak +� += 1. +Secondly, we show the asymptotic behaviour of �βtal. Write β = βo∗+ ν +√ +T for β ∈ Θ with ν := [ν1,0, . . . , ν1,p, ν2,0, +. . . , ν2,p+1]′ ∈ R2p+3, and rewrite the objective function to obtain �βtal as follows: +V (ν) = −L +� +βo∗ + ν +√ +T +; {yt}, {z∗ +t } +� ++ �λk,T +p +� +i=1 +�wk,i +����βo∗ +1,i + ν1,i +√ +T +���� + λσ,T +p+1 +� +j=1 +�wσ,j +����βo∗ +2,j + ν2,j +√ +T +���� . +(78) +We obtain �ν := arg min +ν∈R2p+3 V (ν) − V (0) such that +�βtal = β∗o + +�ν +√ +T +. +(79) +Specifically, +V (ν) − V (0) += − +� +L(βo∗ + ν +√ +T +; {yt}, {z∗ +t }) − L (βo∗; {yt}, {z∗ +t }) +� ++ +�λk,T +√ +T +p +� +i=1 +�wk,i +√ +T +�����βo∗ +1,i + ν1,i +√ +T +���� − +��βo∗ +1,i +�� +� ++ λσ,T +√ +T +p+1 +� +j=1 +�wσ,j +√ +T +�����βo∗ +2,j + ν2,j +√ +T +���� − +��βo∗ +2,j +�� +� +. +(80) +Analogously to (75), we can also get that +lim +T →∞ V (ν) − V (0) = +� +� +� +� +� +� +� +Op(1), +if ν1,i = Op(1), ν2,j = Op(1), for ∀(1, i) ∈ Ak, ∀(2, j) ∈ Aσ, +and ν1,i = 0, ν2,j = 0, for ∀(1, i) /∈ Ak, ∀(2, j) /∈ Aσ, +∞, +otherwise of +ν = Op(1). +(81) +Therefore, minimizing V (ν)−V (0) is equivalent to minimizing − +� +L(βo∗ + +ν +√ +T ; {yt}, {z∗ +t }) − L (βo∗; {yt}, {z∗ +t }) +� +with ν1,i = 0, ν2,j = 0 for any (1, i) /∈ Ak, (2, j) /∈ Aσ, which leads to +√ +T +� +�βtal − βo∗� += �ν = +arg min +ν∈R2p+3∩{ν1,i=0.ν2,j=0,(1,i)/∈Ak,(2,j)/∈Aσ} +V (ν) − V (0) +(82) +as T → ∞ and thereby we obtain the asymptotic behavior of �ν by Theorem 2 with the restriction of +limT →∞ �ν1,i = 0, limT →∞ �ν2,j = 0 for any (1, i) /∈ Ak, (2, j) /∈ Aσ. Moreover, +lim +T →∞ P +� +Atal +T += A +� += lim +T →∞ P +� +{�βtal +1,i = β∗o +1,i + �ν1,i +√ +T +̸= 0, �βtal +2,j = β∗o +2,j + �ν2,j +√ +T +̸= 0, ∀(1, i) ∈ Ak, ∀(2, j) ∈ Aσ} ∩ {�βtal +1,i = 0, �βtal +2,j = 0, ∀(1, i) /∈ Ak, ∀(2, j) /∈ Aσ} +� += P +� +{β∗o +1,i ̸= 0, β∗o +2,j ̸= 0, ∀(1, i) ∈ Ak, ∀(2, j) ∈ Aσ} ∩ {β∗o +1,i = 0, β∗o +2,j = 0, ∀(1, i) /∈ Ak, ∀(2, j) /∈ Aσ} +� += 1 +(83) +which concludes that �βtal is model selection consistent. Together with the limiting distribution of +√ +T +� +�βtal − βo∗� +, +we conclude that �βtal has the oracle property. +30 + diff --git a/ddAzT4oBgHgl3EQfZ_yv/content/tmp_files/load_file.txt b/ddAzT4oBgHgl3EQfZ_yv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96188261ab8eadc406b82751b33af8e7fd9f7e16 --- /dev/null +++ b/ddAzT4oBgHgl3EQfZ_yv/content/tmp_files/load_file.txt @@ -0,0 +1,3347 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf,len=3346 +page_content='Measuring tail risk at high-frequency: An L1-regularized extreme value regression approach with unit-root predictors Julien Hambuckers†, Li Sun†⋆ and Luca Trapin‡ † University of Li`ege - HEC Li`ege, Belgium ‡ University of Bologna, Italy Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We study tail risk dynamics in high-frequency financial markets and their connection with trading activity and market uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We introduce a dynamic extreme value regression model ac- commodating both stationary and local unit-root predictors to appropriately capture the time-varying behaviour of the distribution of high-frequency extreme losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' To characterize trading activity and mar- ket uncertainty, we consider several volatility and liquidity predictors, and propose a two-step adaptive L1-regularized maximum likelihood estimator to select the most appropriate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We establish the oracle property of the proposed estimator for selecting both stationary and local unit-root predictors, and show its good finite sample properties in an extensive simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Studying the high-frequency extreme losses of nine large liquid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' stocks using 42 liquidity and volatility predictors, we find the severity of extreme losses to be well predicted by low levels of price impact in period of high volatility of liquidity and volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Keywords: high-frequency financial data · peaks-over-threshold (POT) · time-varying generalized Pareto distribution· L1-regularized maximum likelihood estimation · nonstationary variable selection 1 Introduction Measuring tail risk at high-frequency has become of utmost importance to market players and regulators (Weller, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' While much efforts have been devoted to the measurement of tail risk at low-frequency (Nieto and Ruiz, 2016), few attempts have been made to measure risk at high-frequency, see Giot (2005), Dionne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' (2009) and Chavez-Demoulin and Davison (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Moreover, although these models can be very accurate, they explain the tail risk evolution in a “reduced form” manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', using autoregressive terms exploiting the persistence of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' They thus fail to provide a deeper structural understanding of the factors driving tail risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' As much as understanding the macroeconomic determinants of tail risk is a relevant problem at low-frequency (Massacci, 2017), it is important to understand how market uncertainty and trading activity impacts tail risk at high-frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' From a market microstructure perspective, though the intensification of high-frequency trading has improved trading costs and liquidity (Hendershott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2011), it is also suspected to be responsible for more frequent extreme price movements over short periods of time (Brogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Such extreme fluctuations are often the result of an aggressive directional market making activity initiated when the market is already under stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Brogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' (2018) find that market wide extreme shocks are likely to trigger the risk controls of high-frequency liquidity providers that thus withdraw from the market to reduce their risk exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Similarly, Kirilenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' (2017) find that during the market turbulence induced by the 2010 Flash Crash, many high- frequency liquidity providers withdrew from the market, thus exacerbating the price fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Studying how market uncertainty and trading activity affect extreme losses can thus provide a deeper understanding of the evolution of tail risk at high-frequency, and this paper proposes appropriate econometric techniques to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We consider a dynamic extreme value regression framework (Chavez-Demoulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Massacci, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Schwaab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2021) where the distribution of extreme losses is assumed to be well approximated by a general- ized Pareto distribution (GPD) with time-varying parameters driven by exogenous preditors and autoregressive ⋆ Corresponding author: ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='sun@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='01362v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='EM] 3 Jan 2023 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' To assess the impact of market uncertainty and trading activity on extreme losses, we consider several volatility predictors, proxing for market uncertainty, and liquidity predictors, characterizing trading activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Despite extreme value regression techniques have been widely applied in finance (Chavez-Demoulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Hambuckers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Bee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2019), our investigation presents new challenges: (i) as the financial litera- ture proposes several volatility and liquidity measures, we face a variable selection problem aimed at identifying predictors capturing the most relevant aspects of trading activity affecting extremes as well as improving the predictive accuracy of tail risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' (ii) volatility and liquidity measures observed at high-frequency exhibit strong persistence and seasonalities, thus violating the classical stationary assumptions required for inference with the maximum likelihood estimator (MLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' To overcome these issues, we develop a two-step adaptive L1-regularized maximum likelihood estimator (ALMLE) that allows performing variable selection with both stationary and local unit-root predictors (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2022), and establish its oracle property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We investigate the impact of 42 liquidity and volatility indicators on the distribution of high-frequency extreme losses of nine large liquid U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' stocks observed from 2006 to 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' We find that the severity of tail risk, as measured by the shape parameter of the GPD, is well predicted by low price impact (Goyenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=', 2009) during periods of high volatility of volatility and high volatility of liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' This finding is coherent with the evidence in Brogaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' (2018) that market markers liquidity supply is outstripped by liquidity demand after large uncertainty shocks, and their rush to leave the market to lower their risk exposures amplify extreme price movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Our two-step ALMLE is necessary to reveal this pattern as the standard MLE finds almost all predictors to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' To validate our estimating strategy, we provide an out-of-sample VaR forecast analysis and find that the estimated model performs well in the out-of-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section 2 presents the time-varying GPD model accom- modating stationary and local unit-root predictors as well as autoregressive components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Section 3 presents the MLE and shows its asymptotic non-normality when local unit-root predictors are included in the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Section 4 introduces the two-step ALMLE and prove the oracle property of this estimator in selecting both stationary and local unit-root predictors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Section 5 provides an extensive simulation study comparing the performance of the two-step ALMLE to those of the MLE, showing the superiority of the former in finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Section 6 discusses the results of the empirical study whereas Section 7 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Additional results and mathematical proofs are relegated to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' 2 Extreme value regression We denote the logarithmic loss and return time series of a financial asset by {lt}T t=1 and {rt}T t=1, respectively, with lt = −rt, and denote zt a vector of exogenous predictors observed at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Assumption M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' {lt}T t=1 and {zt}T t=1 are on a complete probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' At each time t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' , T}, we have an information set Ft−1 available which is the σ-algebra generated by {zt−1, lt−1, zt−2, lt−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Let assume {lt}T t=1 is independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=') with a cumulative distribution function (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=') F(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Probabilistic results from extreme value theory show that if there exist real sequences aT > 0 and βT such that limT →∞ F T (aT x + βT ) converges to a non-degenerate distribution G(·), then F(·) belongs to the max-domain of attraction of G(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' F ∈ D(G), and G(·) must be the generalized extreme value (GEV) distribution (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' of Coles (2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Let {yt}T t=1 be a censored sequence of excess losses above a high threshold u, such that the excess loss yt = lt − u, if lt > u, and yt = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Define the conditional distribution of excess losses, F|u(y) := P {lt − u ≤ y|lt > u} = P {yt ≤ y|yt > 0} , 0 < yt ≤ LF − u, 2 with LF := sup{x : F(x) < 1} the right end point of F(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' Pickands (1975) and Balkema and De Haan (1974) show that if F(·) ∈ D(G) then the limiting distribution of F|u(y) is a GPD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfZ_yv/content/2301.01362v1.pdf'} +page_content=' lim u→+∞ sup 0 10σ lower than both the 1× solar metallicity (588 ppm +for NRS1 and 1,080 ppm for NRS2) and 5× solar metal- +licity (413 ppm for NRS1 and 789 ppm for NRS2) GCM +predictions. Such large discrepancies are unsurprising, +given the small measurement uncertainties and the fact +that the GCMs were not tuned to match the data. Still, +it is notable that the measured nightside brightness tem- +peratures are both substantially lower than predicted +by the GCM simulations. Furthermore, the Parmentier +et al. (2018) GCMs did not include the dissociation and +recombination of hydrogen, which if anything would re- +sult in the predicted nightside temperatures being higher +than those shown in Figure 4 due to the release of la- +tent heat (Bell & Cowan 2018; Tan & Komacek 2019; +Komacek et al. 2022), exacerbating the discrepancy. +Clouds – which were also not included in the Par- +mentier et al. (2018) GCMs – might help explain our +lower-than-predicted nightside brightness temperature +measurements. Previous HST observations have shown +that the nightside temperature profile of WASP-121b +does not have a thermal inversion at the near-infrared +photosphere, instead cooling with decreasing pressure +(Mikal-Evans et al. 2022). +If an optically-thick cloud +deck blankets the nightside hemisphere, it could block +the emission from deeper, hotter layers of the atmo- +sphere, lowering the observed brightness temperature +(Mendon¸ca et al. 2018; Keating et al. 2019; Beatty et al. +2019; Parmentier et al. 2021). Alternatively, Komacek +et al. (2022) have shown that if nightside cloud is patchy +rather than uniform, the resulting brightness tempera- +tures can be higher than cloud-free predictions. How- +ever, even for patchy cloud, Komacek et al. (2022) also +identified scenarios in which nightside brightness tem- +peratures could be lower than predicted by cloud-free +models, depending on the precise assumptions made for +the cloud properties, such as the size distribution and +radiative properties of the cloud particles. +The nightside brightness temperatures that we mea- +sure for WASP-121b in the NRS1 (925+11 +−11 K) and NRS2 +(1127+6 +−6 K) passbands fall below the condensation tem- +peratures of silicates, such as enstatite and forsterite +(Visscher et al. 2010; Wakeford et al. 2017), which are +expected to be abundant in the nightside atmospheres +of hot Jupiters (e.g. Gao & Powell 2021). Therefore, our +measurements reveal nightside conditions for WASP- +121b that do appear to be conducive to the forma- +tion of silicate clouds, strengthening the possibility that +clouds may prove key to understanding why our mea- +sured nightside brightness temperatures are significantly +lower than predicted by the Parmentier et al. (2018) +3.0 +3.5 +4.0 +4.5 +5.0 +3500 +4000 +4500 +5000 +5500 +Dayside +NRS1 +NRS2 +GCM (P2018, 1 × solar) +GCM (P2018, 5 × solar) +G395H data +3.0 +3.5 +4.0 +4.5 +5.0 +0 +250 +500 +750 +1000 +1250 +1500 +Nightside +Wavelength ( m) +Planet-to-star emission (ppm) +Planet-to-star emission (ppm) +Figure 4. Black diamonds show the measured dayside (top +panel) and nightside (bottom panel) planet-to-star emission +levels. Note that the measurement uncertainties are smaller +than the diamond symbols. Predictions from the cloud-free +3D GCM simulations of Parmentier et al. (2018) are also +shown for heavy element enrichments of 1× solar (blue lines) +and 5× solar (orange lines). Circle and square symbols show, +respectively, the 1× and 5× solar GCM predictions binned +to the light curve passbands, which are shown as green lines +in the top panel. +GCMs shown in Figure 4. Further work will be required +to investigate if the difference in the nightside bright- +ness temperatures obtained for the two passbands could +be caused by the wavelength-dependent opacity of atmo- +spheric layers above a cloud deck. For these future anal- +yses, the dayside and nightside emission spectra will be +extracted and interpreted, rather than the broad pass- +bands considered here. +It should also be stressed that the analysis we have +presented here is preliminary. +In particular, we have +adopted a very simple light curve model and it is evi- +dent by eye that low-amplitude correlations still remain +in the time series of residuals (Figure 2). We did exper- +iment with including the x and y pointing coordinates +shown in Figure 1 as additional linear decorrelation vari- +ables in the light curve fits, but found that this did not +appreciably reduce the scatter in the residuals, nor did +it affect the derived parameter distributions shown in +Figure 3. Tidal deformation of WASP-121b could also +subtly affect the observed phase curve (Cowan et al. +2012; Wahl et al. 2021). +We did use starry to per- +form some preliminary light curve fits treating the plan- + +8 +Mikal-Evans et al. +etary oblateness as a free parameter, but found that our +basic conclusions were unaffected. Further work is re- +quired to determine if an oblate shape for the planet +is supported by the data. Additional possibilities yet to +be considered include adding higher-order spherical har- +monics terms for the planetary brightness map and al- +lowing for nonlinear instrumental baseline trends. These +investigations are ongoing and could conceivably affect +the constraints that we ultimately obtain for parameters +of interest, such as the planetary nightside emission and +brightness phase offsets. +In the meantime, our phase curve measurement for +WASP-121b demonstrates the overall high level of sta- +bility that JWST NIRSpec is capable of maintain- +ing for a single-stare observation lasting 37.8 hr. +The +approximately-linear drift observed in the baseline flux +level is extremely mild in comparison to the instrumen- +tal systematics that have affected past HST and Spitzer +datasets (e.g. Kreidberg et al. 2018; Mikal-Evans et al. +2022). +We are hopeful that the additional 80-90 ppm +high-frequency systematics noise observed here in our +preliminary analysis can be further reduced as more on- +sky calibrations become available, along with continued +refinement of the data analysis, such as improved treat- +ment of the 1/f detector noise. Based on other recent +analyses of NIRSpec data (Alderson et al. 2022; Rus- +tamkulov et al. 2022b), we also anticipate that simple +models fitted to light curves generated over narrower +wavelength ranges than the broad passbands we have +considered in the present study will achieve residual +scatters closer to Poisson predictions. +6. CONCLUSION +We have measured a full-orbit phase curve for the ul- +trahot Jupiter WASP-121b using JWST NIRSpec. The +resulting light curves generated across broad passbands +for the NRS1 and NRS2 detectors exhibit minimal sys- +tematics over the 37.8 hr observation. We find that the +phase curve peaks are shifted prior to mid-eclipse by +3.36 ± 0.11 ◦ (NRS1) and 2.66 ± 0.12 ◦ (NRS2), suggest- +ing that the eastern region of the dayside hemisphere is +hotter on average than the western region. The mea- +sured dayside emission in the NRS1 passband is in good +agreement with a cloud-free GCM assuming 5× solar +metallicity; however, the same GCM underpredicts the +dayside emission in the NRS2 passband by 19σ. For the +nightside emission, cloud-free GCM simulations assum- +ing 1× and 5× solar metallicity significantly over-predict +the data. This observation could possibly be explained +by nightside clouds blocking the emission from deeper, +hotter layers of the atmosphere. +The corresponding +nightside brightness temperatures are < 1200 K in both +passbands, which is cool enough for various conden- +sates to form, including silicates such as enstatite and +forsterite. +The authors are grateful to the anonymous referee for +constructive feedback that improved the paper. Support +for JWST program GO-1729 was provided by NASA +through a grant from the Space Telescope Science In- +stitute, which is operated by the Association of Uni- +versities for Research in Astronomy, Inc., under NASA +contract NAS 5-26555. JKB was supported by a Sci- +ence and Technology Facilities Council Ernest Ruther- +ford Fellowship. NM was partly supported by a Science +and Technology Facilities Council Consolidated Grant +[ST/R000395/1], the Leverhulme Trust through a re- +search project grant [RPG2020-82] and a UKRI Future +Leaders Fellowship [grant number MR/T040866/1]. +Software: +NumPy (van der Walt et al. 2011), SciPy +(Virtanen et al. 2020), Matplotlib (Hunter 2007), JWST +Python pipeline (Bushouse et al. 2022), FIREFly (Rus- +tamkulov et al. 2022a), starry (Luger et al. 2019), PyMC3 +(Salvatier et al. 2016), pysynphot (STScI Development +Team2013) +Facilities: +JWST(NIRSpec) +All of the JWST data used in this paper were ob- +tained from the Mikulski Archive for Space Telescopes +(MAST) at the Space Telescope Science Institute. +The specific observations analyzed can be accessed via +10.17909/23j6-ng29 +REFERENCES +Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2022, +arXiv e-prints, arXiv:2211.10488 +Arcangeli, J., D´esert, J.-M., Parmentier, V., et al. 2019, +A&A, 625, A136 +Beatty, T. G., Marley, M. 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A., Kataria, T., et al. 2018, AJ, +155, 83 + diff --git a/fdE1T4oBgHgl3EQfewQV/content/tmp_files/load_file.txt b/fdE1T4oBgHgl3EQfewQV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86a233890a6faa66be93d9f06e6b1f4c0277dcbd --- /dev/null +++ b/fdE1T4oBgHgl3EQfewQV/content/tmp_files/load_file.txt @@ -0,0 +1,861 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf,len=860 +page_content='Draft version January 10, 2023 Typeset using LATEX twocolumn style in AASTeX62 A JWST NIRSpec phase curve for WASP-121b: dayside emission strongest eastward of the substellar point and nightside conditions conducive to cloud formation Thomas Mikal-Evans,1 David K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Sing,2, 3 Jiayin Dong,4, ∗ Daniel Foreman-Mackey,4 Tiffany Kataria,5 Joanna K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 122 Sciences Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' NY 14853,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' USA 9Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Utah Valley University,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' UK 11University of Bristol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' HH Wills Physics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Tyndall Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Bristol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' BS8 1TL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' UK (Received December 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Revised December 27, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Accepted December 29, 2022) Submitted to Astrophysical Journal Letters ABSTRACT We present the first exoplanet phase curve measurement made with the JWST NIRSpec instrument, highlighting the exceptional stability of this newly-commissioned observatory for exoplanet climate studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The target, WASP-121b, is an ultrahot Jupiter with an orbital period of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We analyze two broadband light curves generated for the NRS1 and NRS2 detectors, covering wavelength ranges of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='70-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='72 µm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='82-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Both light curves exhibit minimal systematics, with approximately linear drifts in the baseline flux level of 30 ppm/hr (NRS1) and 10 ppm/hr (NRS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Assuming a simple brightness map for the planet described by a low-order spherical harmonic dipole, our light curve fits suggest that the phase curve peaks coincide with orbital phases 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11◦ (NRS1) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12◦ (NRS2) prior to mid-eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This is consistent with the strongest dayside emission emanating from eastward of the substellar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We measure planet-to-star emission ratios of 3, 924± 7 ppm (NRS1) and 4, 924 ± 9 ppm (NRS2) for the dayside hemisphere, and 136 ± 8 ppm (NRS1) and 630 ± 10 ppm (NRS2) for the nightside hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The latter nightside emission ratios translate to planetary brightness temperatures of 926 ± 12 K (NRS1) and 1, 122 ± 10 K (NRS2), which are low enough for a wide range of refractory condensates to form, including enstatite and forsterite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' A nightside cloud deck may be blocking emission from deeper, hotter layers of the atmosphere, potentially helping to explain why cloud-free 3D general circulation model simulations systematically over-predict the nightside emission for WASP-121b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Keywords: Exoplanet astronomy (486), Exoplanet atmospheres (487) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' INTRODUCTION Corresponding author: Thomas Mikal-Evans tmevans@mpia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='de ∗ Flatiron Research Fellow It is crucial to explore the coupling between the day- side and nightside hemispheres of tidally-locked planets if we are to develop a global understanding of their atmo- spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' One of the most effective means of constraining the chemical, thermal, and dynamical properties of both hemispheres is to measure planetary emission spectra arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='03209v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='EP] 9 Jan 2023 2 Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' over the course of a full planetary orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Arcangeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' To this end, our team observed a spectroscopic phase curve for the ultrahot Jupiter WASP-121b using JWST NIRSpec with the G395H grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' In this letter, we present an analysis of two broadband light curves generated from this dataset, one for each of the NRS1 and NRS2 de- tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' As well as providing some preliminary scientific results, our purpose is to provide a brief report on the stability of JWST and NIRSpec when performing long- stare time series measurements lasting tens of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The target, WASP-121b, is an inflated gas giant with a mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='183+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='064 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='062 MJ, a radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='753 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='036 RJ measured at red-optical wavelengths, and an orbital pe- riod of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='59820 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00001 hr (Delrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Bour- rier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Secondary eclipse measurements made with HST have shown that the dayside atmosphere has a thermal inversion, with a near-infrared brightness tem- perature close to 2,700 K (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Evidence has been uncovered for numerous UV-optical absorbers in the atmosphere that are likely responsible for the dayside thermal inversion, such as VO, SiO, V, Fe, Mg, and Ca (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Hoeijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Phase curve measurements with HST have shown that temperatures on the nightside of WASP-121b are likely low enough for refractory condensates containing V, Fe, Mg, and Ca to form (Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Since these latter elements are observed in the gas-phase at the day-night terminator, if such clouds do form they are presumably recirculated back to the dayside hemisphere where they are vaporised before settling to the deeper layers of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Meanwhile, non-detections of Ti and Al at the day-night terminator reported by Hoei- jmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2020, 2022) are consistent with these el- ements being cold-trapped in deeper layers of the night- side hemisphere as perovskite (CaTiO3) and corundum (Al2O3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' From these previous studies, we are already observ- ing the important connection between the dayside and nightside hemispheres of WASP-121b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' One of the ulti- mate goals of our JWST program will be to provide sig- nificantly tighter constraints on the dayside and night- side properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This should be possible thanks to the JWST data having a much higher signal-to-noise and spectral resolution than the existing HST data, as well as the more favorable planet-to-star emission ratio at the longer wavelengths covered by NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 x jitter (pixel) y jitter (pixel) x jitter (pixel) y jitter (pixel) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Pointing jitter along the x and y axes of the NRS1 (top panel) and NRS2 (bottom panel) detectors, as measured by the FIREFly code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Our team observed the WASP-121 system over a full planetary orbit on 2022 October 14-15 using the NIRSpec instrument as part of program GO-1729 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Mikal-Evans, co-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Kataria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Observations were made without interruption for 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 hr, commencing 145 min prior to secondary eclipse ingress and continuing until 105 min after egress of the following secondary eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We used the G395H grating, providing wavelength cov- erage between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='70-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 µm at R ∼ 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For reading the detector, we employed the SUB2048 subarray option with the NRSRAPID readout pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Each integration consisted of 42 groups, translating to integration times of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 seconds and an overall duty cycle of 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' To extract the target spectra from the data frames, we used the FIREFLy code (Fast InfraRed Exoplanet Fit- ting for Lightcurves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Rustamkulov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The data reduction begins with a customized reduction of stage 1 from the JWST Python pipeline (Bushouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Detector “1/f” noise (Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Rus- JWST NIRSpec phase curve for WASP-121b 3 tamkulov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020) is removed at the group level, masking out the spectral trace pix- els and removing the median value from each column of the remaining background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Bad pixels and cos- mic rays are then flagged as outliers, both in the time series and spatially on the 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Movement of the spectra along the x and y axes of the detector are mea- sured using cross-correlation, exhibiting standard devi- ations at the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='002 pixels in both directions for NRS1 and in the y direction for NRS2, while a higher standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='005 pixels is measured in the x direction for NRS2 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This level of jitter is sim- ilar to previously reported on-sky performance (Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022), though demonstrated here on a much longer timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The 2D data frames are then adjusted using interpolation to account for the small pointing shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We find that this alignment proce- dure can dampen x and y correlations in the subsequent spectrophotometry, as it effectively takes into account the redistribution of flux across adjacent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' How- ever, residual x and y trends may still remain due to intrapixel sentivities (Birkmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Next, a fourth order polynomial was used to trace the center of the point-spread-function (PSF) along the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Counts were then summed within an aperture centered on the trace with a width of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='9 pixels for NRS1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 pixels for NRS2, linearly interpolating the counts in those pixels where only a fraction was contained within the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The aperture widths were selected after trialing a number of values and selecting those that min- imized the scatter of the in-eclipse photometric time se- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' To obtain the wavelength calibration, we extrapo- lated JWST pipeline data products across the detector edge pixels which did not have an assigned wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' LIGHT CURVE ANALYSIS We generated two light curves by summing each ex- tracted 1D spectrum between dispersion columns 300- 2042 for NRS1 and 5-2010 for NRS2, conservatively en- compassing the full passbands of both detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The resulting raw light curves are shown in the top two pan- els of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We modeled these light curves using the starry package (Luger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2019) for the star-planet signal, multiplied by a linear trend in time to account for instrumental drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the planet brightness distri- bution, a simple dipole map was adopted, comprised of the Y0,0 and Y1,0 spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the stellar brightness distribution, a quadratic limb darkening pro- file was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' In defining the data log-likelihood, we assumed that the data were normally distributed with standard deviations given by the pipeline Poisson un- certainties and an additional high-frequency systematics noise term combined in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We fitted the NRS1 and NRS2 light curves jointly, with the following parameters shared across both light curves: the stellar mass (M⋆) and logarithmic radius (log10 R⋆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the differences in the orbital period (∆P) and transit mid-time (∆Tc) from the values reported by Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the orbital inclination (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' and the planetary mass (Mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Separate sets of the remaining model parameters were fitted for each light curve in the joint fit, namely: the stellar limb darkening coefficients (u1, u2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the logarithmic planetary radius (log10 Rp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the overall amplitude of the planetary brightness map (A) and the coefficient of the Y1,0 spherical harmonic (y1,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the rotational offset of the planetary brightness map rel- ative to a map with a brightness peak centered at the substellar point (∆φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the coefficients of the linear in- strumental drift trend (f0, f1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' and the high-frequency systematics noise term (σsyst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The priors we adopted for each parameter are shown by the dashed lines in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For M⋆, log10 R⋆, i, and Mp, these were nor- mal priors based on the posteriors reported in Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Broad normal priors were also adopted for log10 Rp, ∆Tc, u1, u2, A, y1,0, ∆φ, f0, and f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Half- normal priors only allowing positive values were adopted for σsyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We marginalized the resulting posterior distri- bution using the No-U-Turn-Sampling (NUTS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Hoffman & Gelman 2014) method implemented by PyMC3 (Sal- vatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' RESULTS The maximum a posteriori (MAP) light curve mod- els are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Posterior distributions for each of the parameters that were varied in the fitting are shown as dark blue histograms in Figure 3, while the light blue histograms show the corresponding distri- butions for other parameters that were calculated from the latter, namely: the stellar radius R⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the planet- to-star radius ratio Rp/R⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' the planet-to-star flux ratio Fp/F⋆ for the dayside and nightside hemispheres of the planet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' and the corresponding planetary brightness tem- peratures Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The brightness temperatures were calcu- lated by assuming Fp/F⋆ = (Rp/R⋆)2 Bp(Tb)/F⋆, where Bp denotes a black body function for the planetary flux, integrated over the instrument passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' In deriving distributions for Tb, uncertainties in the host star prop- erties were accounted for by first randomly drawing val- ues for the stellar effective temperature (T⋆), surface gravity (log g⋆), and metallicity ([M/H]) from the poste- rior distributions reported in Delrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2016), which were assumed to be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For each of these sets of stellar properties, we used pysynphot (STScI Develop- 4 Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 (b) 1000 0 1000 2000 3000 4000 5000 (c) 1000 0 1000 2000 3000 4000 5000 (d) 500 250 0 250 500 (e) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='4 500 250 0 250 500 (f) BJDTDB 2459860 (day) System emission variation (%) Planet-to-star emission (ppm) Residuals (ppm) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Panels (a) and (b): Raw broadband light curves for the NRS1 and NRS2 detectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Orange lines show the best-fit light curve models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Blue lines show the corresponding instrument baseline trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Grey horizontal lines are calibrated to the bottoms of the first eclipse, to highlight the instrumental drift over the course of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Panels (c) and (d): The same light curves focusing on the planetary emission signal and after correcting for the instrument baseline trends shown by the blue lines in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Panels (e) and (f): Black points show the residuals between the data and best-fit models for the NRS1 and NRS2 detectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Orange lines show the residuals after applying a Gaussian filter to smooth the random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Green and purple shading show the times of transit and eclipse, respectively, with darker ranges corresponding to ingress and egress times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' JWST NIRSpec phase curve for WASP-121b 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='20 Shared: log[R (R )] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1575+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0066 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1547 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='55 Shared: R (R ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='437+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='022 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='458 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='030 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 Shared: M (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='342+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='060 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='358 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='080 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 Shared: Mp(MJ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='157+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='071 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='157 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='070 20 0 20 Shared: P (10 3 s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='04+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='63 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='03 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='96 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 Shared: i ( ) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='14 = 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='49 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 Shared: Tc (s) 28+18 17 = 28 = 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='74 NRS1: log[Rp(RJ)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='754+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='754 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1228 NRS1: Rp/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='122551+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='04 NRS1: u1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='020+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='011 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='041 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='20 NRS1: u2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='169+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='018 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='120 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='500 2020 2030 2040 NRS1: A (ppm) 2030+7 6 = 2032 = 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='815 NRS1: y1, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='809+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='003 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='808 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 NRS1: ( ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0 25 50 75 100 NRS1: syst (ppm) 90+2 2 = 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='74 NRS2: log[Rp(RJ)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='750+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='007 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='751 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='1238 NRS2: Rp/R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='123668+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='02 NRS2: u1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='011+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='013 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='039 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='20 NRS2: u2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='161+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='022 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='229 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='500 2760 2780 2800 NRS2: A (ppm) 2777+8 8 = 2773 = 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='675 NRS2: y1, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='670+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='003 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='672 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 NRS2: ( ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='65 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='00 0 25 50 75 100 NRS2: syst (ppm) 80+4 4 = 20 2360 2340 NRS1: f0 (ppm) 2354+7 7 = 2357 = 1000000 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 NRS1: f1 (ppm/hr) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2 = 1000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 4820 4800 4780 NRS2: f0 (ppm) 4805+9 9 = 4799 = 1000000 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 NRS2: f1 (ppm/hr) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='3 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 = 1000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 3900 3920 3940 NRS1: dayside Fp/F 3924+7 7 (ppm) 120 140 160 NRS1: nightside Fp/F 136+8 8 (ppm) 4900 4920 4940 NRS2: dayside Fp/F 4924+9 9 (ppm) 600 620 640 660 NRS2: nightside Fp/F 630+10 10 (ppm) 2700 2800 NRS1: dayside Tb 2762+30 32 (K) 900 925 950 NRS1: nightside Tb 926+12 12 (K) 2700 2800 2900 NRS2: dayside Tb 2768+39 39 (K) 1100 1120 1140 NRS2: nightside Tb 1122+10 10 (K) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Dark blue histograms show posterior distributions for model parameters that were varied in the light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Light blue histograms show corresponding distributions for other parameters that were subsequently derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Posterior medians and ±34% credible intervals are listed in black font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Red dashed lines show the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the normal priors, the mean µ and standard deviation σ are listed in red font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Half-normal priors were adopted for the σsyst parameters, allowing only positive values and with scale σ listed in red font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Vertical axes are not labeled, as arbitrary normalizations have been applied to all plotted distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 6 Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' ment Team 2013) to obtain a corresponding ATLAS9 stellar model (Castelli & Kurucz 2003) that was then integrated over the instrument passband to obtain an estimate for F⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The resulting set of F⋆ values, along with the Rp/R⋆ and Fp/F⋆ posterior samples obtained from the light curve fits, were used to derive the final distributions for Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We measure a significantly larger Rp/R⋆ value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='123668+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000068 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000069 for NRS2, compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='122551+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000062 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='000063 for NRS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This difference can potentially be explained by the wavelength-dependent opacity of the planetary atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We find that the planetary brightness map is shifted eastward by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11 ◦ (30σ) for NRS1 and by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12 ◦ (22σ) for NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We also make sig- nificant detections of the planetary nightside emission in both passbands: 136 ± 8 ppm (17σ) for NRS1 and 630 ± 10 ppm (63σ) for NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' These relative emission values translate to nightside brightness temperatures of 926+12 −12 K and 1, 122+10 −10 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the dayside hemispheres, we obtain corresponding brightness tem- peratures of 2, 762+30 −32 K for the NRS1 passband and 2, 768+39 −39 K for the NRS2 passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We note that the brightness temperature uncertainties are dominated by the stellar model uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' When we only account for the uncertainties in Fp/F⋆ and Rp/R⋆ derived from the light curve fits while holding the stellar properties fixed to the best-fit values reported by Delrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2016), the uncertainties on the dayside brightness tem- peratures reduce by an order-of-magnitude from around 30-40 K (Figure 3) to 3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' By contrast, the uncertainties associated with dayside brightness temperatures mea- sured previously with the Spitzer Space Telescope in the 3-5 µm wavelength range have been dominated by the Fp/F⋆ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For example, Garhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2020) reported dayside brightness temperatures for WASP- 121b of 2, 490 ± 77 K and 2, 562 ± 66 K in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 µm and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 µm Infrared Array Camera (IRAC) passbands, which cover similar wavelength ranges to NRS1 and NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The long-term instrumental drift is found to be mild for both detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For NRS1, we obtain a negative drift of 1,172 ppm over the course of the observation, corresponding to a rate of approximately 30 ppm/hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' NRS2 is even better-behaved, exhibiting a positive drift of only 416 ppm from start to finish, at a rate of ap- proximately 10 ppm/hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The standard deviation of the light curve residuals are 127 ppm and 161 ppm for the NRS1 and NRS2 detectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This is 39% and 17% higher than the Poisson uncertainties derived from the instrument pipeline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' As such, we derive high-frequency systematics noise values (σsyst) of 90 ± 2 ppm for NRS1 and 80 ± 4 ppm for NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' DISCUSSION The small positive values that we measure for ∆φ in both the NRS1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11 ◦) and NRS2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12 ◦) passbands translate to phase curves with maxima shifted prior to mid-eclipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This in turn suggests that the eastern half of WASP-121b’s dayside hemisphere is hot- ter on average than the western hemisphere, which is qualitatively consistent with the predictions of general circulation model (GCM) simulations of hot Jupiters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Showman & Guillot 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Showman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' However, GCMs typically pre- dict eastward phase curve shifts that are significantly higher than what we have measured here for WAP- 121b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' closer to 10◦ or more (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Parmentier & Cross- field 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Small phase curve shifts relative to GCM predictions have also been measured for a number of other hot Jupiters with properties similar to WASP- 121b (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For example, Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018) measured an east- ward phase shift of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='7 ◦ for WASP-103b in the Spitzer IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='6 µm channel, compared to a shift of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='2◦ predicted by a GCM presented in the same study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For ultrahot planets like WASP-121b and WASP-103b – which both have dayside brightness temperatures well above 2, 500 K – Lorenz drag resulting from the motion of thermally ionized gas through the planetary magnetic field may inhibit the advection of heat throughout the atmosphere, reducing the observed phase curve offsets (Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Alternatively, even if the hottest region of the dayside hemisphere is located significantly eastward of the substellar point, the presence of night- side clouds can reduce the observed phase shift in the re- sulting light curve (Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We plan to investigate the latter possibility further in future analy- ses by including higher-order spherical harmonics terms for the planetary brightness map, which will allow for a sharper delineation in the brightness of the dayside and nightside hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Our dayside and nightside emission measurements are plotted in Figure 4 against the 3D GCM predictions from Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018), which assume 1× and 5× solar metallicity, chemical equilibrium, and do not include clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The models exhibit spectral features of H2O and CO, which are in emission on the day- side due to a thermal inversion and absorption on the nightside due to a cooling temperature profile (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The agreement between the measured dayside emis- sion (3,924±7 ppm) and the 5× solar GCM prediction (3,916 ppm) across the NRS1 passband is very close to 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' However, the dayside emission measured across the NRS2 passband (4,924±9 ppm) is 19σ higher than the 5× solar GCM prediction (3,916 ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the JWST NIRSpec phase curve for WASP-121b 7 nightside emission, the measurements across the NRS1 (136 ± 8 ppm) and NRS2 (630 ± 10 ppm) passbands are > 10σ lower than both the 1× solar metallicity (588 ppm for NRS1 and 1,080 ppm for NRS2) and 5× solar metal- licity (413 ppm for NRS1 and 789 ppm for NRS2) GCM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Such large discrepancies are unsurprising, given the small measurement uncertainties and the fact that the GCMs were not tuned to match the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Still, it is notable that the measured nightside brightness tem- peratures are both substantially lower than predicted by the GCM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Furthermore, the Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018) GCMs did not include the dissociation and recombination of hydrogen, which if anything would re- sult in the predicted nightside temperatures being higher than those shown in Figure 4 due to the release of la- tent heat (Bell & Cowan 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Tan & Komacek 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022), exacerbating the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Clouds – which were also not included in the Par- mentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018) GCMs – might help explain our lower-than-predicted nightside brightness temperature measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Previous HST observations have shown that the nightside temperature profile of WASP-121b does not have a thermal inversion at the near-infrared photosphere, instead cooling with decreasing pressure (Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' If an optically-thick cloud deck blankets the nightside hemisphere, it could block the emission from deeper, hotter layers of the atmo- sphere, lowering the observed brightness temperature (Mendon¸ca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Keating et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Beatty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Alternatively, Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2022) have shown that if nightside cloud is patchy rather than uniform, the resulting brightness tempera- tures can be higher than cloud-free predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' How- ever, even for patchy cloud, Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2022) also identified scenarios in which nightside brightness tem- peratures could be lower than predicted by cloud-free models, depending on the precise assumptions made for the cloud properties, such as the size distribution and radiative properties of the cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The nightside brightness temperatures that we mea- sure for WASP-121b in the NRS1 (925+11 −11 K) and NRS2 (1127+6 −6 K) passbands fall below the condensation tem- peratures of silicates, such as enstatite and forsterite (Visscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2017), which are expected to be abundant in the nightside atmospheres of hot Jupiters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Gao & Powell 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Therefore, our measurements reveal nightside conditions for WASP- 121b that do appear to be conducive to the forma- tion of silicate clouds, strengthening the possibility that clouds may prove key to understanding why our mea- sured nightside brightness temperatures are significantly lower than predicted by the Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 3500 4000 4500 5000 5500 Dayside NRS1 NRS2 GCM (P2018, 1 × solar) GCM (P2018, 5 × solar) G395H data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='0 0 250 500 750 1000 1250 1500 Nightside Wavelength ( m) Planet-to-star emission (ppm) Planet-to-star emission (ppm) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Black diamonds show the measured dayside (top panel) and nightside (bottom panel) planet-to-star emission levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Note that the measurement uncertainties are smaller than the diamond symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Predictions from the cloud-free 3D GCM simulations of Parmentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' (2018) are also shown for heavy element enrichments of 1× solar (blue lines) and 5× solar (orange lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Circle and square symbols show, respectively, the 1× and 5× solar GCM predictions binned to the light curve passbands, which are shown as green lines in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' GCMs shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Further work will be required to investigate if the difference in the nightside bright- ness temperatures obtained for the two passbands could be caused by the wavelength-dependent opacity of atmo- spheric layers above a cloud deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For these future anal- yses, the dayside and nightside emission spectra will be extracted and interpreted, rather than the broad pass- bands considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' It should also be stressed that the analysis we have presented here is preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' In particular, we have adopted a very simple light curve model and it is evi- dent by eye that low-amplitude correlations still remain in the time series of residuals (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We did exper- iment with including the x and y pointing coordinates shown in Figure 1 as additional linear decorrelation vari- ables in the light curve fits, but found that this did not appreciably reduce the scatter in the residuals, nor did it affect the derived parameter distributions shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Tidal deformation of WASP-121b could also subtly affect the observed phase curve (Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We did use starry to per- form some preliminary light curve fits treating the plan- 8 Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' etary oblateness as a free parameter, but found that our basic conclusions were unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Further work is re- quired to determine if an oblate shape for the planet is supported by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Additional possibilities yet to be considered include adding higher-order spherical har- monics terms for the planetary brightness map and al- lowing for nonlinear instrumental baseline trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' These investigations are ongoing and could conceivably affect the constraints that we ultimately obtain for parameters of interest, such as the planetary nightside emission and brightness phase offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' In the meantime, our phase curve measurement for WASP-121b demonstrates the overall high level of sta- bility that JWST NIRSpec is capable of maintain- ing for a single-stare observation lasting 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The approximately-linear drift observed in the baseline flux level is extremely mild in comparison to the instrumen- tal systematics that have affected past HST and Spitzer datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Mikal-Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We are hopeful that the additional 80-90 ppm high-frequency systematics noise observed here in our preliminary analysis can be further reduced as more on- sky calibrations become available, along with continued refinement of the data analysis, such as improved treat- ment of the 1/f detector noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Based on other recent analyses of NIRSpec data (Alderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Rus- tamkulov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022b), we also anticipate that simple models fitted to light curves generated over narrower wavelength ranges than the broad passbands we have considered in the present study will achieve residual scatters closer to Poisson predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' CONCLUSION We have measured a full-orbit phase curve for the ul- trahot Jupiter WASP-121b using JWST NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The resulting light curves generated across broad passbands for the NRS1 and NRS2 detectors exhibit minimal sys- tematics over the 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='8 hr observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' We find that the phase curve peaks are shifted prior to mid-eclipse by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='11 ◦ (NRS1) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content='12 ◦ (NRS2), suggest- ing that the eastern region of the dayside hemisphere is hotter on average than the western region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The mea- sured dayside emission in the NRS1 passband is in good agreement with a cloud-free GCM assuming 5× solar metallicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' however, the same GCM underpredicts the dayside emission in the NRS2 passband by 19σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' For the nightside emission, cloud-free GCM simulations assum- ing 1× and 5× solar metallicity significantly over-predict the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' This observation could possibly be explained by nightside clouds blocking the emission from deeper, hotter layers of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The corresponding nightside brightness temperatures are < 1200 K in both passbands, which is cool enough for various conden- sates to form, including silicates such as enstatite and forsterite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The authors are grateful to the anonymous referee for constructive feedback that improved the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Support for JWST program GO-1729 was provided by NASA through a grant from the Space Telescope Science In- stitute, which is operated by the Association of Uni- versities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' JKB was supported by a Sci- ence and Technology Facilities Council Ernest Ruther- ford Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' NM was partly supported by a Science and Technology Facilities Council Consolidated Grant [ST/R000395/1], the Leverhulme Trust through a re- search project grant [RPG2020-82] and a UKRI Future Leaders Fellowship [grant number MR/T040866/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' Software: NumPy (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2011), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2020), Matplotlib (Hunter 2007), JWST Python pipeline (Bushouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022), FIREFly (Rus- tamkulov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2022a), starry (Luger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2019), PyMC3 (Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' 2016), pysynphot (STScI Development Team2013) Facilities: JWST(NIRSpec) All of the JWST data used in this paper were ob- tained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} +page_content=' The specific observations analyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE1T4oBgHgl3EQfewQV/content/2301.03209v1.pdf'} 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--git a/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/2301.03669v1.pdf.txt b/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/2301.03669v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2955d738a01f374bd1e031950e44c8da5cb57f9 --- /dev/null +++ b/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/2301.03669v1.pdf.txt @@ -0,0 +1,1141 @@ +Draft version January 11, 2023 +Typeset using LATEX default style in AASTeX631 +Moist convection is most vigorous at intermediate atmospheric humidity +Jacob T. Seeley1 and Robin D. Wordsworth1 +1Department of Earth and Planetary Sciences +Harvard University +Cambridge, MA 02461, USA +ABSTRACT +In Earth’s current climate, moist convective updraft speeds increase with surface warming. This +trend suggests that very vigorous convection might be the norm in extremely hot and humid atmo- +spheres, such as those undergoing a runaway greenhouse transition. However, theoretical and numerical +evidence suggests that convection is actually gentle in water vapor-dominated atmospheres, implying +that convective vigor may peak at some intermediate humidity level. Here, we perform small-domain +convection-resolving simulations of an Earth-like atmosphere over a wide range of surface temperatures +and confirm that there is indeed a peak in convective vigor, which we show occurs near Ts ≃ 330 K. +We show that a similar peak in convective vigor exists when the relative abundance of water vapor +is changed by varying the amount of background (non-condensing) gas at fixed Ts, which may have +implications for Earth’s climate and atmospheric chemistry during the Hadean and Archean. We also +show that Titan-like thermodynamics (i.e., a thick nitrogen atmosphere with condensing methane and +low gravity) produce a peak in convective vigor at Ts ≃ 95 K, which is curiously close to the current +surface temperature of Titan. Plotted as functions of the saturation specific humidity at cloud base, +metrics of convective vigor from both Earth-like and Titan-like experiments all peak when cloud-base +air contains roughly 10% of the condensible gas by mass. Our results point to a potentially common +phenomenon in terrestrial atmospheres: that moist convection is most vigorous when the condensible +component is between dilute and non-dilute abundance. +1. INTRODUCTION +Moist convection produces some of the most impactful weather and climate phenomena on Earth, from reflective +stratocumulus layers to drenching seasonal monsoons. A longstanding challenge in climate science is to improve the +crude treatment of such convective phenomena in general circulation models, which hampers predictions of critical +quantities such as changes in local precipitation and the sensitivity of Earth’s climate to CO2. Looking beyond Earth, +there is strong evidence that moist convection — that is, convection coupled to phase changes of a condensible substance +— factors into the past and present evolution of the majority of solar system atmospheres, including Venus (Kasting +1988), Mars (Wordsworth 2016; Yamashita et al. 2016), Jupiter (Gierasch et al. 2000), Saturn (Li & Ingersoll 2015), +the ice giants (Hueso et al. 2020), and Titan (Schneider et al. 2012). Therefore, in order improve our understanding +of contemporary Earth climate and planetary climate more generally, there is a clear need to deepen and generalize +our understanding of moist convective physics. +In this paper, we take a step toward a more generalized understanding of moist convection by focusing on one of +its most basic characteristics: convective vigor, or the buoyancy and vertical velocity of cloud updrafts (e.g., Zipser +et al. 2006; Hansen & Back 2015; Hansen et al. 2020). +Updraft speeds affect instantaneous surface precipitation +rates (Muller & Takayabu 2020) and the fraction of cloud condensate that is lofted instead of falling to the surface, +which in turn exerts a strong influence on cloud cover and climate sensitivity (Zhao 2014, 2016). +More vigorous +updrafts are more likely to overshoot the tropopause and inject near-surface air into the stratosphere, thereby affecting +stratospheric humidity and chemistry (Liu et al. 2020; O’Neill et al. 2021) and planetary water loss on geologic +timescales (Wordsworth & Pierrehumbert 2013). +Lightning, which is closely associated with high updraft speeds +(Deierling & Petersen 2008), further modulates atmospheric chemistry. Focusing on a basic property such as convective +vigor, therefore, may reveal patterns in moist convective behavior with widespread implications. +arXiv:2301.03669v1 [astro-ph.EP] 9 Jan 2023 + +2 +Currently, there are two paradigms for how the vigor of moist convection depends on thermodynamic conditions. +The first paradigm, which we can call “warming-driven invigoration”, states that moist convective vigor increases +with the surface temperature. This paradigm has emerged from numerical modeling of Earth’s contemporary tropics: +in idealized cloud-resolving simulations, mean and extreme convective updraft speeds increase with warming (Romps +2011; Muller et al. 2011; Seeley & Romps 2015; Singh & O’Gorman 2015; Abbott et al. 2020). These changes are +consistent with, and typically attributed to, the increases in convective available potential energy (CAPE) that occur in +both global and convection-resolving models (e.g., Del Genio et al. 2007; Sobel & Camargo 2011; Romps 2011; Muller +et al. 2011; Singh & O’Gorman 2013; Seeley & Romps 2015). Although actual updrafts do not attain the velocities +implied by CAPE due to drag and entrainment of unsaturated environmental air, CAPE is nevertheless an extremely +useful proxy for intense convection (e.g., Johns & Doswell 1992; Romps et al. 2014). As shown by Romps (2016), the +increase of CAPE with warming under contemporary conditions is itself attributable to the Clausius-Clapeyron scaling +of near-surface saturation specific humidity. Therefore, the chain of causality in the “warming-driven invigoration” +paradigm is: +increasing Ts → increasing humidity → increasing CAPE → increasing convective vigor. +The second paradigm for convective vigor is relevant to atmospheres in which the condensible component is highly +non-dilute, with the limiting case being a pure steam atmosphere. Ding & Pierrehumbert (2016) and Pierrehumbert & +Ding (2016) argued that convection should be gentle in such cases by reasoning that, when the atmosphere is saturated, +the Clausius-Clapeyron curve dictates a one-to-one relationship between pressure and temperature, which precludes +appreciable temperature anomalies between parcels at the same pressure level1. These theoretical predictions gained +support from the work of Tan et al. (2021), who conducted idealized convection-resolving simulations of pure steam +atmospheres with surface temperatures of 600–800 K and found quiescent condensing layers with small temperature +variability and weak updrafts. Hence the line of reasoning in the “gentle pure-steam limit” is: +very high humidity → unique relationship between Tand p → low convective vigor. +Clearly, “warming-driven invigoration”, when extrapolated to very warm temperatures, conflicts with the “gentle pure- +steam limit”. Taken together, these paradigms suggest that convective vigor should reach a peak at some intermediate +humidity level. +In this paper, we seek to generalize our understanding of moist convective vigor by filling the gap between these +two paradigms. +Our approach is to simulate a wide range of planetary atmospheres that differ appreciably from +contemporary Earth in terms of their surface temperature, surface pressure, gravitational constant, and composition. +These atmospheres span the parameter space from highly dilute to non-dilute conditions, allowing us to continuously +probe the transitional behavior between the two paradigms for convective vigor. To circumvent the inherent uncertainty +of convective parameterizations and allow detailed analysis of convective dynamics, we performed these simulations +with a flexible convection-resolving model (described in detail in section 2). We analyze metrics of convective vigor +in these atmospheres, and show that our results match the predictions of an analytical theory for radiative-convective +equilibrium (RCE) originally developed in the context of Earth’s tropics (Romps 2016) (hereafter, R16). The analytical +theory of R16 predicts the mean CAPE of a convecting atmosphere, and we show that this theory provides a unifying +theoretical framework that links the two paradigms for convective vigor. +The outline of the paper is as follows. In section 2, we describe the flexible convection-resolving model and our +numerical experiments. Section 3 analyzes convective vigor in the simulated atmospheres. In section 4, we apply the +theory of R16 to our results. We conclude in section 5 with a discussion of the implications of our work for early Earth +and other planetary atmospheres. +2. EXPERIMENTAL METHODS +2.1. Core experiments +Our core suite of convection-resolving RCE simulations consists of three experiments. The first core experiment +used the Earth-like model configuration, with the total surface pressure fixed at the contemporary value of 105 Pa and +the surface temperature varying from 275 K to 365 K (experiment name EarthTemp). In the second core Earth-like +1 This argument neglects “virtual” effects of compositional differences and condensates on density. + +3 +experiment (EarthPressure), the surface temperature was instead fixed at 300 K while the surface pressure varied +from (1/16)× to 8× the contemporary value. The third core experiment (Titan) used the Titan-like configuration of +the model with the surface pressure fixed at the contemporary value of 1.467×105 Pa and the surface temperature +varying from 80 to 110 K. The parameters distinguishing the Earth-like and Titan-like configurations will be described +in detail below; see also Table 1. A number of additional simulations were performed as sensitivity tests, which we +will describe as they come up in the course of the main text. +2.2. Convection-resolving model +For all experiments, we simulated nonrotating radiative-convective equilibrium on square, doubly periodic domains +with the convection-resolving model DAM (Romps 2008). DAM has a finite-volume, fully-compressible dynamical core +and uses the implicit approach to sub-grid diffusion. The vertical grid had a variable spacing, transitioning smoothly +from ∆z = 50 m below an altitude of 500 m to ∆z = 1000 m at altitudes between 10 km and the model top. The +model top was at a variable height because our simulated tropospheres vary widely in geometric depth. The default +horizontal resolution was ∆x = ∆y = 2 km and the default horizontal domain size was Lx = Ly = 96 km. We also +ran a small subset of simulations at a higher resolution of ∆x = 500 m and with free-tropospheric ∆z = 500 m (the +EarthTemp hr experiment). The default model time step was ∆t = 20 s, sub-stepped to satisfy a CFL condition; +this time step was used for all simulations but EarthTemp hr, which used ∆t = 5 s. Overall, the model domains are +similar in size and resolution to the “RCE small” protocol from the RCEMIP project (Wing et al. 2017) in which +DAM participated. +2.3. Radiative transfer +To avoid inessential complexity and focus attention on convective dynamics, we used a simplified treatment of +radiative transfer in the majority of our simulations (as in, e.g., Tan et al. 2021). +Specifically, we prescribed an +idealized tropospheric radiative cooling using an equation of the form: +− ∂ +∂T F = α(T − Tt), +(1) +where T is the temperature, the temperature derivative ∂/∂T is a vertical derivative, α (W/m2/K2) is a constant +setting the magnitude of radiative forcing, F (W/m2) is the net upward radiative flux, and Tt (K) is the prescribed +tropopause temperature. Numerically, we approximated the vertical derivative ∂/∂T with a centered finite difference +on the model’s vertical grid. Equation 1 was proposed by Jeevanjee & Romps (2018) as a fit to the invariant radiative +flux divergence curve found for Earth, and has been used as an idealized representation of tropospheric radiative +cooling in more recent work (Jeevanjee & Zhou 2022). We used equation (1) to prescribe radiative forcing for both +Earth-like and Titan-like experiments, although we used different values of the parameters (Table 1). At altitudes +above the tropopause, temperatures were simply nudged to Tt on a timescale of 6 hours; since our focus is on convective +dynamics in the troposphere, this simplified approach to the stratosphere does not affect our results. +One deficiency of equation (1) is that it does not predict the transition to lower-tropospheric radiative heating in very +warm climates, a regime which is known to affect convective dynamics (Seeley & Wordsworth 2021). Therefore, we also +re-ran the EarthTemp experiment with interactive clear-sky radiative transfer (the EarthTemp realrad experiment) to +check that our main results are not sensitive to the simplified treatment of radiation we employ in our core experiments. +We focused on interactive clear-sky radiation because that is sufficient to produce the low-level radiative heating regime +of Seeley & Wordsworth (2021), and because DAM is not coupled to a cloud-radiation scheme that is validated up to +the very high temperatures we simulate. Our approach to clear-sky radiative transfer was identical to the line-by-line +method of Seeley & Wordsworth (2021), and we refer the reader to that paper for a complete description. +2.4. Microphysics +The default microphysics scheme in DAM is a bulk scheme with six water classes (vapor, cloud liquid, cloud ice, +rain, snow, and graupel). +However, to avoid relying on an overly Earth-centric parameterization, we conducted +our simulations using a simplified microphysics scheme that has been described and used in previous work (Seeley +& Wordsworth 2021). In the simplified microphysics scheme, only three bulk classes of condensible substance are +modeled: vapor, non-precipitating cloud liquid, and rain, with associated mass fractions qv, qc, and qr, respectively. +Microphysical transformations between vapor and cloud condensate are handled by a saturation adjustment routine, + +4 +parameter +description +Earth-like +Titan-like +g +gravitational constant (m/s2) +10. +1.35 +Tt +tropopause temperature (K) +200. +70. +α +radiative forcing constant (W/m2/K) +0.025 +0.5 +Ra +specific gas constant of dry air (J/kg/K) +287.04 +296.8 +Rv +specific gas constant of condensible gas (J/kg/K) +461. +518.28 +cva +specific heat capacity at constant volume of dry air (J/kg/K) +719. +707.2 +cvv +specific heat capacity at constant volume of condensible gas (J/kg/K) +1418. +1707.4 +cvl +specific heat capacity at constant volume of condensible liquid (J/kg/K) +4119. +3381.55 +cpv +specific heat capacity at constant pressure of condensible gas (J/kg/K) +cvv + Rv +cvv + Rv +E0v +internal energy difference between vapor and liquid at the triple point (J/kg) +2.374×106 +4.9×105 +ptrip +pressure at condensible’s triple point (Pa) +611.65 +11700. +Ttrip +temperature at condensible’s triple point (K) +273.16 +90.68 +Table 1. Parameters for the convection-resolving experiments in the Earth-like and Titan-like model configurations. +which prevents relative humidity from exceeding 100% (i.e., abundant cloud condensation nuclei are assumed to be +present) and evaporates cloud condensate in subsaturated air. Conversion of non-precipitating cloud condensate to +rain is modeled as autoconversion according to +a = −qc/τa, +(2) +where a (s−1) is the sink of cloud condensate from autoconversion and τa (s) is an autoconversion timescale. We use +τa = 25 minutes, which was found in prior work to produce a similar mean cloud fraction profile as the default bulk +scheme in DAM (Seeley et al. 2020). We did not set an autoconversion threshold for qc. Rain is given a fixed freefall +speed with a default value of 8 m/s, but we also checked the sensitivity of our results to this value when appropriate +(the VaryGrav fs experiment). When rain falls through subsaturated air, it is allowed to evaporate according to +e = (q∗ +v − qv)/τr, +(3) +where e (s−1) is the rate of rain evaporation, q∗ +v is the saturation specific humidity, and τr (s) is a rain-evaporation +timescale. We set τr = 50 hours, which was found in prior work to produce a tropospheric relative humidity profile +similar to that of the bulk scheme (Seeley et al. 2020). Since microphysics on Titan is very poorly constrained, we used +the same values for these microphysical constants in both the Earth-like and Titan-like model configurations; future +work could attempt to use first-principles theories to tune microphysical parameters for the Titan regime (Lorenz 1993; +Loftus & Wordsworth 2021). +2.5. Thermodynamics +The principal difference between our Earth-like and Titan-like model configurations pertains to the atmospheric +thermodynamics. The thermodynamics of moist air in DAM is based on a standard set of approximations applying +to mixtures of dry air and a condensible component which may be present in vapor, liquid, and solid form. These +approximations are 1) both dry air and the condensible vapor are treated as ideal gases, 2) the heat capacities of all +components are assumed not to depend on temperature, and 3) condensates are assumed to have zero specific volume +(Ambaum 2010; Romps 2008, 2021). +In our convection-resolving experiments, we use a simplified treatment of the condensible component by neglecting +the solid phase, which is not of leading importance for convective dynamics in Earth’s tropics (Seeley & Romps 2016). +Therefore, at all temperatures saturation is evaluated via the expression for the saturation vapor pressure over liquid, +which is derived from the above approximations and implemented in DAM as +p∗ +v = ptrip +� T +Ttrip +�(cpv−cvl)/Rv +exp +�Le(Ttrip) +RvTtrip +− Le(T) +RvT +� +, +(4) +where Le is the temperature-dependent latent enthalpy of evaporation given by +Le(T) = E0v + RvT + (cvv − cvl)(T − Ttrip), +(5) + +5 +and where the other physical constants appearing in equations (4–5) are defined in Table 1. +The values assigned to these and other physical constants determine whether DAM simulates Earth-like or Titan-like +atmospheric composition and moist thermodynamics (Table 1). For our Earth-like configuration, dry air is assumed to +be a mixture of 80% N2 and 20% O2, while the condensible component is water (H2O). For the Titan-like configuration, +dry air is assumed to be entirely N2 and the condensible component is methane (CH4). +2.6. Surface fluxes +Surface fluxes were modeled with bulk aerodynamic formulae. Specifically, the surface latent and sensible heat fluxes +(LHF and SHF) were given by +LHF(x, y) = ρ1(x, y)CD +� +u1(x, y)2 + v1(x, y)2 + V 2Le [q∗ +s − q1(x, y)] ; +(6) +SHF(x, y) = ρ1(x, y)CD +� +u1(x, y)2 + v1(x, y)2 + V 2cp [SST − T1(x, y)] , +(7) +where ρ1, q1, u1, v1, and T1 are the density, specific humidity, horizontal winds, and temperature at the first model +level, CD = 1.5×10−3 is a drag coefficient, V = 5 m/s is a background “gustiness”, Le is the latent heat of evaporation, +cp is the specific heat capacity at constant pressure of moist air, and q∗ +s is the saturation specific humidity at the sea +surface temperature and surface pressure. Since the time-mean surface enthalpy flux is constrained by the (imposed) +tropospheric radiative cooling, the values of CD and V determine the near-surface air-sea enthalpy disequilibrium but +do not otherwise affect our results. +3. RESULTS +200 +225 +250 +275 +300 +325 +350 +375 +temperature (K) +0 +20 +40 +60 +80 +100 +120 +height (km) +environment +adiabatic parcel +a +Temperature profiles +270 +290 +310 +330 +350 +370 +SST (K) +0 +8 +16 +24 +32 +(kJ/kg) +b +CAPE +Figure 1. (a) Temperature profiles of the environment (dashed) and adiabatically-lifted near-surface parcels (solid) from the +EarthTemp experiment; the area between these two profiles is shaded where the parcel is warmer than the environment. For +visual clarity, a subset of surface temperature cases are plotted, and the environmental temperature profile is only plotted where +it is cooler than the adiabatic parcel. (b) Convective available potential energy (CAPE), defined as the vertically-integrated +positive buoyancy of the lifted parcels whose temperature profiles are shown in panel (a). The condensate mass fraction in the +lifted parcels was subjected to an exponential-decay sink term with a vertical length scale of L = 5 km; see Figure B1 for the +effect of different condensate fallout assumptions. +We begin our study of convective vigor with a focus on CAPE. CAPE is the maximum specific vertical kinetic +energy, w2/2 (where w is vertical velocity), that clouds can attain while rising, so it is a useful summary statistic + +6 +for the potential vigor of convection. Here we calculate CAPE as the vertically-integrated positive buoyancy b of an +adiabatically-lifted parcel between its lifted condensation level (LCL) and its level of neutral buoyancy (LNB): +CAPE = +� LNB +LCL +max(b, 0) dz, +(8) +where the buoyancy of the parcel is b = g(ρe/ρp − 1) for parcel density ρp and environmental density ρe. +We first examine CAPE in the EarthTemp experiment, for which the surface temperature was varied in the Earth-like +model configuration. Figure 1a shows, for a subset of surface temperature cases, the environmental temperature profile +(time- and horizontal-mean) compared to the temperature profile of an adiabatically-lifted parcel that is initialized +with the thermodynamic properties of mean near-surface air. There is a clear pattern in these temperature profiles: +at low and high surface temperatures, the environment and the adiabat2 are nearly indistinguishable, whereas for +intermediate surface temperatures the adiabat is significantly warmer than the environment, especially in the upper +troposphere. We will describe the physical explanation for this behavior in section 4. Since CAPE measures the +integrated buoyancy of an adiabatic parcel, CAPE is therefore small at both low and high surface temperatures, with +a peak in between (Fig. 1b). While the quantitative magnitude and location of this peak in CAPE are somewhat +sensitive to assumptions about condensate fallout in the lifted adiabatic parcel (Fig. B1), the existence of a peak is +robust to these details. +280 +300 +320 +340 +360 +SST (K) +0 +15 +30 +45 +wmax (m/s) +max CAPE +a +EarthTemp +104 +105 +106 +ps (Pa) +0 +15 +30 +45 +max CAPE +updraft-mean +99.9th %ile +b +EarthPressure +80 +90 +100 +110 +SST (K) +0 +3 +6 +9 +12 +max CAPE +c +Titan +Figure 2. Metrics of actual convective vigor for the (a) EarthTemp, (b) EarthPressure, and (c) Titan experiments. The metric +wmax refers to the maximum tropospheric value of the profile of mean updraft velocity (diamond markers) or the profile of +99.9th percentile of vertical velocity at each altitude (circle markers). In each panel, the location on the x-axis with maximum +CAPE is marked by the triangle at the top of the plot +. In panel (a), the black open circles and black open squares show w99.9 +max from the simulations at 275, 325, and 355 K with high +resolution (EarthTemp hr) and with interactive clear-sky radiation (EarthTemp realrad), respectively. Note that panel (c) has +a different y-axis limit than panels (a) and (b). +The growth and decline of CAPE with warming in the EarthTemp experiment appears to connect the previously +mentioned “warming-driven invigoration” and “gentle pure-steam limit” regimes. The near-surface specific humidity +in these simulations ranges from about 0.2% in the coldest simulation to about 60% in the warmest, confirming that +water vapor transitions from being a very minor trace gas to a dominant component. However, CAPE only measures +the potential vigor of convection. How do actual convective updraft speeds vary in these simulations? Figure 2a +shows two metrics of actual convective vigor from the EarthTemp experiment. The first of these metrics, wmean +max , +is calculated by conditionally sampling all grid cells with w > 1 m/s and cloud condensate qc > 10−5 kg/kg. The +horizontal-mean vertical velocity of these cloudy-updraft grid cells is calculated at each tropospheric vertical grid level, +and the maximum value of this profile is what we refer to as wmean +max (labeled “updraft-mean” in Fig. 2b). The second +metric of convective vigor is calculated from the histogram of vertical velocities at each tropospheric vertical grid +2 For brevity, we will refer to the temperature profile of a lifted, undiluted near-surface parcel as an “adiabat”, ignoring the small effect of +buoyancy on the parcel’s lapse rate (e.g., Riehl & Malkus 1958; Romps 2015). Note that the “adiabats” plotted in Figure 1 are intermediate +between the pseudo-adiabatic process (all condensates removed) and the reversible process (all condensates retained). Specifically, the +condensate sink term from fallout is modeled as ∂qc +∂z |fall = −qc/L + +7 +level; these histograms are not conditionally sampled based on cloud condensate. We calculate the 99.9th percentile +of each tropospheric grid level’s vertical velocity histogram, and the maximum value of this vertical profile is what +we refer to as w99.9 +max (labeled “99.9th percentile” in Fig. 2b). Similar to CAPE, these metrics also show an initial +growth and eventual decline with increasing surface temperature, with the warmest simulation actually having slower +wmax values than the coldest. Note that although the qualitative behavior of wmax is similar to CAPE, the peaks in +the wmax metrics occur at a lower surface temperature than the CAPE peak, and wmax is right-skewed while CAPE +is left-skewed. +Other choices of summary statistics for convective vigor lead to peaks at slightly different surface +temperatures, but the overall phenomenon is robust to these choices. The peak in convective vigor is also robust to +increased horizontal and vertical resolution (the EarthTemp hr experiment; Fig. 2a, open circles) and use of realistic +clear-sky radiation (the EarthTemp realrad experiment; Fig. 2a, open squares). +200 +220 +240 +260 +280 +300 +temperature (K) +0 +10 +20 +30 +40 +50 +60 +70 +height (km) +environment +adiabatic parcel +a +Temperature profiles +104 +105 +106 +ps (Pa) +0 +5 +10 +15 +20 +(kJ/kg) +b +CAPE +Figure 3. As in Figure 1, but for the EarthPressure experiment. Note that in (b) the horizontal axis is inverted so that specific +humidity increases toward the right. +How general is this peak in convective vigor with respect to atmospheric humidity? In the EarthTemp experiment, +the increase in q∗ +v is driven by the increasing surface temperature and associated Clausius-Clapeyron scaling of the +saturation vapor pressure. However, specific humidity can also be increased, at fixed temperature, by lowering the +amount of non-condensing background gas (Wordsworth & Pierrehumbert 2013). Would varying the surface pressure, +therefore, also produce variations in CAPE and convective vigor? To test this, we turn to the EarthPressure experiment, +in which we fixed the surface temperature at 300 K but varied the surface pressure from 8×105 Pa to 6250 Pa +(between a factor of 8× and 1/16× the contemporary value). Figure 3 shows that CAPE varies in EarthPressure in +qualitatively the same manner as in EarthTemp, reaching a peak at an intermediate surface pressure (around 2×104 +Pa, approximately 20% of the contemporary value). Likewise, Figure 2b shows that actual convective vigor in the +EarthPressure experiment also peaks at intermediate surface pressures, although there is again an offset between the +peak CAPE and the peak in actual convective vigor. Since Earth’s surface pressure is relatively unconstrained during +the Hadean and Archean (Kavanagh & Goldblatt 2015; Som et al. 2016), these results may have implications for the +early evolution of Earth’s climate and atmospheric chemistry; we will return to this topic in section 5. +If the boom-bust evolution of CAPE seen in the EarthTemp and EarthPressure experiments is attributable to +the transition from condensible-poor (dilute) to condensible-rich (non-dilute) conditions, this recipe is not specific to +Earth-like moist convection: in any atmosphere with a sufficiently large surface reservoir of a condensible species, the +condensible will become increasingly volatile with warming and eventually come to dominate atmospheric composition. +To what other planetary atmospheres, then, might this boom-bust CAPE behavior apply? A natural candidate is + +8 +70 +80 +90 +100 +110 +temperature (K) +0 +20 +40 +60 +80 +100 +120 +height (km) +environment +adiabatic parcel +a +Temperature profiles +80 +85 +90 +95 +100 +105 +110 +SST (K) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +(kJ/kg) +b +CAPE +Figure 4. As in Figures 1 and 3, but for the Titan experiment. +Saturn’s moon Titan, which is often regarded as the closest current hydrological analog to Earth due to its active +methane precipitation cycle. Therefore, to further generalize our understanding of convective vigor, we next turn to +the Titan experiment. This experiment assumes Titan-like thermodynamic conditions and atmospheric composition +(i.e., a thick N2 atmosphere with condensing CH4 and low gravity; Table 1). Similar to the Earth-like experiments, +Figure 4 shows a peak in CAPE as the simulated Titan-like atmospheres transition from dilute to non-dilute methane +abundance. Figure 2c shows that metrics of actual convective vigor in this experiment peak at a surface temperature +of about 95 K. Intriguingly, this is very close to the current surface temperature of Titan. +To better compare our three core experiments, it is helpful to recast the results in terms of variations in atmospheric +humidity rather than surface temperature or pressure. Figure 5 plots CAPE and high-percentile updraft speeds from +the core experiments as a function of the specific humidity at the lifted condensation level, q∗ +v,LCL. +This reveals +that CAPE and convective vigor in all three experiments peaks when cloud base air contains roughly 10% of the +condensible component by mass, give or take a factor of about 2. Therefore, we can conclude that the “warming- +driven invigoration” regime comes to an end at intermediate humidity, well before these atmospheres approach the +steam limit. +4. THE PHYSICAL ORIGIN OF THE CAPE PEAK +Taken all together, our three core experiments point to a potentially common phenomenon in terrestrial atmospheres: +moist convection is most vigorous at intermediate atmospheric humidity. What is the physical explanation for this +behavior? In this section, we show that recent advances in the theory of convection provide a quantitative explanation +for the peak in CAPE with respect to atmospheric humidity. +The basic ingredient required to understand climatological variations in CAPE is a theory for the tropospheric lapse +rate, which we denote by Γ(z) = −∂T/∂z (K/km). Note that the standard assumption in idealized 1-dimensional +radiative-convective modeling, which is that Γ(z) is given by the moist adiabat, is useless for the purpose of predicting +CAPE: the CAPE of a moist-adiabatic atmosphere is zero by definition. +It is the systematic deviations from a +moist-adiabatic thermal structure that a successful theory for CAPE must predict. +The key theoretical breakthrough in this regard was made by Singh & O’Gorman (2013) (hereafter, SO13), who +emphasized the role of entrainment. Entrainment refers to the turbulent mixing with environmental air that cloudy +updrafts experience as they ascend. Because entrainment of subsaturated air reduces condensation and latent heating +in ascending parcels, entraining parcels cool more rapidly with height than otherwise identical undiluted parcels. The +insight of SO13 is that the troposphere can be approximated as being neutrally buoyant with respect to such entraining + +9 +10−3 +10−2 +10−1 +100 +q * +v, LCL (kg/kg) +a +Earth-like, Ts varies +Earth-like, ps varies +Titan-like, Ts varies +CAPE +10−3 +10−2 +10−1 +100 +q * +v, LCL (kg/kg) +b +High-percentile updrafts +Figure 5. (a) Normalized CAPE from the EarthTemp, EarthPressure, and Titan experiments, plotted as a function of the +saturation specific humidity at the lifted condensation level q∗ +v,LCL. (b) As in (a), but for normalized high-percentile updraft +speeds (w99.9 +max). +convective parcels, rather than with respect to an undiluted parcel; this assumption has come to be known as the +“zero-buoyancy” (ZB) approximation3. Defining Γm as the lapse rate of an undiluted parcel and Γe as the lapse rate +of an entraining parcel (which, by the ZB approximation, is equal to Γ), we can state that Γm < Γ = Γe. According +to this picture, then, entrainment is the wedge that drives Γm and Γ apart, allowing for potentially large reservoirs of +CAPE even in steady-state RCE. +To make this discussion quantitative, we turn to the simplest model of a convecting atmosphere that incorporates +entrainment: the “bulk-plume model”4. The bulk-plume model divides the atmosphere into two plumes: ascending, +saturated (cloudy) air, and descending, subsaturated environmental air; the “bulk” descriptor refers to the fact that +the thermodynamic properties of these two plumes are assumed to be homogeneous in the horizontal at each altitude. +Mass exchange between the two plumes is represented by specified entrainment and detrainment rates, such that +conservation of mass in the bulk-plume model is expressed as +∂M +∂z = e − d += M(ε − δ), +(9) +where M (kg/m2/s) is the upward convective mass flux (equal and opposite, in RCE, to the subsidence mass flux in +the environmental plume), e and d (kg/m3/s) are the mass entrainment and detrainment rates, and ε and δ (m−1) +are known as the fractional entrainment and detrainment rates, defined as e/M and d/M, respectively. Equation (9) +implies that the convective mass flux increases with height if entrainment outpaces detrainment, and vice versa. +The second bulk-plume equation we will use is the conservation equation for moist static energy h, which is conven- +tionally defined as h = cpT + gz + Lqv. Here cp (J/kg/K) is the specific heat capacity of air at constant pressure, L +(J/kg) is the latent heat of evaporation, and the other symbols take their usual meaning. While this is an approxi- +mate expression for moist static energy — neglecting, for instance, the temperature-dependence of the latent heat of +evaporation (Romps 2015) — it is sufficiently accurate to form the basis of a theory for CAPE. The conservation of +moist static energy in the entraining convective plume is expressed as +∂(Mh∗) +∂z += ehe − dh∗, +(10) +3 We stress that this zero-buoyancy assumption is not a zero-CAPE assumption: it is the entraining convective parcels that are assumed to +have zero buoyancy with respect to the mean environment, not the adiabatic parcel that is used to calculate CAPE. +4 This discussion of the zero-buoyancy bulk-plume theory for CAPE is based on that in Romps (2016) and Romps (2020); we refer the reader +to these references for a more thorough treatment. + +10 +where h∗ = cpT + gz + Lq∗ +v is the saturation moist static energy (appropriate for the convective plume because it is +saturated). The environmental moist static energy can be expressed as he = cpT +gz +LRq∗ +v, where we have used the +same T as in the convective plume (invoking the ZB approximation) and where the environmental specific humidity +is given by the product of R, the environmental relative humidity, and the saturation specific humidity, q∗ +v. +Using the product rule on the left-hand side of equation (10), substituting in equation (9), and using the definitions +of h∗ and he given above, we arrive at an important result from SO13: +∂h∗ +∂z = −ε (1 − R) Lq∗ +v, +(11) +Equation 11 states that the entraining plume’s moist static energy decreases with height at a rate proportional to the +saturation deficit of the environment, (1 − R)q∗ +v (Seeley & Romps 2015). +Although SO13 treated the environmental relative humidity, R, as an external parameter that must be specified, +analysis of the bulk-plume water budget can yield a self-consistent analytic expression for R (Romps 2014): +R = δ + αγ − αε +δ + γ − αε . +(12) +In equation 12, α is a nondimensional parameter specifying the fraction of condensates formed at a given height that +evaporate at that height instead of precipitating out of the atmosphere (i.e., the precipitation efficiency of the bulk- +plume convection is 1 − α). The quantity γ is the water vapor lapse rate, defined as γ ≡ −∂ ln q∗ +v/∂z and expressed in +terms of thermodynamic parameters as +γ = +LΓ +RvT 2 − +g +RT , +(13) +where Rv and R (J/kg/K) are the specific gas constants for water vapor and dry air, respectively. The expression +(13) is straightforward to derive by combining the Clausius-Clapeyron equation for the saturation vapor pressure with +hydrostatic balance (Romps 2014). +The final step toward the theory for CAPE was taken by Romps (2016). For analytic solubility, that work considered +a simplified case and assumed that M, R, and α are all constant with height. The constancy of M implies ε = δ, by +equation (9), while the constancy of R and α imply that the relative humidity and entrainment rate can be expressed +in terms of another constant, a ≥ 0, as follows: +R = a + α +1 + a , +(14) +ε = a +� +γ +1 − α +� +. +(15) +Note that a, which we will refer to as the “bulk-plume parameter”, is proportional to the entrainment rate, so that +a = 0 corresponds to undiluted convection. Plugging in equations (14–15) to the right-hand side of equation (11) +yields +∂h∗ +∂z = − +a +1 + aγLq∗ +v. +(16) +We can obtain a second equation for ∂h∗/∂z by differentiating the expression for h∗ directly: +∂h∗ +∂z = −cpΓ + g − γLq∗ +v += g +� +1 + q∗ +vL +RT +� ++ Γ +� +cp + q∗ +vL2 +RvT 2 +� +, +(17) +where the second line follows from substituting equation (13) for γ. Equating the right-hand sides of equations (16) +and (17) and solving for Γ, we obtain +Γ = +� g +cp +� � +1 + a + q∗ +vL/(RT) +1 + a + q∗vL2/(cpRvT 2) +� +. +(18) +Equation (18) is a generalization of the moist lapse rate that accounts for the effect of entrainment (Romps 2020). +In the limit of no entrainment (a → 0), equation 18 reduces to the standard expression for the moist adiabat, Γm: +Γm = +� g +cp +� � +1 + q∗ +vL/(RT) +1 + q∗vL2/(cpRvT 2) +� +. +(19) + +11 +We can use equation (18) to analyse the difference between Γ and Γm — and, therefore, CAPE — in the limit of very +dry and very moist atmospheres. In the dry limit (q∗ +v → 0), which is approached by reducing the surface temperature +or increasing the amount of background dry air, we can drop all terms multiplied by q∗ +v inside the brackets in equation +(18), which allows the factors of (1 + a) in the numerator and denominator to cancel and yields Γm = g/cp = Γd, the +dry-adiabatic lapse rate. Hence entrainment has no effect on the lapse rate in the dry limit, Γ and Γm are both equal +to the dry adiabat, and CAPE is zero. On the other hand, in the very moist limit (q∗ +v → 1), which is approached +by increasing the surface temperature or decreasing the amount of background dry air, the terms multiplied by q∗ +v +dominate over the factors of a for typical Earth and Titan-like conditions. For example, for Earth-like conditions5, +the factor L/(RT) ranges from about 25–50, while the factor L2/(cpRvT 2) ranges from about 100–500, in both cases +dominating over a, which typically takes on values of O(0.1) − O(1) (Romps 2016). Hence the entraining lapse rate +asymptotes to Γ ≃ gT/L — which, as in the dry limit, is independent of the entrainment rate, and also equal to the +moist limit of Γm. To summarize, equation (18) suggests that we should expect minimal CAPE in both the dry and +moist limits, with a peak in between for the intermediate values of q∗ +v that allow entrainment to drive a wedge between +Γ and Γm. +270 +290 +310 +330 +350 +370 +SST (K) +0 +5 +10 +15 +20 +25 +30 +35 +40 +a=0.2 +a=0.8 +a +EarthTemp +104 +105 +106 +ps (Pa) +0 +7 +14 +21 +R16 theory +a=0.2 +a=0.8 +b +EarthPressure +CRM +80 +85 +90 +95 +100 +105 +110 +SST (K) +0 +1 +2 +3 +4 +CAPE (kJ/kg) +a=0.5 +a=1.5 +c +Titan +Figure 6. +CAPE from the (a) EarthTemp, (b) EarthPressure, and (c) Titan experiments from the convection-resolving +simulations (colored markers) and from the analytical predictions for CAPE given by equation (A1) (gray shaded areas). The +analytical predictions are shown for values of the bulk-plume parameter a ranging from 0.2 to 0.8 for the Earth-like experiments, +and for a ranging from 0.5 to 1.5 for the Titan-like experiment. The analytical solutions are initialized with the temperature +and humidity of the lifted condensation level from the corresponding simulation. +Quantitatively, the magnitude and atmospheric humidity of the peak in CAPE can be predicted using the analytic +formalism of Romps (2016). Figure 6 shows analytical predictions for CAPE using the theory of R16 (reproduced in our +equation A1), in comparison to the results from our convection-resolving simulations. The theory clearly captures the +boom-bust evolution of CAPE in all three core experiments, providing the theoretical bridge between the “warming- +driven invigoration” and the “gentle pure-steam limit” regimes. While R16 applied their analytic theory for CAPE +to the context of surface warming on Earth, here we have shown that the same essential physics explains convective +vigor in convection-resolving simulations with varied surface temperature, varied surface pressure, and with Titan-like +thermodynamics. The notion that entraining and adiabatic temperature profiles converge in non-dilute atmospheres +resembles the arguments given by Ding & Pierrehumbert (2016) and Pierrehumbert & Ding (2016), although our +results suggest that this physics begins to operate well before the steam limit is reached. Ding & Pierrehumbert +(2016) and Pierrehumbert & Ding (2016) also argued that environmental relative humidity should approach 1 as the +atmosphere becomes increasingly non-dilute, a prediction that is confirmed by our results (Fig. B2). +In addition to explaining the behavior of CAPE in our convection-resolving simulations, the theory of R16 provides +clarity on which planetary parameters control CAPE. An interesting attribute of equation A1 is that it has no explicit +dependence on a planet’s gravitational constant, g. Physically, this can be understood as a cancellation between two +5 We leave investigation of whether this limit holds in more exotic circumstances, such as exoplanet silicate atmospheres with temperatures +in the thousands of kelvins (e.g., Kang et al. 2021), to future work. + +12 +factors: 1) for a given parcel temperature anomaly, there is a direct proportionality between the parcel’s buoyancy and +g; and 2) for a given surface temperature and tropopause temperature, there is an inverse proportionality between +the geometric depth of the atmosphere and g; hence reducing gravity lowers the integrand in equation 8 but increases +the domain of vertical integration by a compensating amount. +To test this prediction of an insensitivity to g, we ran a subset of surface temperature cases from the EarthTemp +experiment with either enhanced (g = 25 m/s2) or reduced (g = 3.5 m/s2) gravitational constant (the VaryGrav +experiment); these values of g approximately cover surface conditions for solar system planets ranging in mass from +Mercury to Jupiter. We find a weak dependence of CAPE on g (i.e., CAPE varies by a factor of ≃2 when g varies by a +factor of ≃7), in rough accordance with the theoretical prediction, and also find a correspondingly small sensitivity in +our metrics of actual convective vigor to g. The small variations in CAPE and convective vigor likely result from the +effect of g on the nature of turbulence in the simulations, which would affect both the entrainment rate that enters into +the theory for CAPE (through the bulk-plume parameter a) and the drag experienced by actual convecting parcels. +These experiments with varied gravity also afford an opportunity to test the sensitivity of our results to the precipi- +tation fall speed parameter, which some studies have suggested is a key control on updraft speeds (Parodi & Emanuel +2009). Assuming a conservative6 linear dependence of precipitation fall speed on g, we re-ran the cases with g = 3.5 +m/s2 and g = 25 m/s2 with the precipitation fall speeds modified to 2.8 m/s and 20 m/s, respectively. We found this +had a minimal effect, with changes in CAPE and convective vigor generally limited to ±5%. +5. DISCUSSION +Using an idealized convection-resolving model, we have demonstrated that convective vigor is expected to peak at +intermediate concentrations of the condensing substance in a diversity of planetary circumstances. However, more work +is needed to build our results into a universal understanding of convective vigor. Our simulations of local radiative- +convective equilibrium are highly idealized, and many questions remain about how the peak in convective vigor with +respect to atmospheric humidity would manifest in more realistic modeling setups that include coupling to large-scale +circulations (Pierrehumbert & Ding 2016) or a diurnal cycle. Another promising avenue for extension concerns our +Titan-like simulations, for which we assumed a limitless supply of surface evaporation. In reality, the surface of Titan is +quite arid (Schneider et al. 2012), and future work could explore convective vigor in the regime where the atmospheric +condensible inventory is comparable to the total inventory. +Additionally, although we have successfully applied the theory of R16 to our results, that theory is limited in its +general applicability because it approximates the specific gas constant and heat capacity of moist air by those of the dry +component. This is a tolerable source of error when the dry and condensing components do not differ too much in molar +mass, as in the case of H2O condensing in an N2/O2 mixture or CH4 condensing in N2, but this approximation breaks +down when the background gas is very light (e.g., mainly H2). In that case, the atmospheric scale height can collapse +with warming as the atmosphere comes to be dominated by the relatively heavier condensing component (Koll & Cronin +2019). Since many planets form with a primordial hydrogen envelope, this is an important class of atmospheres to +which the R16 theory and our simulation results may not apply. Additional theoretical and computational work could +investigate this regime, for which the “virtual” effects of compositional differences on buoyancy may be crucial. +In additional to building fundamental understanding of moist convection, our results may also have implications for +planetary evolution on long time scales. The rate at which a terrestrial planet loses water depends on stratospheric +humidity, because water transported to the stratosphere becomes vulnerable to photolysis and subsequent loss of +H to space (Kasting 1988). +Purely thermodynamic arguments suggest that stratospheric moistening in planetary +atmospheres depends on both surface temperature and surface pressure, with the transition to a moist stratosphere +occurring at intermediate humidities (Wordsworth & Pierrehumbert 2013). However, on present-day Earth, injection +of water into the stratosphere by intense convective storms plays an important role in setting the average stratospheric +water content (Corti et al. 2008). It may be that in real atmospheres, the increase in convective vigor at intermediate +humidity causes the “moist greenhouse” transition (Kasting 1988) to be approached more rapidly than either one- +dimensional radiative-convective models or 3D general circulation model simulations (e.g., Leconte et al. 2013) would +suggest. Further research using models that couple convection to large-scale dynamical and radiative processes is +required to investigate this possibility. +6 Theoretical results for monodisperse droplets predict a square-root dependence of raindrop terminal velocity on g (Loftus & Wordsworth +2021). + +13 +Furthermore, because the rate of lightning strikes in planetary atmospheres is believed to depend in part on convective +vigor, these results may also have important implications for lightning-driven atmospheric chemistry. Romps et al. +(2014) proposed that the lightning flash rate on modern Earth is proportional to the product of CAPE and precipitation +rate. If this relation is robust across wide ranges of planetary conditions, it implies that the importance of lightning +chemistry would be strongly enhanced in atmospheres with intermediate specific humidity. +Because both surface +temperature and atmospheric pressure may have varied significantly on Earth in the Hadean, this has interesting +implications for the rate of lightning-driven formation of important prebiotic molecules such as HCN during this +period (Chameides & Walker 1981; Ardaseva et al. 2017). +Data availability: Cloud-resolving model output and the code that generates the figures in this manuscript is +available at https://doi.org/10.5281/zenodo.7331932. +1 +2 +APPENDIX +A. ANALYTICAL EXPRESSION FOR CAPE FROM ROMPS (2016) +The solutions of R16 yield a closed-form expression for the CAPE of an atmosphere in RCE: +CAPE = R +2f {W(ya)[2 − 2f(Ts − Tt) + W(ya)] − W(e−f(Ts−Tt)ya)[2 + W(e−f(Ts−Tt)ya)] +− W(y0)2 − 2f(Ts − Tt) + W(y0)] + W(e−f(Ts−Tt)y0)[2 + W(e−f(Ts−Tt)y0)]}. +(A1) +where W is the Lambert W function defined by W(xex) = x, and where +f = +L +RvT 2 +0 +− cp +RT0 +, +and +(A2) +ya = +Lq∗ +vs +(1 + a)RT0 +exp +� +Lq∗ +vs +(1 + a)RT0 +� +. +(A3) +Here, R (J/kg/K) is the specific gas constant of dry (background) air, Rv (J/kg/K) is the specific gas constant of +the condensible vapor, L (J/kg) is the latent heat of condensation (assumed constant), cp (J/kg/K) is the specific +heat capacity at constant pressure of dry air, q∗ +vs (kg/kg) is the saturation specific humidity at the surface, and T0 +(K) is a constant reference temperature chosen to be the average of the surface temperature Ts and the tropopause +temperature Tt. The dimensionless parameter a characterizes the bulk-plume convection. In equation A1, y0 is given +by equation A3 with a = 0. +The expression for CAPE given above depends only on known physical constants, observable planetary conditions, +and a summary parameter characterizing the bulk-plume convection. The bulk-plume parameter a is proportional to +the entrainment rate7, and it is easy to verify that setting a = 0 (the limit of non-entraining convection) produces a +moist-adiabatic atmosphere with zero CAPE, as expected. We can also verify that equation A1 makes quantitatively +accurate predictions for Earth’s tropics today. We set a = 0.2, which corresponds to typical values of precipitation +efficiency and entrainment rate as diagnosed in cloud-resolving simulations, as discussed in R16. Additionally using +Ts = 300 K, q∗ +vs = 20 g/kg, L = 2.5 × 106 J/kg, and Tt = 200 K, equation A1 predicts CAPE ≃ 2500 J/kg, which is +indeed a typical value for convectively-active parts of Earth’s tropics (Riemann-Campe et al. 2009). +B. SUPPLEMENTAL FIGURES +7 Specifically, a = ϵPE/γ, where ϵ (1/m) is the fractional entrainment rate, the precipitation efficiency PE is defined as the ratio of net +condensation to gross condensation (assumed constant throughout the troposphere), and γ ≡ −∂z log q∗ +v (1/m) is the saturation water-vapor +lapse rate. One of the assumptions made by R16 for analytical tractability is that ϵ ∝ γ, so that a is constant. + +14 +275 +285 +295 +305 +315 +325 +335 +345 +355 +365 +SST (K) +0 +5 +10 +15 +20 +25 +30 +35 +CAPE (kJ/kg) +pseudoadiabatic +L = 5 km +L = 25 km +reversible +Figure B1. The effect of varying condensate fallout assumptions on parcel lifting calculations for computation of CAPE. The +condensed water mass fraction is assumed to have a sink term due to fallout that manifests as an exponential decay with height, +on a length scale L, during each discrete lifting step. The main text figures use L = 5 km, but here we also show the cases +L → 0 (pseudoadiabatic), L = 25 km, and L → ∞ (reversible). In all cases there is a peak in CAPE. +0.6 +0.8 +1.0 +relative humidity +200 +220 +240 +260 +280 +300 +320 +340 +360 +T (K) +EarthTemp +0.6 +0.8 +1.0 +relative humidity +200 +220 +240 +260 +280 +300 +EarthPressure +0.8 +0.9 +1.0 +relative humidity +70 +75 +80 +85 +90 +95 +100 +105 +110 +Titan +Figure B2. Profiles of environmental relative humidity in the (left) EarthTemp, (center) EarthPressure, and (right) Titan +experiments. Relative humidity is plotted as a function of mean temperature in the troposphere, as in Romps (2014). 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P. 2006, Bulletin of the American Meteorological +Society, 87, 1057, doi: 10.1175/BAMS-87-8-1057 + diff --git a/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/load_file.txt b/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa5cfe6b87529d5e1d1e5a636ff933470e3a63db --- /dev/null +++ b/hdE2T4oBgHgl3EQfHgaB/content/tmp_files/load_file.txt @@ -0,0 +1,761 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf,len=760 +page_content='Draft version January 11, 2023 Typeset using LATEX default style in AASTeX631 Moist convection is most vigorous at intermediate atmospheric humidity Jacob T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Seeley1 and Robin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Wordsworth1 1Department of Earth and Planetary Sciences Harvard University Cambridge, MA 02461, USA ABSTRACT In Earth’s current climate, moist convective updraft speeds increase with surface warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' This trend suggests that very vigorous convection might be the norm in extremely hot and humid atmo- spheres, such as those undergoing a runaway greenhouse transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, theoretical and numerical evidence suggests that convection is actually gentle in water vapor-dominated atmospheres, implying that convective vigor may peak at some intermediate humidity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Here, we perform small-domain convection-resolving simulations of an Earth-like atmosphere over a wide range of surface temperatures and confirm that there is indeed a peak in convective vigor, which we show occurs near Ts ≃ 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We show that a similar peak in convective vigor exists when the relative abundance of water vapor is changed by varying the amount of background (non-condensing) gas at fixed Ts, which may have implications for Earth’s climate and atmospheric chemistry during the Hadean and Archean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We also show that Titan-like thermodynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', a thick nitrogen atmosphere with condensing methane and low gravity) produce a peak in convective vigor at Ts ≃ 95 K, which is curiously close to the current surface temperature of Titan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Plotted as functions of the saturation specific humidity at cloud base, metrics of convective vigor from both Earth-like and Titan-like experiments all peak when cloud-base air contains roughly 10% of the condensible gas by mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Our results point to a potentially common phenomenon in terrestrial atmospheres: that moist convection is most vigorous when the condensible component is between dilute and non-dilute abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' INTRODUCTION Moist convection produces some of the most impactful weather and climate phenomena on Earth, from reflective stratocumulus layers to drenching seasonal monsoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' A longstanding challenge in climate science is to improve the crude treatment of such convective phenomena in general circulation models, which hampers predictions of critical quantities such as changes in local precipitation and the sensitivity of Earth’s climate to CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Looking beyond Earth, there is strong evidence that moist convection — that is, convection coupled to phase changes of a condensible substance — factors into the past and present evolution of the majority of solar system atmospheres, including Venus (Kasting 1988), Mars (Wordsworth 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Yamashita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2016), Jupiter (Gierasch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2000), Saturn (Li & Ingersoll 2015), the ice giants (Hueso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020), and Titan (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore, in order improve our understanding of contemporary Earth climate and planetary climate more generally, there is a clear need to deepen and generalize our understanding of moist convective physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In this paper, we take a step toward a more generalized understanding of moist convection by focusing on one of its most basic characteristics: convective vigor, or the buoyancy and vertical velocity of cloud updrafts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Zipser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Hansen & Back 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Updraft speeds affect instantaneous surface precipitation rates (Muller & Takayabu 2020) and the fraction of cloud condensate that is lofted instead of falling to the surface, which in turn exerts a strong influence on cloud cover and climate sensitivity (Zhao 2014, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' More vigorous updrafts are more likely to overshoot the tropopause and inject near-surface air into the stratosphere, thereby affecting stratospheric humidity and chemistry (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' O’Neill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2021) and planetary water loss on geologic timescales (Wordsworth & Pierrehumbert 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Lightning, which is closely associated with high updraft speeds (Deierling & Petersen 2008), further modulates atmospheric chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Focusing on a basic property such as convective vigor, therefore, may reveal patterns in moist convective behavior with widespread implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='03669v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='EP] 9 Jan 2023 2 Currently, there are two paradigms for how the vigor of moist convection depends on thermodynamic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The first paradigm, which we can call “warming-driven invigoration”, states that moist convective vigor increases with the surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' This paradigm has emerged from numerical modeling of Earth’s contemporary tropics: in idealized cloud-resolving simulations, mean and extreme convective updraft speeds increase with warming (Romps 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Muller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Seeley & Romps 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Singh & O’Gorman 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' These changes are consistent with, and typically attributed to, the increases in convective available potential energy (CAPE) that occur in both global and convection-resolving models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Sobel & Camargo 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Romps 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Muller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Singh & O’Gorman 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Seeley & Romps 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Although actual updrafts do not attain the velocities implied by CAPE due to drag and entrainment of unsaturated environmental air, CAPE is nevertheless an extremely useful proxy for intense convection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Johns & Doswell 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Romps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' As shown by Romps (2016), the increase of CAPE with warming under contemporary conditions is itself attributable to the Clausius-Clapeyron scaling of near-surface saturation specific humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore, the chain of causality in the “warming-driven invigoration” paradigm is: increasing Ts → increasing humidity → increasing CAPE → increasing convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The second paradigm for convective vigor is relevant to atmospheres in which the condensible component is highly non-dilute, with the limiting case being a pure steam atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Ding & Pierrehumbert (2016) and Pierrehumbert & Ding (2016) argued that convection should be gentle in such cases by reasoning that, when the atmosphere is saturated, the Clausius-Clapeyron curve dictates a one-to-one relationship between pressure and temperature, which precludes appreciable temperature anomalies between parcels at the same pressure level1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' These theoretical predictions gained support from the work of Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (2021), who conducted idealized convection-resolving simulations of pure steam atmospheres with surface temperatures of 600–800 K and found quiescent condensing layers with small temperature variability and weak updrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Hence the line of reasoning in the “gentle pure-steam limit” is: very high humidity → unique relationship between Tand p → low convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Clearly, “warming-driven invigoration”, when extrapolated to very warm temperatures, conflicts with the “gentle pure- steam limit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Taken together, these paradigms suggest that convective vigor should reach a peak at some intermediate humidity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In this paper, we seek to generalize our understanding of moist convective vigor by filling the gap between these two paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Our approach is to simulate a wide range of planetary atmospheres that differ appreciably from contemporary Earth in terms of their surface temperature, surface pressure, gravitational constant, and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' These atmospheres span the parameter space from highly dilute to non-dilute conditions, allowing us to continuously probe the transitional behavior between the two paradigms for convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To circumvent the inherent uncertainty of convective parameterizations and allow detailed analysis of convective dynamics, we performed these simulations with a flexible convection-resolving model (described in detail in section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We analyze metrics of convective vigor in these atmospheres, and show that our results match the predictions of an analytical theory for radiative-convective equilibrium (RCE) originally developed in the context of Earth’s tropics (Romps 2016) (hereafter, R16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The analytical theory of R16 predicts the mean CAPE of a convecting atmosphere, and we show that this theory provides a unifying theoretical framework that links the two paradigms for convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In section 2, we describe the flexible convection-resolving model and our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Section 3 analyzes convective vigor in the simulated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In section 4, we apply the theory of R16 to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We conclude in section 5 with a discussion of the implications of our work for early Earth and other planetary atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' EXPERIMENTAL METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Core experiments Our core suite of convection-resolving RCE simulations consists of three experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The first core experiment used the Earth-like model configuration, with the total surface pressure fixed at the contemporary value of 105 Pa and the surface temperature varying from 275 K to 365 K (experiment name EarthTemp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In the second core Earth-like 1 This argument neglects “virtual” effects of compositional differences and condensates on density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 3 experiment (EarthPressure), the surface temperature was instead fixed at 300 K while the surface pressure varied from (1/16)× to 8× the contemporary value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The third core experiment (Titan) used the Titan-like configuration of the model with the surface pressure fixed at the contemporary value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='467×105 Pa and the surface temperature varying from 80 to 110 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The parameters distinguishing the Earth-like and Titan-like configurations will be described in detail below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' see also Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' A number of additional simulations were performed as sensitivity tests, which we will describe as they come up in the course of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Convection-resolving model For all experiments, we simulated nonrotating radiative-convective equilibrium on square, doubly periodic domains with the convection-resolving model DAM (Romps 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' DAM has a finite-volume, fully-compressible dynamical core and uses the implicit approach to sub-grid diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The vertical grid had a variable spacing, transitioning smoothly from ∆z = 50 m below an altitude of 500 m to ∆z = 1000 m at altitudes between 10 km and the model top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The model top was at a variable height because our simulated tropospheres vary widely in geometric depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The default horizontal resolution was ∆x = ∆y = 2 km and the default horizontal domain size was Lx = Ly = 96 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We also ran a small subset of simulations at a higher resolution of ∆x = 500 m and with free-tropospheric ∆z = 500 m (the EarthTemp hr experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The default model time step was ∆t = 20 s, sub-stepped to satisfy a CFL condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' this time step was used for all simulations but EarthTemp hr, which used ∆t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Overall, the model domains are similar in size and resolution to the “RCE small” protocol from the RCEMIP project (Wing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2017) in which DAM participated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Radiative transfer To avoid inessential complexity and focus attention on convective dynamics, we used a simplified treatment of radiative transfer in the majority of our simulations (as in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Specifically, we prescribed an idealized tropospheric radiative cooling using an equation of the form: − ∂ ∂T F = α(T − Tt), (1) where T is the temperature, the temperature derivative ∂/∂T is a vertical derivative, α (W/m2/K2) is a constant setting the magnitude of radiative forcing, F (W/m2) is the net upward radiative flux, and Tt (K) is the prescribed tropopause temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Numerically, we approximated the vertical derivative ∂/∂T with a centered finite difference on the model’s vertical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Equation 1 was proposed by Jeevanjee & Romps (2018) as a fit to the invariant radiative flux divergence curve found for Earth, and has been used as an idealized representation of tropospheric radiative cooling in more recent work (Jeevanjee & Zhou 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We used equation (1) to prescribe radiative forcing for both Earth-like and Titan-like experiments, although we used different values of the parameters (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' At altitudes above the tropopause, temperatures were simply nudged to Tt on a timescale of 6 hours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' since our focus is on convective dynamics in the troposphere, this simplified approach to the stratosphere does not affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' One deficiency of equation (1) is that it does not predict the transition to lower-tropospheric radiative heating in very warm climates, a regime which is known to affect convective dynamics (Seeley & Wordsworth 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore, we also re-ran the EarthTemp experiment with interactive clear-sky radiative transfer (the EarthTemp realrad experiment) to check that our main results are not sensitive to the simplified treatment of radiation we employ in our core experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We focused on interactive clear-sky radiation because that is sufficient to produce the low-level radiative heating regime of Seeley & Wordsworth (2021), and because DAM is not coupled to a cloud-radiation scheme that is validated up to the very high temperatures we simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Our approach to clear-sky radiative transfer was identical to the line-by-line method of Seeley & Wordsworth (2021), and we refer the reader to that paper for a complete description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Microphysics The default microphysics scheme in DAM is a bulk scheme with six water classes (vapor, cloud liquid, cloud ice, rain, snow, and graupel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, to avoid relying on an overly Earth-centric parameterization, we conducted our simulations using a simplified microphysics scheme that has been described and used in previous work (Seeley & Wordsworth 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In the simplified microphysics scheme, only three bulk classes of condensible substance are modeled: vapor, non-precipitating cloud liquid, and rain, with associated mass fractions qv, qc, and qr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Microphysical transformations between vapor and cloud condensate are handled by a saturation adjustment routine, 4 parameter description Earth-like Titan-like g gravitational constant (m/s2) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='35 Tt tropopause temperature (K) 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' α radiative forcing constant (W/m2/K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 Ra specific gas constant of dry air (J/kg/K) 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='04 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 Rv specific gas constant of condensible gas (J/kg/K) 461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='28 cva specific heat capacity at constant volume of dry air (J/kg/K) 719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2 cvv specific heat capacity at constant volume of condensible gas (J/kg/K) 1418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='4 cvl specific heat capacity at constant volume of condensible liquid (J/kg/K) 4119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 3381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='55 cpv specific heat capacity at constant pressure of condensible gas (J/kg/K) cvv + Rv cvv + Rv E0v internal energy difference between vapor and liquid at the triple point (J/kg) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='374×106 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9×105 ptrip pressure at condensible’s triple point (Pa) 611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='65 11700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Ttrip temperature at condensible’s triple point (K) 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='16 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='68 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Parameters for the convection-resolving experiments in the Earth-like and Titan-like model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' which prevents relative humidity from exceeding 100% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', abundant cloud condensation nuclei are assumed to be present) and evaporates cloud condensate in subsaturated air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Conversion of non-precipitating cloud condensate to rain is modeled as autoconversion according to a = −qc/τa, (2) where a (s−1) is the sink of cloud condensate from autoconversion and τa (s) is an autoconversion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We use τa = 25 minutes, which was found in prior work to produce a similar mean cloud fraction profile as the default bulk scheme in DAM (Seeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We did not set an autoconversion threshold for qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Rain is given a fixed freefall speed with a default value of 8 m/s, but we also checked the sensitivity of our results to this value when appropriate (the VaryGrav fs experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' When rain falls through subsaturated air, it is allowed to evaporate according to e = (q∗ v − qv)/τr, (3) where e (s−1) is the rate of rain evaporation, q∗ v is the saturation specific humidity, and τr (s) is a rain-evaporation timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We set τr = 50 hours, which was found in prior work to produce a tropospheric relative humidity profile similar to that of the bulk scheme (Seeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Since microphysics on Titan is very poorly constrained, we used the same values for these microphysical constants in both the Earth-like and Titan-like model configurations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' future work could attempt to use first-principles theories to tune microphysical parameters for the Titan regime (Lorenz 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Loftus & Wordsworth 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Thermodynamics The principal difference between our Earth-like and Titan-like model configurations pertains to the atmospheric thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The thermodynamics of moist air in DAM is based on a standard set of approximations applying to mixtures of dry air and a condensible component which may be present in vapor, liquid, and solid form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' These approximations are 1) both dry air and the condensible vapor are treated as ideal gases, 2) the heat capacities of all components are assumed not to depend on temperature, and 3) condensates are assumed to have zero specific volume (Ambaum 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Romps 2008, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In our convection-resolving experiments, we use a simplified treatment of the condensible component by neglecting the solid phase, which is not of leading importance for convective dynamics in Earth’s tropics (Seeley & Romps 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' at all temperatures saturation is evaluated via the expression for the saturation vapor pressure over liquid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' which is derived from the above approximations and implemented in DAM as p∗ v = ptrip � T Ttrip �(cpv−cvl)/Rv exp �Le(Ttrip) RvTtrip − Le(T) RvT � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (4) where Le is the temperature-dependent latent enthalpy of evaporation given by Le(T) = E0v + RvT + (cvv − cvl)(T − Ttrip),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (5) 5 and where the other physical constants appearing in equations (4–5) are defined in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The values assigned to these and other physical constants determine whether DAM simulates Earth-like or Titan-like atmospheric composition and moist thermodynamics (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' For our Earth-like configuration, dry air is assumed to be a mixture of 80% N2 and 20% O2, while the condensible component is water (H2O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' For the Titan-like configuration, dry air is assumed to be entirely N2 and the condensible component is methane (CH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Surface fluxes Surface fluxes were modeled with bulk aerodynamic formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Specifically, the surface latent and sensible heat fluxes (LHF and SHF) were given by LHF(x, y) = ρ1(x, y)CD � u1(x, y)2 + v1(x, y)2 + V 2Le [q∗ s − q1(x, y)] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (6) SHF(x, y) = ρ1(x, y)CD � u1(x, y)2 + v1(x, y)2 + V 2cp [SST − T1(x, y)] , (7) where ρ1, q1, u1, v1, and T1 are the density, specific humidity, horizontal winds, and temperature at the first model level, CD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5×10−3 is a drag coefficient, V = 5 m/s is a background “gustiness”, Le is the latent heat of evaporation, cp is the specific heat capacity at constant pressure of moist air, and q∗ s is the saturation specific humidity at the sea surface temperature and surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Since the time-mean surface enthalpy flux is constrained by the (imposed) tropospheric radiative cooling, the values of CD and V determine the near-surface air-sea enthalpy disequilibrium but do not otherwise affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' RESULTS 200 225 250 275 300 325 350 375 temperature (K) 0 20 40 60 80 100 120 height (km) environment adiabatic parcel a Temperature profiles 270 290 310 330 350 370 SST (K) 0 8 16 24 32 (kJ/kg) b CAPE Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (a) Temperature profiles of the environment (dashed) and adiabatically-lifted near-surface parcels (solid) from the EarthTemp experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' the area between these two profiles is shaded where the parcel is warmer than the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' For visual clarity, a subset of surface temperature cases are plotted, and the environmental temperature profile is only plotted where it is cooler than the adiabatic parcel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (b) Convective available potential energy (CAPE), defined as the vertically-integrated positive buoyancy of the lifted parcels whose temperature profiles are shown in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The condensate mass fraction in the lifted parcels was subjected to an exponential-decay sink term with a vertical length scale of L = 5 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' see Figure B1 for the effect of different condensate fallout assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We begin our study of convective vigor with a focus on CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' CAPE is the maximum specific vertical kinetic energy, w2/2 (where w is vertical velocity), that clouds can attain while rising, so it is a useful summary statistic 6 for the potential vigor of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Here we calculate CAPE as the vertically-integrated positive buoyancy b of an adiabatically-lifted parcel between its lifted condensation level (LCL) and its level of neutral buoyancy (LNB): CAPE = � LNB LCL max(b, 0) dz, (8) where the buoyancy of the parcel is b = g(ρe/ρp − 1) for parcel density ρp and environmental density ρe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We first examine CAPE in the EarthTemp experiment, for which the surface temperature was varied in the Earth-like model configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 1a shows, for a subset of surface temperature cases, the environmental temperature profile (time- and horizontal-mean) compared to the temperature profile of an adiabatically-lifted parcel that is initialized with the thermodynamic properties of mean near-surface air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' There is a clear pattern in these temperature profiles: at low and high surface temperatures, the environment and the adiabat2 are nearly indistinguishable, whereas for intermediate surface temperatures the adiabat is significantly warmer than the environment, especially in the upper troposphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We will describe the physical explanation for this behavior in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Since CAPE measures the integrated buoyancy of an adiabatic parcel, CAPE is therefore small at both low and high surface temperatures, with a peak in between (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' While the quantitative magnitude and location of this peak in CAPE are somewhat sensitive to assumptions about condensate fallout in the lifted adiabatic parcel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' B1), the existence of a peak is robust to these details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 280 300 320 340 360 SST (K) 0 15 30 45 wmax (m/s) max CAPE a EarthTemp 104 105 106 ps (Pa) 0 15 30 45 max CAPE updraft-mean 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9th %ile b EarthPressure 80 90 100 110 SST (K) 0 3 6 9 12 max CAPE c Titan Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Metrics of actual convective vigor for the (a) EarthTemp, (b) EarthPressure, and (c) Titan experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The metric wmax refers to the maximum tropospheric value of the profile of mean updraft velocity (diamond markers) or the profile of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9th percentile of vertical velocity at each altitude (circle markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In each panel, the location on the x-axis with maximum CAPE is marked by the triangle at the top of the plot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In panel (a), the black open circles and black open squares show w99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9 max from the simulations at 275, 325, and 355 K with high resolution (EarthTemp hr) and with interactive clear-sky radiation (EarthTemp realrad), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Note that panel (c) has a different y-axis limit than panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The growth and decline of CAPE with warming in the EarthTemp experiment appears to connect the previously mentioned “warming-driven invigoration” and “gentle pure-steam limit” regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The near-surface specific humidity in these simulations ranges from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2% in the coldest simulation to about 60% in the warmest, confirming that water vapor transitions from being a very minor trace gas to a dominant component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, CAPE only measures the potential vigor of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' How do actual convective updraft speeds vary in these simulations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 2a shows two metrics of actual convective vigor from the EarthTemp experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The first of these metrics, wmean max , is calculated by conditionally sampling all grid cells with w > 1 m/s and cloud condensate qc > 10−5 kg/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The horizontal-mean vertical velocity of these cloudy-updraft grid cells is calculated at each tropospheric vertical grid level, and the maximum value of this profile is what we refer to as wmean max (labeled “updraft-mean” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The second metric of convective vigor is calculated from the histogram of vertical velocities at each tropospheric vertical grid 2 For brevity, we will refer to the temperature profile of a lifted, undiluted near-surface parcel as an “adiabat”, ignoring the small effect of buoyancy on the parcel’s lapse rate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Riehl & Malkus 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Romps 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Note that the “adiabats” plotted in Figure 1 are intermediate between the pseudo-adiabatic process (all condensates removed) and the reversible process (all condensates retained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Specifically, the condensate sink term from fallout is modeled as ∂qc ∂z |fall = −qc/L 7 level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' these histograms are not conditionally sampled based on cloud condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We calculate the 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9th percentile of each tropospheric grid level’s vertical velocity histogram, and the maximum value of this vertical profile is what we refer to as w99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9 max (labeled “99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9th percentile” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Similar to CAPE, these metrics also show an initial growth and eventual decline with increasing surface temperature, with the warmest simulation actually having slower wmax values than the coldest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Note that although the qualitative behavior of wmax is similar to CAPE, the peaks in the wmax metrics occur at a lower surface temperature than the CAPE peak, and wmax is right-skewed while CAPE is left-skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Other choices of summary statistics for convective vigor lead to peaks at slightly different surface temperatures, but the overall phenomenon is robust to these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The peak in convective vigor is also robust to increased horizontal and vertical resolution (the EarthTemp hr experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2a, open circles) and use of realistic clear-sky radiation (the EarthTemp realrad experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2a, open squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 200 220 240 260 280 300 temperature (K) 0 10 20 30 40 50 60 70 height (km) environment adiabatic parcel a Temperature profiles 104 105 106 ps (Pa) 0 5 10 15 20 (kJ/kg) b CAPE Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' As in Figure 1, but for the EarthPressure experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Note that in (b) the horizontal axis is inverted so that specific humidity increases toward the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' How general is this peak in convective vigor with respect to atmospheric humidity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In the EarthTemp experiment, the increase in q∗ v is driven by the increasing surface temperature and associated Clausius-Clapeyron scaling of the saturation vapor pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, specific humidity can also be increased, at fixed temperature, by lowering the amount of non-condensing background gas (Wordsworth & Pierrehumbert 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Would varying the surface pressure, therefore, also produce variations in CAPE and convective vigor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To test this, we turn to the EarthPressure experiment, in which we fixed the surface temperature at 300 K but varied the surface pressure from 8×105 Pa to 6250 Pa (between a factor of 8× and 1/16× the contemporary value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 3 shows that CAPE varies in EarthPressure in qualitatively the same manner as in EarthTemp, reaching a peak at an intermediate surface pressure (around 2×104 Pa, approximately 20% of the contemporary value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Likewise, Figure 2b shows that actual convective vigor in the EarthPressure experiment also peaks at intermediate surface pressures, although there is again an offset between the peak CAPE and the peak in actual convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Since Earth’s surface pressure is relatively unconstrained during the Hadean and Archean (Kavanagh & Goldblatt 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Som et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2016), these results may have implications for the early evolution of Earth’s climate and atmospheric chemistry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' we will return to this topic in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' If the boom-bust evolution of CAPE seen in the EarthTemp and EarthPressure experiments is attributable to the transition from condensible-poor (dilute) to condensible-rich (non-dilute) conditions, this recipe is not specific to Earth-like moist convection: in any atmosphere with a sufficiently large surface reservoir of a condensible species, the condensible will become increasingly volatile with warming and eventually come to dominate atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To what other planetary atmospheres, then, might this boom-bust CAPE behavior apply?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' A natural candidate is 8 70 80 90 100 110 temperature (K) 0 20 40 60 80 100 120 height (km) environment adiabatic parcel a Temperature profiles 80 85 90 95 100 105 110 SST (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 (kJ/kg) b CAPE Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' As in Figures 1 and 3, but for the Titan experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Saturn’s moon Titan, which is often regarded as the closest current hydrological analog to Earth due to its active methane precipitation cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore, to further generalize our understanding of convective vigor, we next turn to the Titan experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' This experiment assumes Titan-like thermodynamic conditions and atmospheric composition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', a thick N2 atmosphere with condensing CH4 and low gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Similar to the Earth-like experiments, Figure 4 shows a peak in CAPE as the simulated Titan-like atmospheres transition from dilute to non-dilute methane abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 2c shows that metrics of actual convective vigor in this experiment peak at a surface temperature of about 95 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Intriguingly, this is very close to the current surface temperature of Titan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To better compare our three core experiments, it is helpful to recast the results in terms of variations in atmospheric humidity rather than surface temperature or pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 5 plots CAPE and high-percentile updraft speeds from the core experiments as a function of the specific humidity at the lifted condensation level, q∗ v,LCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' This reveals that CAPE and convective vigor in all three experiments peaks when cloud base air contains roughly 10% of the condensible component by mass, give or take a factor of about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Therefore, we can conclude that the “warming- driven invigoration” regime comes to an end at intermediate humidity, well before these atmospheres approach the steam limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' THE PHYSICAL ORIGIN OF THE CAPE PEAK Taken all together, our three core experiments point to a potentially common phenomenon in terrestrial atmospheres: moist convection is most vigorous at intermediate atmospheric humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' What is the physical explanation for this behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In this section, we show that recent advances in the theory of convection provide a quantitative explanation for the peak in CAPE with respect to atmospheric humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The basic ingredient required to understand climatological variations in CAPE is a theory for the tropospheric lapse rate, which we denote by Γ(z) = −∂T/∂z (K/km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Note that the standard assumption in idealized 1-dimensional radiative-convective modeling, which is that Γ(z) is given by the moist adiabat, is useless for the purpose of predicting CAPE: the CAPE of a moist-adiabatic atmosphere is zero by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' It is the systematic deviations from a moist-adiabatic thermal structure that a successful theory for CAPE must predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The key theoretical breakthrough in this regard was made by Singh & O’Gorman (2013) (hereafter, SO13), who emphasized the role of entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Entrainment refers to the turbulent mixing with environmental air that cloudy updrafts experience as they ascend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Because entrainment of subsaturated air reduces condensation and latent heating in ascending parcels, entraining parcels cool more rapidly with height than otherwise identical undiluted parcels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The insight of SO13 is that the troposphere can be approximated as being neutrally buoyant with respect to such entraining 9 10−3 10−2 10−1 100 q * v, LCL (kg/kg) a Earth-like, Ts varies Earth-like, ps varies Titan-like, Ts varies CAPE 10−3 10−2 10−1 100 q * v, LCL (kg/kg) b High-percentile updrafts Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (a) Normalized CAPE from the EarthTemp, EarthPressure, and Titan experiments, plotted as a function of the saturation specific humidity at the lifted condensation level q∗ v,LCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (b) As in (a), but for normalized high-percentile updraft speeds (w99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' convective parcels, rather than with respect to an undiluted parcel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' this assumption has come to be known as the “zero-buoyancy” (ZB) approximation3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Defining Γm as the lapse rate of an undiluted parcel and Γe as the lapse rate of an entraining parcel (which, by the ZB approximation, is equal to Γ), we can state that Γm < Γ = Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' According to this picture, then, entrainment is the wedge that drives Γm and Γ apart, allowing for potentially large reservoirs of CAPE even in steady-state RCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To make this discussion quantitative, we turn to the simplest model of a convecting atmosphere that incorporates entrainment: the “bulk-plume model”4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The bulk-plume model divides the atmosphere into two plumes: ascending, saturated (cloudy) air, and descending, subsaturated environmental air;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' the “bulk” descriptor refers to the fact that the thermodynamic properties of these two plumes are assumed to be homogeneous in the horizontal at each altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Mass exchange between the two plumes is represented by specified entrainment and detrainment rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' such that conservation of mass in the bulk-plume model is expressed as ∂M ∂z = e − d = M(ε − δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (9) where M (kg/m2/s) is the upward convective mass flux (equal and opposite,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' in RCE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' to the subsidence mass flux in the environmental plume),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' e and d (kg/m3/s) are the mass entrainment and detrainment rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' and ε and δ (m−1) are known as the fractional entrainment and detrainment rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' defined as e/M and d/M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Equation (9) implies that the convective mass flux increases with height if entrainment outpaces detrainment, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The second bulk-plume equation we will use is the conservation equation for moist static energy h, which is conven- tionally defined as h = cpT + gz + Lqv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Here cp (J/kg/K) is the specific heat capacity of air at constant pressure, L (J/kg) is the latent heat of evaporation, and the other symbols take their usual meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' While this is an approxi- mate expression for moist static energy — neglecting, for instance, the temperature-dependence of the latent heat of evaporation (Romps 2015) — it is sufficiently accurate to form the basis of a theory for CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The conservation of moist static energy in the entraining convective plume is expressed as ∂(Mh∗) ∂z = ehe − dh∗, (10) 3 We stress that this zero-buoyancy assumption is not a zero-CAPE assumption: it is the entraining convective parcels that are assumed to have zero buoyancy with respect to the mean environment, not the adiabatic parcel that is used to calculate CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 4 This discussion of the zero-buoyancy bulk-plume theory for CAPE is based on that in Romps (2016) and Romps (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' we refer the reader to these references for a more thorough treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 10 where h∗ = cpT + gz + Lq∗ v is the saturation moist static energy (appropriate for the convective plume because it is saturated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The environmental moist static energy can be expressed as he = cpT +gz +LRq∗ v, where we have used the same T as in the convective plume (invoking the ZB approximation) and where the environmental specific humidity is given by the product of R, the environmental relative humidity, and the saturation specific humidity, q∗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Using the product rule on the left-hand side of equation (10), substituting in equation (9), and using the definitions of h∗ and he given above, we arrive at an important result from SO13: ∂h∗ ∂z = −ε (1 − R) Lq∗ v, (11) Equation 11 states that the entraining plume’s moist static energy decreases with height at a rate proportional to the saturation deficit of the environment, (1 − R)q∗ v (Seeley & Romps 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Although SO13 treated the environmental relative humidity, R, as an external parameter that must be specified, analysis of the bulk-plume water budget can yield a self-consistent analytic expression for R (Romps 2014): R = δ + αγ − αε δ + γ − αε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (12) In equation 12, α is a nondimensional parameter specifying the fraction of condensates formed at a given height that evaporate at that height instead of precipitating out of the atmosphere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', the precipitation efficiency of the bulk- plume convection is 1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The quantity γ is the water vapor lapse rate, defined as γ ≡ −∂ ln q∗ v/∂z and expressed in terms of thermodynamic parameters as γ = LΓ RvT 2 − g RT , (13) where Rv and R (J/kg/K) are the specific gas constants for water vapor and dry air, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The expression (13) is straightforward to derive by combining the Clausius-Clapeyron equation for the saturation vapor pressure with hydrostatic balance (Romps 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The final step toward the theory for CAPE was taken by Romps (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' For analytic solubility, that work considered a simplified case and assumed that M, R, and α are all constant with height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The constancy of M implies ε = δ, by equation (9), while the constancy of R and α imply that the relative humidity and entrainment rate can be expressed in terms of another constant, a ≥ 0, as follows: R = a + α 1 + a , (14) ε = a � γ 1 − α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (15) Note that a, which we will refer to as the “bulk-plume parameter”, is proportional to the entrainment rate, so that a = 0 corresponds to undiluted convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Plugging in equations (14–15) to the right-hand side of equation (11) yields ∂h∗ ∂z = − a 1 + aγLq∗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (16) We can obtain a second equation for ∂h∗/∂z by differentiating the expression for h∗ directly: ∂h∗ ∂z = −cpΓ + g − γLq∗ v = g � 1 + q∗ vL RT � + Γ � cp + q∗ vL2 RvT 2 � , (17) where the second line follows from substituting equation (13) for γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Equating the right-hand sides of equations (16) and (17) and solving for Γ, we obtain Γ = � g cp � � 1 + a + q∗ vL/(RT) 1 + a + q∗vL2/(cpRvT 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (18) Equation (18) is a generalization of the moist lapse rate that accounts for the effect of entrainment (Romps 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In the limit of no entrainment (a → 0), equation 18 reduces to the standard expression for the moist adiabat, Γm: Γm = � g cp � � 1 + q∗ vL/(RT) 1 + q∗vL2/(cpRvT 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (19) 11 We can use equation (18) to analyse the difference between Γ and Γm — and, therefore, CAPE — in the limit of very dry and very moist atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In the dry limit (q∗ v → 0), which is approached by reducing the surface temperature or increasing the amount of background dry air, we can drop all terms multiplied by q∗ v inside the brackets in equation (18), which allows the factors of (1 + a) in the numerator and denominator to cancel and yields Γm = g/cp = Γd, the dry-adiabatic lapse rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Hence entrainment has no effect on the lapse rate in the dry limit, Γ and Γm are both equal to the dry adiabat, and CAPE is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' On the other hand, in the very moist limit (q∗ v → 1), which is approached by increasing the surface temperature or decreasing the amount of background dry air, the terms multiplied by q∗ v dominate over the factors of a for typical Earth and Titan-like conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' For example, for Earth-like conditions5, the factor L/(RT) ranges from about 25–50, while the factor L2/(cpRvT 2) ranges from about 100–500, in both cases dominating over a, which typically takes on values of O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='1) − O(1) (Romps 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Hence the entraining lapse rate asymptotes to Γ ≃ gT/L — which, as in the dry limit, is independent of the entrainment rate, and also equal to the moist limit of Γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To summarize, equation (18) suggests that we should expect minimal CAPE in both the dry and moist limits, with a peak in between for the intermediate values of q∗ v that allow entrainment to drive a wedge between Γ and Γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 270 290 310 330 350 370 SST (K) 0 5 10 15 20 25 30 35 40 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 a EarthTemp 104 105 106 ps (Pa) 0 7 14 21 R16 theory a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 b EarthPressure CRM 80 85 90 95 100 105 110 SST (K) 0 1 2 3 4 CAPE (kJ/kg) a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 c Titan Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' CAPE from the (a) EarthTemp, (b) EarthPressure, and (c) Titan experiments from the convection-resolving simulations (colored markers) and from the analytical predictions for CAPE given by equation (A1) (gray shaded areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The analytical predictions are shown for values of the bulk-plume parameter a ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 for the Earth-like experiments, and for a ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 for the Titan-like experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The analytical solutions are initialized with the temperature and humidity of the lifted condensation level from the corresponding simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Quantitatively, the magnitude and atmospheric humidity of the peak in CAPE can be predicted using the analytic formalism of Romps (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Figure 6 shows analytical predictions for CAPE using the theory of R16 (reproduced in our equation A1), in comparison to the results from our convection-resolving simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The theory clearly captures the boom-bust evolution of CAPE in all three core experiments, providing the theoretical bridge between the “warming- driven invigoration” and the “gentle pure-steam limit” regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' While R16 applied their analytic theory for CAPE to the context of surface warming on Earth, here we have shown that the same essential physics explains convective vigor in convection-resolving simulations with varied surface temperature, varied surface pressure, and with Titan-like thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The notion that entraining and adiabatic temperature profiles converge in non-dilute atmospheres resembles the arguments given by Ding & Pierrehumbert (2016) and Pierrehumbert & Ding (2016), although our results suggest that this physics begins to operate well before the steam limit is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Ding & Pierrehumbert (2016) and Pierrehumbert & Ding (2016) also argued that environmental relative humidity should approach 1 as the atmosphere becomes increasingly non-dilute, a prediction that is confirmed by our results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In addition to explaining the behavior of CAPE in our convection-resolving simulations, the theory of R16 provides clarity on which planetary parameters control CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' An interesting attribute of equation A1 is that it has no explicit dependence on a planet’s gravitational constant, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Physically, this can be understood as a cancellation between two 5 We leave investigation of whether this limit holds in more exotic circumstances, such as exoplanet silicate atmospheres with temperatures in the thousands of kelvins (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2021), to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 12 factors: 1) for a given parcel temperature anomaly, there is a direct proportionality between the parcel’s buoyancy and g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' and 2) for a given surface temperature and tropopause temperature, there is an inverse proportionality between the geometric depth of the atmosphere and g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' hence reducing gravity lowers the integrand in equation 8 but increases the domain of vertical integration by a compensating amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' To test this prediction of an insensitivity to g, we ran a subset of surface temperature cases from the EarthTemp experiment with either enhanced (g = 25 m/s2) or reduced (g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 m/s2) gravitational constant (the VaryGrav experiment);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' these values of g approximately cover surface conditions for solar system planets ranging in mass from Mercury to Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We find a weak dependence of CAPE on g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', CAPE varies by a factor of ≃2 when g varies by a factor of ≃7), in rough accordance with the theoretical prediction, and also find a correspondingly small sensitivity in our metrics of actual convective vigor to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The small variations in CAPE and convective vigor likely result from the effect of g on the nature of turbulence in the simulations, which would affect both the entrainment rate that enters into the theory for CAPE (through the bulk-plume parameter a) and the drag experienced by actual convecting parcels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' These experiments with varied gravity also afford an opportunity to test the sensitivity of our results to the precipi- tation fall speed parameter, which some studies have suggested is a key control on updraft speeds (Parodi & Emanuel 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Assuming a conservative6 linear dependence of precipitation fall speed on g, we re-ran the cases with g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 m/s2 and g = 25 m/s2 with the precipitation fall speeds modified to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 m/s and 20 m/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We found this had a minimal effect, with changes in CAPE and convective vigor generally limited to ±5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' DISCUSSION Using an idealized convection-resolving model, we have demonstrated that convective vigor is expected to peak at intermediate concentrations of the condensing substance in a diversity of planetary circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, more work is needed to build our results into a universal understanding of convective vigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Our simulations of local radiative- convective equilibrium are highly idealized, and many questions remain about how the peak in convective vigor with respect to atmospheric humidity would manifest in more realistic modeling setups that include coupling to large-scale circulations (Pierrehumbert & Ding 2016) or a diurnal cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Another promising avenue for extension concerns our Titan-like simulations, for which we assumed a limitless supply of surface evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In reality, the surface of Titan is quite arid (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2012), and future work could explore convective vigor in the regime where the atmospheric condensible inventory is comparable to the total inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Additionally, although we have successfully applied the theory of R16 to our results, that theory is limited in its general applicability because it approximates the specific gas constant and heat capacity of moist air by those of the dry component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' This is a tolerable source of error when the dry and condensing components do not differ too much in molar mass, as in the case of H2O condensing in an N2/O2 mixture or CH4 condensing in N2, but this approximation breaks down when the background gas is very light (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', mainly H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In that case, the atmospheric scale height can collapse with warming as the atmosphere comes to be dominated by the relatively heavier condensing component (Koll & Cronin 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Since many planets form with a primordial hydrogen envelope, this is an important class of atmospheres to which the R16 theory and our simulation results may not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Additional theoretical and computational work could investigate this regime, for which the “virtual” effects of compositional differences on buoyancy may be crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In additional to building fundamental understanding of moist convection, our results may also have implications for planetary evolution on long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The rate at which a terrestrial planet loses water depends on stratospheric humidity, because water transported to the stratosphere becomes vulnerable to photolysis and subsequent loss of H to space (Kasting 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Purely thermodynamic arguments suggest that stratospheric moistening in planetary atmospheres depends on both surface temperature and surface pressure, with the transition to a moist stratosphere occurring at intermediate humidities (Wordsworth & Pierrehumbert 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' However, on present-day Earth, injection of water into the stratosphere by intense convective storms plays an important role in setting the average stratospheric water content (Corti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' It may be that in real atmospheres, the increase in convective vigor at intermediate humidity causes the “moist greenhouse” transition (Kasting 1988) to be approached more rapidly than either one- dimensional radiative-convective models or 3D general circulation model simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Leconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2013) would suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Further research using models that couple convection to large-scale dynamical and radiative processes is required to investigate this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 6 Theoretical results for monodisperse droplets predict a square-root dependence of raindrop terminal velocity on g (Loftus & Wordsworth 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 13 Furthermore, because the rate of lightning strikes in planetary atmospheres is believed to depend in part on convective vigor, these results may also have important implications for lightning-driven atmospheric chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Romps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (2014) proposed that the lightning flash rate on modern Earth is proportional to the product of CAPE and precipitation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' If this relation is robust across wide ranges of planetary conditions, it implies that the importance of lightning chemistry would be strongly enhanced in atmospheres with intermediate specific humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Because both surface temperature and atmospheric pressure may have varied significantly on Earth in the Hadean, this has interesting implications for the rate of lightning-driven formation of important prebiotic molecules such as HCN during this period (Chameides & Walker 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Ardaseva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Data availability: Cloud-resolving model output and the code that generates the figures in this manuscript is available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='7331932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 1 2 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' ANALYTICAL EXPRESSION FOR CAPE FROM ROMPS (2016) The solutions of R16 yield a closed-form expression for the CAPE of an atmosphere in RCE: CAPE = R 2f {W(ya)[2 − 2f(Ts − Tt) + W(ya)] − W(e−f(Ts−Tt)ya)[2 + W(e−f(Ts−Tt)ya)] − W(y0)2 − 2f(Ts − Tt) + W(y0)] + W(e−f(Ts−Tt)y0)[2 + W(e−f(Ts−Tt)y0)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (A1) where W is the Lambert W function defined by W(xex) = x, and where f = L RvT 2 0 − cp RT0 , and (A2) ya = Lq∗ vs (1 + a)RT0 exp � Lq∗ vs (1 + a)RT0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' (A3) Here, R (J/kg/K) is the specific gas constant of dry (background) air, Rv (J/kg/K) is the specific gas constant of the condensible vapor, L (J/kg) is the latent heat of condensation (assumed constant), cp (J/kg/K) is the specific heat capacity at constant pressure of dry air, q∗ vs (kg/kg) is the saturation specific humidity at the surface, and T0 (K) is a constant reference temperature chosen to be the average of the surface temperature Ts and the tropopause temperature Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The dimensionless parameter a characterizes the bulk-plume convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In equation A1, y0 is given by equation A3 with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The expression for CAPE given above depends only on known physical constants, observable planetary conditions, and a summary parameter characterizing the bulk-plume convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The bulk-plume parameter a is proportional to the entrainment rate7, and it is easy to verify that setting a = 0 (the limit of non-entraining convection) produces a moist-adiabatic atmosphere with zero CAPE, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We can also verify that equation A1 makes quantitatively accurate predictions for Earth’s tropics today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' We set a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='2, which corresponds to typical values of precipitation efficiency and entrainment rate as diagnosed in cloud-resolving simulations, as discussed in R16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Additionally using Ts = 300 K, q∗ vs = 20 g/kg, L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='5 × 106 J/kg, and Tt = 200 K, equation A1 predicts CAPE ≃ 2500 J/kg, which is indeed a typical value for convectively-active parts of Earth’s tropics (Riemann-Campe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' SUPPLEMENTAL FIGURES 7 Specifically, a = ϵPE/γ, where ϵ (1/m) is the fractional entrainment rate, the precipitation efficiency PE is defined as the ratio of net condensation to gross condensation (assumed constant throughout the troposphere), and γ ≡ −∂z log q∗ v (1/m) is the saturation water-vapor lapse rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' One of the assumptions made by R16 for analytical tractability is that ϵ ∝ γ, so that a is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 14 275 285 295 305 315 325 335 345 355 365 SST (K) 0 5 10 15 20 25 30 35 CAPE (kJ/kg) pseudoadiabatic L = 5 km L = 25 km reversible Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The effect of varying condensate fallout assumptions on parcel lifting calculations for computation of CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The condensed water mass fraction is assumed to have a sink term due to fallout that manifests as an exponential decay with height, on a length scale L, during each discrete lifting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The main text figures use L = 5 km, but here we also show the cases L → 0 (pseudoadiabatic), L = 25 km, and L → ∞ (reversible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' In all cases there is a peak in CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 relative humidity 200 220 240 260 280 300 320 340 360 T (K) EarthTemp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 relative humidity 200 220 240 260 280 300 EarthPressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content='0 relative humidity 70 75 80 85 90 95 100 105 110 Titan Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Profiles of environmental relative humidity in the (left) EarthTemp, (center) EarthPressure, and (right) Titan experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' Relative humidity is plotted as a function of mean temperature in the troposphere, as in Romps (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' The line colors are as in the main text figures, with relative humidity generally increasing as the condensible species becomes more prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' REFERENCES Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', Cronin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfHgaB/content/2301.03669v1.pdf'} +page_content=', & Beucler, T.' 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sha256:ca9268ba6aa6a70876488a061f3861326815c66f23a7241ae76878a92bf0ba08 +size 582562 diff --git a/i9FRT4oBgHgl3EQfVze1/vector_store/index.pkl b/i9FRT4oBgHgl3EQfVze1/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2693194c3b072035c55121898e217c57404280fe --- /dev/null +++ b/i9FRT4oBgHgl3EQfVze1/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:18e0331299a70ccafa3da05eb43752bbafdf27131bea4047d48d1558c89308d5 +size 356497 diff --git a/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/2301.13364v1.pdf.txt b/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/2301.13364v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5af79e50a3ef2d32da24e04b11825e51bee369fe --- /dev/null +++ b/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/2301.13364v1.pdf.txt @@ -0,0 +1,1556 @@ +A Counterfactual Collaborative Session-based +Recommender System +Wenzhuo Song +Northeast Normal University +China +wzsong@nenu.edu.cn +Shoujin Wang +University of Technology Sydney +Australia +shoujin.wang@uts.edu.au +Yan Wang +Macquarie University +Australia +yan.wang@mq.edu.au +Kunpeng Liu +Portland State University +United States +kunpeng@pdx.edu +Xueyan Liu∗ +Jilin University +China +xueyanliu@jlu.edu.cn +Minghao Yin +Northeast Normal University +China +ymh@nenu.edu.cn +ABSTRACT +Most session-based recommender systems (SBRSs) focus on extract- +ing information from the observed items in the current session +of a user to predict a next item, ignoring the causes outside the +session (called outer-session causes, OSCs) that influence the user’s +selection of items. However, these causes widely exist in the real +world, and few studies have investigated their role in SBRSs. In this +work, we analyze the causalities and correlations of the OSCs in +SBRSs from the perspective of causal inference. We find that the +OSCs are essentially the confounders in SBRSs, which leads to spu- +rious correlations in the data used to train SBRS models. To address +this problem, we propose a novel SBRS framework named COCO- +SBRS (COunterfactual COllaborative Session-Based Recommender +Systems) to learn the causality between OSCs and user-item interac- +tions in SBRSs. COCO-SBRS first adopts a self-supervised approach +to pre-train a recommendation model by designing pseudo-labels of +causes for each user’s selection of the item in data to guide the train- +ing process. Next, COCO-SBRS adopts counterfactual inference to +recommend items based on the outputs of the pre-trained recom- +mendation model considering the causalities to alleviate the data +sparsity problem. As a result, COCO-SBRS can learn the causalities +in data, preventing the model from learning spurious correlations. +The experimental results of our extensive experiments conducted +on three real-world datasets demonstrate the superiority of our +proposed framework over ten representative SBRSs. +CCS CONCEPTS +• Information systems → Recommender systems. +∗Corresponding author. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +KEYWORDS +session-based recommendation, self-supervised learning, counter- +factuals +1 +INTRODUCTION +Existing studies on Recommender Systems (RSs) mainly focus on +modeling and predicting all user-item interactions to learn users’ +static item preferences, ignoring the dynamic nature of user pref- +erences in the real world. To bridge this gap, session-based rec- +ommender systems (SBRSs) have recently emerged and gained in- +creasing attention [35]. Most existing SBRS models formalize the +recommendation of next user-item interactions (next item for short) +in sessions as a supervised learning task. For a current session of +a user to be predicted (called target session), an SBRS model takes +the items of the observed interactions in this session (called the +session’s context) as input and outputs a list of items as predicted +next items [35]. Most of these SBRSs imply a strong assumption +that a user selects an item as the next item in a session only because +the item correlates with the items in the same session. However, +they ignore the selection of the next items caused by factors other +than items in the same session [11]. For example, on an e-commerce +site, a user may select a product, i.e., an item, because of his/her +short-term intent or previous interest, a discount for this product, or +a short-term item popularity trend. In either case, the item selected +is not related to other items in the same session. +In contrast, in this work, we suggest that it is not appropriate +to train SBRSs without considering the influence of causality of +the factors outside the session context. Let us consider that a user +selects the next items in a session based on two types of causes, i.e., +inner-session causes and outer-session causes. Inner-session causes +(ISCs for short) refer to the selection of the next items caused by the +context of the same session, which may be due to the co-occurrence +pattern and sequential pattern between items in the same session. +In contrast, outer-session causes (OSCs for short) refer to the selec- +tion of the next item caused by factors outside the target session, +such as long-term user preferences, product popularity trends and +discounts for products on e-commerce sites. To facilitate an insight- +ful analysis, we first present a graph in Figure 1 (a) showing the +correlations between variables considered in existing SBRS models. +For example, when camera and fan are on discount at the same +time in a supermarket, many users buy them together because they +are cheaper than usual. Suppose camera and fan are ISCs 𝑀 and +arXiv:2301.13364v1 [cs.IR] 31 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +W. Song, et al. +correlation +causality +spurious +correlation +camera +fan +camera +fan +camera +lens +( c ) +( a ) +discount +( b ) +Figure 1: (a) Existing SBRSs make recommendations based +on the co-occurrence-based correlation between camera and +fan. (b) However, the correlation between camera and fan +is spurious since the purchase of fan is not because of the +purchase of camera. Instead, the co-purchase of camera and +fan rely on the fact that they have discounted prices at the +same time. (c) Actually, there are causalities between items +in SBRS data. For example, a user buys a lens because he/she +has bought a camera. +next item 𝐼 depicted in Figure 1 (a), respectively. The SBRS models +learn the co-occurrence pattern 𝑀 ↔ 𝐼 in the sessions. If a user +buys a camera, the models will recommend a fan to him/her even +if the discount has ended, because there are a large number of ses- +sions/transactions containing camera and fan together. However, +camera and fan are not related in terms of purposes. A user buys +fan is not because he/she has bought camera, but because fan is on +discount. Therefore, this co-occurrence correlation may result in +the mis-modeling problem in SBRSs. +To further explain this problem, we present two causal graphs +to describe the causalities in Figure 1 (b) and (c). The causal graphs +are directed acyclic graphs defined based on the structural causal +model [8]. We use the causal graph in Figure 1 (b) to represent +the causalities in the above-mentioned example, where 𝑁 denotes +the discount of a product, which is one of OSCs in SBRSs. 𝑁 → 𝐼 +and 𝑁 → 𝑀 denote a user buys a fan and a camera due to their +discounted price respectively. Hence, the discounted price is the +common cause of the purchase of camera and fan, and it is called +a confounder1 in SBRSs [21, 45]. Clearly, the confounder results +in a spurious correlation between camera and fan as depicted in +Figure 1 (b) since the purchase of fan is not because the purchase of +camera even though they have been purchased successively in one +session. A reliable SBRS should not make recommendations based +on such spurious correlation since it cannot reflect the true casual +relations between items. Actually, there are some other reliable +causality relations between items in SBRS data. For example, as +depicted in Figure 1 (c), the purchase of a lens is often because of +the prior purchase of a camera. Such kind of relations should be the +basis for making reliable recommendations. From above examples, +we can conclude that the OSCs can be the confounders in SBRSs, +and the relationship between items in session contexts and the +next item in the data is a mixture of the causality, e.g., camera and +lens in Figure 1 (c), and the spurious correlations, e.g., camera and +fan in Figure 1 (b). Therefore, (Problem 1) without considering the +1Confounders are variables that affect both the treatment variable, e.g., camera and +the outcome variable, e.g., fan and lens. +causalities of the OSCs, the SBRSs will learn both the causality and +the spurious correlations caused by the confounder and thus may +generate incorrect recommendations. +One possible way to solve this problem is to list all possible causes +of the next item selection in sessions and then train a deep learning +model in an end-to-end manner to learn the complex relationship +between the next item and the causes. However, the true cause, i.e., +which of ISCs and OSCs is the reason why a user selects a specific +item, is a latent variable in the model, and they are not provided +in the dataset. Thus, (Problem 2) it is difficult for deep learning +models to identify the true causes of the item selections and learn the +causalities shown in Figure 1 (b) and (c). +In this paper, we address Problem 1 by proposing a novel SBRS +framework considering both OSCs and ISCs for the next item recom- +mendation in sessions. The proposed model is named COunterfactual +COllaborative Session-Based Recommender System (COCO-SBRS, or +COCO for short). COCO is inspired by the ideas of counterfactu- +als [45] and collaborative filtering for sessions [6, 19]. Specifically, +COCO first pre-trains a recommendation model to learn the causal- +ities among ISCs, OSCs and user-item interactions in SBRSs, and +then predicts the next item for a session based on some neighbor +sessions with the same ISCs and OSCs as this session and the rec- +ommendation model used to simulate the user’s selection in the +neighbor sessions. To address Problem 2, in the pre-training phase, +we adopt a self-supervised approach to train the recommendation +model by designing pseudo-labels of causes for each user-item inter- +action to guide the training of the model. To alleviate the problem +of the lack of sessions with required ISCs and OSCs in the predic- +tion phase, we adopt counterfactual inference to generate sessions +using required ISCs and OSCs, and simulate users’ decision-making +process with the pre-trained recommendation model to recommend +items. In summary, the main contributions of this work are: +(1) We propose an SBRS framework named counterfactual col- +laborative session-based recommender system (COCO-SBRS +or COCO) for effective next item recommendations in ses- +sions. To the best of our knowledge, this is the first work in +the literature to address the problem of spurious correlations +caused by the confounder in SBRSs. +(2) We are the first to formulate the next item recommendation +in sessions in the framework of counterfactual computing +and collaborative filtering. Specifically, we first develop a self- +supervised approach to pre-train a recommendation model +to learn causalities in SBRSs. Then, we recommend items for +a given session using the model, taking other sessions with +the same ISCs and OSCs as this session as input. +(3) To evaluate the effectiveness of COCO, we conduct exten- +sive experiments on three real-world datasets from various +domains. We compare COCO with the representative and +state-of-the-art SBRSs, and the experimental results show +COCO’s superiority over baseline SBRSs. +2 +RELATED WORK +2.1 +Session-based Recommender Systems +In this section, we introduce two groups of SBRSs: (1) SBRSs that +consider inner-session causes only, and (2) SBRSs that consider +both inner-session and outer-session causes. + +A Counterfactual Collaborative Session-based +Recommender System +Conference’17, July 2017, Washington, DC, USA +SBRSs that consider inner-session causes only make recom- +mendations based on the session context. These algorithms aim to +learn users’ short-term preferences reflected by the items in the +session context and the complex relationships between items in +sessions. According to the employed technology, these SBRSs can +be classified into conventional approaches, latent representation +approaches, and deep neural network approaches [35]. KNN-based +SBRSs and Markov chain-based SBRSs are the most popular conven- +tional approaches. KNN-based SBRSs such as SKNN recommend +items in sessions similar to the target session [6, 19]. Markov chain- +based SBRSs such as FPMC make recommendations by modeling +the transition patterns of items in sessions [14, 25]. The latent rep- +resentation SBRSs utilize the technique of latent factors models or +matrix factorization to make recommendations [4, 18]. In recent +years, deep learning-based SBRSs are becoming popular, and re- +searchers have developed SBRSs based on various deep learning +techniques. Recurrent neural network (RNN) based SBRSs model +the sequential pattern of items in sessions [9, 10, 41]. However, +they are based on a rigid assumption that adjacent items in each +session are sequentially dependent. Attention-based methods relax +this assumption by emphasizing those more informative items in +sessions to reduce the interference of uninformative ones [3, 46]. To +model the high-order transition among items, the graph neural net- +work (GNN) based SBRSs first represent sessions with graphs and +then employ GNN models to make recommendations [23, 42]. Deep +learning-based SBRSs can model the complex patterns of items +in sessions and therefore achieve better performance over other +approaches in many recent studies. However, the above-mentioned +SBRSs can only predict the next item based on the limited informa- +tion in the session context and ignore the global information and +factors outside the session influence users’ selection of items. +SBRSs that consider both inner-session and outer-session +causes aim to extract and fuse information from both the target +session and other sessions to make recommendations. The most +common information from other sessions considered in existing +works includes global information such as item dependency and +long-term item preferences of users. Global information is essential +for SBRSs because the lengths of sessions in the real world tend +to be very short and contain limited information, and the global +information can be used as a supplement to the session context +[22, 34]. User preferences are outer-session information that has +great potential to improve the performance of recommendation +algorithms because, in many real-world scenarios, it is easy to +obtain user behavioral data reflecting user preferences that belong +to the same user of the target session [15, 27, 30]. Unlike the above +algorithms, this work aims to eliminate the spurious correlations +caused by the confounder in SBRSs introduced in Section 1. +2.2 +Counterfactuals in Recommender Systems +Counterfactual computing has been introduced into the research +of recommendation systems in the recent years. Compared to the +rapid development in other machine learning fields, the studies on +counterfactual-based recommender systems (CFBRSs for short) are +still limited. According to the problem to be addressed, existing +CFBRSs can be classified into algorithms for data bias, algorithms +for data missing, and algorithms for other tasks [5, 36]. Among +the algorithms for data bias, researchers have proposed the MACR +algorithm for popularity bias, and the CR algorithm for clickbait +bias [2, 16, 20, 37, 40]. The data missing problem stems from the +large number of inactive users and items in RSs. Existing works use +counterfactual-based data generation to alleviate the problem of +missing data in RSs [39, 43, 47]. Other problems studied in CFBRSs +include explainability, diversity and fairness in recommender sys- +tems [7, 31, 32, 38, 48]. Different from the existing CFBRSs, we study +the counterfactual-based model for learning causality in session- +based recommender systems. +3 +PROBLEM STATEMENT +In this work, we use 𝑈 = {𝑢1, ...,𝑢|𝑈 |} to denote the set of |𝑈 | users +and 𝑉 = {𝑣1, ..., 𝑣 |𝑉 |} to denote the set of |𝑉 | items in the dataset. A +user 𝑢 ∈ 𝑈 has a sequence of user-item interactions, which can be +represented using a list of items corresponding to each interaction. +In SBRSs, the interactions of each user form a series of sessions, +where each session is a list of the user’s interactions, i.e., item list +𝑠 = {𝑣𝑠 +1, ..., 𝑣𝑠 +𝑡 } in a short period of time. The subscript of each item +in 𝑠 denotes its order in 𝑠 w.r.t its occurring time. +Given a target session 𝑠 to be predicted, the goal of an SBRS is to +predict the next interactions in 𝑠, i.e., the next item 𝑣𝑠 +𝑡+1, based on +known information such as user preferences and session context. +In this work, we consider two causes for a user to select the next +item in sessions, i.e., ISCs and OSCs. Specifically, we formulate +the SBRS model as a probabilistic classifier 𝑝(𝑣|𝑀, 𝑁) over a set of +items in the dataset conditioning on the ISCs variable 𝑀 and the +OSCs variable 𝑁. The prediction result of 𝑝(𝑣|𝑀, 𝑁) represents the +probability distribution of the item which user will select as the +next item for 𝑠. Finally, items with top-K probabilities are selected +as the recommendation result. +4 +COUNTERFACTUAL COLLABORATIVE +SESSION-BASED RECOMMENDER SYSTEM +In this section, we introduce the proposed SBRS method named +COunterfactual COllaborative Session-Based Recommender System +(COCO-SBRS, or COCO for short). COCO is inspired by counterfac- +tuals [8, 45] and collaborative filtering for sessions [19, 27]. COCO +first pre-trains a recommendation model to learn the causalities +among ISCs, OSCs and user-item interactions in SBRSs, and then +predicts the next item for a session based on some neighbor sessions +with the same ISCs and OSCs as this session and the recommen- +dation model used to simulate the user’s selection in the neighbor +sessions. Specifically, COCO has the following three steps: (i) Ab- +duction: In this step, we design a generative process to describe +the users’ decision-making process in SBRSs, and then implement +the generative process with an attention-based neural network. +Next, we adopt a self-supervised approach to pre-train the neu- +ral network using the sessions in the training set to determine its +optimal parameters. After pre-training the BRM, the framework +makes recommendations with the steps of action and prediction, +the process of which is shown in Figure 2. (ii) Action: For a target +session 𝑠 of user 𝑢, we first sample a session 𝑠′ of user 𝑢′ from +the dataset, then replace its ISCs, i.e., 𝐼𝑆𝐶(𝑠′) with the ISCs of the +target session, i.e., 𝐼𝑆𝐶(𝑠). Next, we use the modified session as the +input to the model trained in the abduction step to simulate the + +Conference’17, July 2017, Washington, DC, USA +W. Song, et al. +decision-making process of 𝑢′ to answer a counterfactual question +[44]: “Given a target session 𝑠 of 𝑢, which item will be selected as the +next item by another user𝑢′ based on the similar ISCs and OSCs as 𝑠?” +We sample a few sessions of similar users to the user of the target +session, and then perform the action step for each of them. (iii) +Prediction: In this step, we make recommendations by combining +the outputs of all the models, i.e., the answers to the counterfactual +question, and the users’ similarities. +4.1 +Abduction: Base Recommendation Model +and its Self-Supervised based Training +Generative Process for SBRSs: Based on the causal graphs in +Figure 1 (b) and (c), we present a generative process to describe +a user’s decision-making process given specific ISCs and OSCs as +follows: +Given a target session 𝑠 of user 𝑢: +(1) generate 𝑂𝑆𝐶(𝑠) according to the user 𝑢 of 𝑠; +(2) generate 𝐼𝑆𝐶(𝑠) according to the items in 𝑠’s context 𝑐; +(3) sample a cause 𝐶(𝑢,𝑐) from 𝑂𝑆𝐶(𝑠) and 𝐼𝑆𝐶(𝑠) according to +𝑢 and 𝑐 for an item 𝑣 to predict; +(4) generates the score of the item based on 𝐶(𝑢,𝑐). +Next, we implement the generative process for SBRSs based on +attention neural networks. We call the model the base recommen- +dation model (BRM). BRM’s architecture is depicted in Figure 3. +Modeling OSCs in BRM: We consider two OSCs in SBRSs: +static preference and dynamic preference [35]. (1) Static preference +is a long-term user preference that does not change with time. +In this work, we use a fixed-length vector, i.e., user embedding +e𝑢 ∈ R𝑑, to represent the static preference of a user 𝑢. (2) Dynamic +preference is the short-term user preference that changes with time. +We use an attention-based neural network to learn the embedding +of a user’s dynamic preference. Specifically, given a user 𝑢’s session +𝑠, the embedding of dynamic preference h𝑁 (𝑠) for 𝑠 is calculated +by: +h𝑁 (𝑠) = 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(e𝑢, E(𝑅(𝑠)), E(𝑅(𝑠))), +(1) +where 𝑅(𝑠) = {𝑣𝑅 +1 , ..., 𝑣𝑅 +|𝑅|} is a set containing the most recent in- +teractions of 𝑢 before 𝑠, |𝑅(𝑠)| is a hyper-parameter and is set as +10 for all sessions in our experiments, E(𝑅(𝑠)) ∈ R|𝑅(𝑠) |×𝑑 denotes +the matrix containing all the embeddings of items in 𝑅(𝑠). Given +a query vector q′ ∈ R𝑑, a matrix of 𝜅 key vectors K′ ∈ R𝜅×𝑑 and +a matrix of 𝜅 value vectors V′ ∈ R𝜅×𝑑, 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(q′, K′, V′) is an +attention network defined as: +𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(q, K′, V′) = +∑︁ +𝑖 +𝛼(q, K𝑖) × V𝑖, +(2) +where 𝛼(q, K′) is a vector containing the attention scores between +q and each vector in K′ calculated by: +𝛼(q, K𝑖) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥𝑖 (q · K𝑇 +𝑖 / +√ +𝑑). +(3) +Finally, we use h𝑁 (𝑠) as the embedding of the OSCs for 𝑠. +Modeling ISCs in BRM: Given 𝑠’s context 𝑐, the representation +of ISCs, i.e., h𝑀 (𝑠) for 𝑠 is calculated by: +h𝑀 (𝑠) = 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑎𝑣𝑔(E(𝑐)), E(𝑐), E(𝑐)), +(4) +where E(𝑐) ∈ R|𝑐 |×𝑑 denotes a matrix containing the embeddings +of all items in 𝑐, 𝑎𝑣𝑔(E(𝑐)) = +1 +|𝑐 | +� +𝑣∈𝑐 e𝑣 is the mean vector of the +item embeddings in 𝑐. +Next-Item Prediction in BRM: In the generation process, the +user selects one of the OSCs and ISCs as the cause for the next item +selection in 𝑠 according to the user and the session. We implement +this process with an attention-based neural network to learn a soft +weight of the two causes, i.e., 𝜆 ∈ [0, 1]: +𝜆 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(W𝑇 (h𝑀 (𝑠)||h𝑁 (𝑠)||e𝑣) + 𝑏), +(5) +where 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑧) = 1/(1 + 𝑒−𝑧), || denotes the concatenate oper- +ation, 𝑣 ∈ 𝑉 is the item to be predicted, and W ∈ R𝑑 and 𝑏 ∈ R +are parameters to learn. A large 𝜆 means that the OSCs have more +influence than the ISCs on the next item selection, and a small 𝜆 +means the ISCs have more influence than the OSCs for 𝑠. +Next, we incorporate the two causes by: +h(𝑠) = 𝜆h𝑁 (𝑠) + (1 − 𝜆)h𝑀 (𝑠), +(6) +and predict the next item for 𝑠 by: +𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(e(Ω)h(𝑠)𝑇 ), +(7) +where Ω is the set of items to be predicted, and e(Ω) ∈ R|Ω|×𝑑 is a +matrix containing the embeddings of all items in Ω. +Training BRM with Cross Entropy Loss: By regarding the +next item prediction in sessions as a multi-class classification task, +we can train the model using the Cross Entropy Loss: +𝓁1 = +1 +|𝑆𝑏| +∑︁ +𝑠 ∈𝑆𝑏 +∑︁ +𝑣′∈𝑉𝑏 +1[𝑣′ ∈ 𝑐] log𝑝(𝑣 = 𝑣′|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)), +(8) +where 𝑐 is the context of 𝑠, 𝑆𝑏 denotes the set of all sessions in the +training batch, 𝑉𝑏 is the set of all items in the training batch, and +1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] is defined as: +1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] = +� +1, +if 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 is True, +0, +if 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 is False. +(9) +Improve BRM with Self-Supervised Learning: In the pro- +posed generation process, an item is selected as the next item due +to one of the ISCs and OSCs, which we call the true cause for the +corresponding user-item interaction. However, the true cause is +a latent variable in the model since they are not provided in the +dataset. To address this problem, in BRM, we use a linear model, i.e., +Equation (5), to learn the relationship between the latent variable 𝜆 +and the observational variables 𝑀 and 𝑁 so that the model gets the +ability to identify the true cause corresponding to each interaction. +To improve the accuracy of true cause identification, we propose +a self-supervised learning approach to guide the training of the +parameters in Equation (5). Our main idea is to construct a pseudo- +label for the soft weights of two causes, i.e., 𝜆. Specifically, we +assume that: given each item of each session, (1) if an item appears +in the interaction history of 𝑢, the user is more likely to select the +item due to the OSCs, i.e., 𝜆 = 1; (2) if an item appears in the context +of the target session, the user is more likely to select the item due +to the ISCs, i.e., (1 − 𝜆) = 1. Based on these two assumptions, we +obtain the definitions of 𝜆’s pseudo-labels: +� +𝑦𝑁 (𝑠) = 1[𝑣 ∈ 𝑉𝑢], +𝑦𝑀 (𝑠) = 1[𝑣 ∈ 𝑐], +(10) + +A Counterfactual Collaborative Session-based +Recommender System +Conference’17, July 2017, Washington, DC, USA +: sessions +similar to   +Action: +Replace session context with +  +BRM +BRM +BRM +calculate +similarity +Action: +Predict with BRMs +target session +Top-K List +Item 1 +Item 2 +Item 3 +... +Prediction: +Combine BRM outputs  +Top-K List +Item 1 +Item 3 +Item 2 +... +Enhance with +Boost Factor +Inputs: +interactions +before session +the context of +the session ++ +Output: +replace +weighted +sum +  +  +  +  +add to relevant items  +... +... +Figure 2: The steps of Action and Prediction of COCO-SBRS. The BRMs are pre-trained with the sessions in the training set in +the Abduction step, which is not shown in this figure. +... +User +Input +Layer +Embed- +ding +attention network +attention network +Cause +Repre- +sentation +Cause +Weights +Recent Interactions +Context of the Session +Candidate +Item +OSCs +ISCs +Next Item Prediction +Cause +Modeling +Output: +score of +the item +... +... +... +Figure 3: Base Recommendation Model (BRM). +where 𝑉𝑢 is the set of items the user of 𝑠 has interacted before. +Next, we consider the true cause prediction problem as a binary +classification problem and present a self-supervised loss based on +the Binary Cross Entropy Loss: +𝓁2 = +1 +|𝑆𝑏| +∑︁ +𝑠 ∈𝑆𝑏 +𝐵𝐶𝐸(𝜆,𝑦𝑁 (𝑠)) + 𝐵𝐶𝐸((1 − 𝜆),𝑦𝑀 (𝑠)), +(11) +where given a prediction score 𝑥 and the corresponding label 𝑦, +Binary Cross Entropy Loss 𝐵𝐶𝐸(𝑥,𝑦) is defined as: +𝐵𝐶𝐸(𝑥,𝑦) = 𝑦 · log𝑥 + (1 − 𝑦) · log(1 − 𝑥). +(12) +Training BRM according to Equation (11) encourages the model +to generate soft weights of the OSCs and ISCs that are consistent +with pseudo-labels, improving the model’s ability to untangle the +ISCs and OSCs, and identify the true causes of the interactions. +Finally, the loss function for the training of BRM is defined as a +trade-off of two loss functions: +𝓁 = 𝓁1 + 𝛽 ∗ 𝓁2, +(13) +where 𝛽 ∈ [0, +∞) is the trade-off hyper-parameter. +4.2 +Action and Prediction: Counterfactual and +Collaborative Next-Item Recommendation +The key idea of collaborative filtering for sessions is predicting +items in other sessions similar to the target session [6, 19]. However, +when calculating session similarities, these methods only consider +the items in the session context, and ignore outer-session causes +such as static user preference. In addition, these methods assume +that every session in the dataset has similar sessions, which may +not be true when the dataset is sparse. As a result, they may treat +sessions of users who have a completely different preference as +similar sessions, and thus make incorrect recommendations. +To address the above problems, given a target session 𝑠 of a +user 𝑢, we use the BRM model to simulate users’ decision-making +process for the counterfactual question: “which next item will be +selected by another user 𝑢′, who has similar preferences to 𝑢 (i.e., +𝑂𝑆𝐶(𝑠) in BRM) in the same session context (i.e., 𝐼𝑆𝐶(𝑠) in BRM)?” +In this way, we can ensure that the next item for each similar session +is generated when the user has similar ISCs and OSCs to the target +session. Specifically, the process contains two steps: +Action: Given a target session 𝑠 of a user 𝑢, the action step aims +to find a user𝑢′ who has a similar preference to𝑢 and then compute +the probability distribution of the next item under the same ISCs of +𝑠. Specifically, we first find a session 𝑠′ with similar OSCs to 𝑠 by +calculating the similarity of the recent interaction sets2: +𝑠𝑖𝑚(𝑠,𝑠′) = |𝑅(𝑠) ∩ 𝑅(𝑠′)| +|𝑅(𝑠) ∪ 𝑅(𝑠′)|, +(14) +where 𝑠′ ≠ 𝑠 is a session sampled from the dataset. A large 𝑠𝑖𝑚(𝑠,𝑠′) +means that 𝑢′ has similar OSCs to 𝑢 when the session 𝑠′ occurred. +However, a larger value of 𝑠𝑖𝑚(𝑠,𝑠′) does not mean that the ISCs +of 𝑠′ are similar to the ISCs of 𝑠, and direct use of BRM to predict for +2The most recent interaction set of a session, i.e., 𝑅(𝑠), can reflect the dynamic user +preferences when the session occurs, which is the main part of the OSCs for 𝑠. + +Conference’17, July 2017, Washington, DC, USA +W. Song, et al. +Table 1: The detailed statistical information of the three +datasets used in the experiments. +Last.fm +Delicious +Reddit +#sessions +5915 +45,772 +1,122,150 +#interactions +38,367 +249,919 +2,874,671 +#users +1,101 +1,752 +19,878 +#items +711 +5,047 +13,742 +#interactions per user +34.85 +142.65 +144.62 +#interactions per session +6.49 +5.46 +2.56 +#sessions per user +5.37 +26.13 +56.45 +𝑠 based on 𝑠′ will produce erroneous results. We replace the 𝐼𝑆𝐶(𝑠′) +with 𝐼𝑆𝐶(𝑠) as an action on the BRM’s prediction in Equation (7) +to address this problem: +𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠′)). +(15) +In this way, the BRM can predict the next item as it simulates 𝑢′ in +the context of session 𝑠, which does not exist in the dataset. +Prediction: In the action step, we only consider one similar +user. Based on the idea of collaborative filtering, in the prediction +step, we consider that increasing the number of similar users could +improve the recommendation performance of our model. +Specifically, we first select the most similar sessions of 𝑠 from +the dataset to form a session set 𝜋(𝑠) according to the session +similarities of Equation (14). Next, for each user 𝑢𝑖 of session 𝑠𝑖 ∈ +𝜋(𝑠), we replace the 𝐼𝑆𝐶(𝑠𝑖) in its corresponding BRM with 𝐼𝑆𝐶(𝑠) +while keeping its 𝑂𝑆𝐶(𝑠𝑖) unchanged, and then we weighted sum +the results of all BRMs of each session in 𝜋(𝑠) to obtain the result: +𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = +1 +𝐶 +∑︁ +𝑠𝑖 ∈𝜋 (𝑠) +𝑠𝑖𝑚(𝑠,𝑠𝑖) × 𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠𝑖)), +(16) +where 𝐶 is the normalization factor. +4.3 +Enhancement with Boost Factor +After obtaining the item distribution 𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀, 𝑁) for the next +item recommendation, we find that emphasizing the recently seen +items in the target session 𝑠 can further improve the performance +of our model. Specifically, for each item 𝑣𝑖 ∈ (𝑅(𝑠) ∩ 𝑐), we first +modify the weights of recently seen items by adding a boost factor +𝜖 > 0 and then obtain the final result of COCO after enhancement +with the boost factor : +𝑝′ +𝑐𝑜𝑐𝑜 (𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = +1 +𝐶′ (𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀, 𝑁) + 𝜖 ∗ 1[𝑣 ∈ (𝑅(𝑠) ∩ 𝑐)]), +(17) +where 1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] is a vector containing the results of Equation +(10) for all 𝑣 ∈ 𝑉 , and 𝐶′ is the normalization factor. +5 +EXPERIMENTS +5.1 +Experimental Settings +Baseline Algorithms: In the comparison experiments, we select +(1) five SBRSs that consider inner-session causes only (SKNN, GRU4Rec, +STAMP, CSRM and SRGNN) and (2) five SBRSs that consider both +inner-session and outer-session causes (HGRU, II-RNN, SASRec, +BERT4Rec and INSERT) as baseline algorithms. These baselines are +based on various techniques, including collaborative filtering, RNN, +attention neural networks, memory neural networks and graph +neural networks. +• SKNN: A collaborative filtering SBRS built on the idea of +recommending the items in neighbor sessions which are +similar to the target session [19]. +• GRU4Rec: GRU4Rec employs an RNN to extract dynamic +user preference in the session context for the next item rec- +ommendation [10]. +• STAMP: A memory neural network-based SBRS uses atten- +tion to capture users’ short-term preferences for the next +item recommendations in sessions [17]. +• CSRM: A memory neural network-based SBRS uses atten- +tion networks to learn the similarities between sessions and +make recommendations based on the information extracted +from neighbor sessions and their similarities [33]. +• SR-GNN: A graph neural network-based SBRS first repre- +sents the sessions with graphs and then employs a graph +neural network to recommend items based on the item tran- +sition patterns extracted from the graph [42]. +The following five SBRSs consider both inter-session causes, i.e., +the information from the session context, and outer-session causes, +e.g., users preference: +• HGRU: HGRU employs a hierarchical RNN where one RNN +models the sequential patterns in the session context, and +the other RNN learns a user’s preference across his sessions +[24]. +• II-RNN: II-RNN utilizes the information extracted from the +most recent session to complement and initialize the RNN +modeling the target session [26]. +• SASRec: SASRec is a self-attention based sequential RS de- +signed to model users’ interaction sequences. In this work, +we concatenate all interactions of each user to form the user +sequences [12]. +• BERT4Rec: BERT4Rec is a deep bidirectional self-attention +based sequential RS model. BERT4Rec adopts the Cloze ob- +jective and predicts an item based on its context and the +user’s historical interactions [29]. +• INSERT: INSERT is the state-of-the-art SBRS considering +both user preference and item patterns in sessions. It is de- +signed for next item recommendations in short sessions +based on few-shot learning and meta-learning [27]. +Datasets: We conduct the experiments on the following three +publicly available real-world datasets used in previous SBRS works: +• Last.fm3 used in [13] contains the logs of users’ music lis- +tening behaviors in Last.fm online music service [1]. +• Delicious is a dataset used in [28] which contains the user +tagging records in a social network-based bookmarking sys- +tem named Delicious. +3Last.fm and Delicious are from https://grouplens.org/datasets/hetrec-2011/. + +A Counterfactual Collaborative Session-based +Recommender System +Conference’17, July 2017, Washington, DC, USA +Table 2: Recommendation performance of all compared methods on three datasets. R@5 and R@20 are short for Recall@5 +and Recall@20. N@5 and N@20 are short for NDCG@5 and NDCG@20. We adopt 5-fold cross-validation and report the av- +erage value for each metric. For all evaluation metrics, higher numbers represent better method performance. The bold and +underlined numbers under each metric represent the best and the second-best performing method, respectively. * means the +improvement is significant at 𝑝 < 0.05. +Last.fm +Delicious +Reddit +R@5 +N@5 +R@20 +N@20 +R@5 +N@5 +R@20 +N@20 +R@5 +N@5 +R@20 +N@20 +SKNN +0.235 +0.116 +0.536 +0.202 +0.111 +0.055 +0.293 +0.107 +0.202 +0.107 +0.383 +0.159 +GRU4Rec +0.331 +0.238 +0.542 +0.298 +0.234 +0.172 +0.390 +0.216 +0.225 +0.163 +0.397 +0.212 +STAMP +0.269 +0.190 +0.500 +0.256 +0.150 +0.104 +0.294 +0.105 +0.192 +0.133 +0.323 +0.170 +CSRM +0.342 +0.250 +0.562 +0.312 +0.197 +0.144 +0.346 +0.186 +0.200 +0.150 +0.366 +0.197 +SR-GNN +0.265 +0.186 +0.477 +0.246 +0.205 +0.149 +0.354 +0.191 +0.251 +0.186 +0.422 +0.235 +HGRU +0.340 +0.242 +0.576 +0.309 +0.218 +0.160 +0.377 +0.205 +0.337 +0.257 +0.518 +0.309 +II-RNN +0.359 +0.259 +0.586 +0.323 +0.257 +0.189 +0.424 +0.236 +0.365 +0.277 +0.560 +0.333 +SASRec +0.346 +0.206 +0.651 +0.201 +0.193 +0.130 +0.385 +0.184 +0.310 +0.209 +0.537 +0.274 +BERT4Rec +0.304 +0.182 +0.622 +0.273 +0.207 +0.101 +0.369 +0.186 +0.409 +0.271 +0.623 +0.338 +INSERT +0.364 +0.258 +0.589 +0.323 +0.264 +0.196 +0.436 +0.245 +0.391 +0.301 +0.561 +0.350 +COCO-SBRS +0.504* +0.289* +0.793* +0.374* +0.359* +0.215* +0.520* +0.263* +0.412* +0.248 +0.699* +0.331 +• Reddit4 used in II-RNN [26] records the user’s history of +visiting the subreddits, i.e., discussion topics, in Reddit. +Data Preparation: We prepare the datasets based on the process +in previous SBRS works [13, 26, 27]. For each dataset, we remove +inactive users and items with the number of interactions less than +10. Then, we put two successive items in a user’s interaction history +into one session if the interval between them is less than 6 hours. In +contrast, if the interval is greater than 6 hours, they are put into two +different sessions. Next, we remove the sessions containing only +one item and the sessions with more than 20 items. The reason for +not considering sessions with one item is that these sessions cannot +have the context together with a item to be predicted [27]. Removing +long sessions is a common practice in SBRSs [13, 27], because there +are few long sessions, so removing them will not affect the results +of SBRSs, but keeping them will greatly increase the running time +of many baseline algorithms. After the pre-processing, we show +the basic information of the three datasets in Table 1. +Evaluation Protocol and Metrics: We use 5-fold cross-validation +to obtain reliable experimental results. Specifically, we randomly +divide all sessions in the dataset into five equal parts. We select one +of the five parts as the test set and the remaining as the training +set for each experiment. The experiments for each dataset will be +conducted five times so that the test sets cover all the data, and the +average experimental results of the five parts are reported. Note +that in each fold, we ensure that the training set contains all users +and all items to avoid abnormal results. Besides, we randomly select +half of the sessions in the test set to form the validation set. +We use commonly used ranking measures in previous works, +i.e., recall and NDCG, to evaluate the performance of each SBRS +[12, 27, 29]. Specifically, for each test session, we iteratively select +each item as the target item and the items before this item is the +session’s context. For each algorithm, we sort all items based on +their recommendation scores output by the model and select the +top-K scored items to calculate recall@K and NDCG@K. +4https://www.kaggle.com/datasets/colemaclean/subreddit-interactions +To obtain the best performance for each algorithm, we first ini- +tialize its hyper-parameters using the settings given in the paper +for each algorithm and then fine-tune the most important hyper- +parameters based on the performance of each algorithm on the +validation set. We report the performance of the next item rec- +ommendations in short sessions since some SBRSs will perform +better (see Appendix A for details). The main parameters of the +baselines are set as follows: In KNN, the number of similar sessions +is 500; In CSRM, the number of memory slots is 256; In II-RNN, +the dimension of embeddings is 50; In INSERT, the dimension of +the hidden state is 50, and the number of similar sessions is set to +10; In both SASRec and BERT4Rec, we set the number of attention +layers 𝐿 = 2 and the number of attention heads ℎ = 2. In the pro- +posed COCO-SBRS, the number of other sessions, i.e., |𝜋|, is 10, +and the trade-off parameter 𝛽 is set to 1 for all experiments. We +implement COCO-SBRS using PyTorch and GPU to accelerate the +model’s training. The source code of COCO-SBRS is available in +https://github.com/wzsong17/COCO-SBRS. +5.2 +Recommendation Performance Evaluation +and Analysis +In this section, we conduct comparison experiments to evaluate the +performance of the proposed COCO-SBRS and all baseline algo- +rithms to answer the question: “How does the proposed COCO- +SBRS perform compared with the baseline SBRSs?” +Table 2 presents the recommendation performance of all com- +pared algorithms on three datasets. In general, the first five SBRSs, +i.e., SKNN, GRU4Rec, STAMP, CSRM and SR-GNN, do not perform +well compared with other SBRSs, which consider the outer-session +causes. SKNN does not consider the complex item patterns and +user preferences when calculating session similarity. STAMP uses +attention neural networks to model sessions and is good at extract- +ing information from sessions with uninformative items. CSRM is +a collaborative filtering-based SBRS, but its performance relies on +the session similarities learned with attention networks and the + +Conference’17, July 2017, Washington, DC, USA +W. Song, et al. +quality of session embedding obtained by the memory neural net- +work. GRU4Rec considers the sequential pattern among the items +in sessions, while SR-GNN models the item transition patterns in +sessions. Thus, their performance depends on the proportion of the +corresponding patterns present in the dataset. +Different from the above five algorithms, HGRU, II-RNN, SASRec, +BERT4Rec, and INSERT consider both inner-session causes and +outer-session causes, i.e., user preference across sessions. HGRU +considers the dynamic change of user preferences between succes- +sive sessions of the same user and models it with a user-level RNN. +In contrast, II-RNN considers that user preferences are constant be- +tween successive sessions of the same user, so II-RNN directly uses +the information of the most recent session to supplement the lim- +ited information in the target session. For SASRec and BERT4Rec, +we concatenate the target session and all the historical interactions +of the user as the interaction sequence, which allows the models +to fully extract information from both the ISCs and OSCs based +on their deep neural networks. We can see that BERT4Rec can +obtain good performance on Reddit due to many repeat items in +user sequences. INSERT is the state-of-the-art SBRS for next item +recommendations in short sessions, which uses a specially designed +session similarity calculation module to incorporate information +from other users. From Table 2, we can see that INSERT performs +better than the above methods w.r.t. half of metrics. However, IN- +SERT still needs to find similar sessions from the training set. It has +difficulty finding similar sessions from the data when considering +both inner-session causes and outer-session causes. +The proposed COCO-SBRS achieves a significant improvement +in both Recall and NDCG compared to the baseline methods (except +for the NDCG in Reddit). The reason is that the counterfactual +computing framework alleviates the difficulty of finding similar +sessions due to data sparsity while considering both inner-session +causes and outer-session causes. Besides, we pre-train BRM in +the framework so that the model can better untangle and identify +the true causes and model the causalities in SBRSs. Finally, the +performance of COCO is further enhanced by the boost factor. +5.3 +Ablation Analysis +In this experiment, we test the performance of several simplified +versions of COCO-SBRS to answer the question: “How does the +proposed counterfactual computing framework benefit the +next item recommendation in SBRS?” +We design three simplified variants of COCO-SBRS: (1) BRM, +the base recommendation model predicts the next item in sessions +without the counterfactual computing framework; (2) COCO w/o +BF, which removes the boost factor by setting 𝛽 = 0; (3) COCO +w/o SSL, which removes the self-supervised loss, i.e., Equation (11). +Table 3 shows the performance of the three simplified variants +and the full version of COCO-SBRS on Delicious and Lastfm. The +table shows that BRM performs the worst, showing the effective- +ness of the proposed collaborative filtering-based counterfactual +framework proposed in this paper. It also shows that even consid- +ering all the causes for the selection of the next item, it is difficult +to model the causalities in SBRSs well using deep learning models +only. The poor performance of COCO w/o BF and COCO w/o SSL +compared to the full version COCO-SBRS indicates that both the +Table 3: Recommendation performance of COCO-SBRS and +its simplified variants on Delicious and Lastfm. +dataset +Variant +R@5 +N@5 +R@20 +N@20 +Lastfm +BRM +0.325 +0.201 +0.643 +0.292 +COCO w/o BF +0.444 +0.253 +0.727 +0.335 +COCO w/o SSL +0.412 +0.247 +0.756 +0.346 +COCO-SBRS +0.504 +0.289 +0.793 +0.374 +Delicious +BRM +0.214 +0.137 +0.397 +0.190 +COCO w/o BF +0.271 +0.169 +0.458 +0.223 +COCO w/o SSL +0.273 +0.170 +0.463 +0.225 +COCO-SBRS +0.359 +0.215 +0.520 +0.263 +0 0.51.0 +2.0 +5.0 +0.44 +0.45 +0.46 +0.47 +0.48 +Recall@20 +Recall@20 +0.21 +0.22 +0.23 +0.24 +NDCG@20 +NDCG@20 +010 +50 +100 +150 +200 +| | +0.46 +0.47 +0.48 +Recall@20 +Recall@20 +0.22 +0.23 +0.24 +NDCG@20 +NDCG@20 +Figure 4: Sensitivity of 𝛽 and |𝜋| on Delicious. +boost factor and the pre-training with self-supervised loss can ef- +fectively improve the performance of the proposed COCO-SBRS. +The booster factor can improve the performance of COCO-SBRS +by emphasizing the most recently seen items since, in real-world +data, users often interact with preferred items repeatedly [13]. In +addition, the self-supervised loss can help COCO-SBRS learn un- +tangled representations of ISCs and OSCs and improve the ability +to identify the true causes in Equation (5). +5.4 +Hyper-parameters Sensitivity Test +We use two groups of experiments to test COCO’s hyper-parameter +sensitivity: (1) The process of pre-training BRM in COCO-SBRS +considers two loss functions, i.e., Equation (8) and Equation (11), so +in the first group of experiments we test the sensitivity of the model +to the balance of two losses. (2) In collaborative filtering-based +models, an important hyper-parameter affecting the algorithms’ +performance is the number of similar neighbors. In COCO-SBRS, +this parameter refers to the number of other sessions that answer +the counterfactual question, i.e., |𝜋|. Thus, in the second group +of experiments, we test the performance of COCO-SBRS under +different |𝜋| while keeping other hyper-parameters unchanged. +The experimental results of the two groups of experiments are +shown in Figure 4. +From Figure 4, we can see that the performance of COCO-SBRS +increases as 𝛽 goes from 0 to 1 but gradually decreases after the 𝛽 +is greater than 1. The experimental results indicate that the optimal +trade-off weight for balancing the two losses is 1. Besides, the +model performs best when |𝜋| = 50 but decreases as |𝜋| continues +to increase. Note that for COCO-SBRS a large |𝜋| will result in +more running time of the model, so in the previous experiments, + +A Counterfactual Collaborative Session-based +Recommender System +Conference’17, July 2017, Washington, DC, USA +we set |𝜋| = 10 to balance the recommended performance and +model efficiency. +6 +CONCLUSION +The present study was designed to address the problem that the +confounder in SBRSs can cause the SBRS models to learn spurious +correlations in the data. This work proposes a counterfactual-based +framework named COCO-SBRS for next item recommendation in +SBRSs. COCO-SBRS first adopts a self-supervised approach to pre- +train a recommendation model to learn the causalities in SBRSs, and +then make recommendations for a session based on some neighbor +sessions with the same causes as this session and the recommen- +dation model used to simulate the user’s selection in sessions. We +conduct extensive experiments on three real-world datasets, and +the results show that the proposed COCO-SBRS can eliminate the +influence of spurious correlations caused by the confounder in +SBRSs and make accurate next item recommendations. A limitation +of this study is that COCO only considers user preference and item +co-occurrence when pre-training the base recommendation model. +In the future, we will explore more inner-session causes and outer- +session causes such as social influence and their impacts on SBRSs +for better recommendation performance. +ACKNOWLEDGMENTS +This work is supported by NSFC (under Grant No.62202200, 61976050, +61972384), the Fundamental Research Funds for the Central Univer- +sities 2412019ZD013, and Jilin Science and Technology Department +20200201280JC. + +Conference’17, July 2017, Washington, DC, USA +W. 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The shorter +the session length, the less information contained in the context +of the session, and the lower the probability that the user select +an item because of the ISCs, i.e., the information in the session +context. Therefore, the performance of each algorithm on sessions +with different lengths can reflect their ability to learn the causality +and the correlation among ISCs, OSCs and the interactions in SBRS. +We conduct comparison experiments on three datasets, and the +results are depicted in Figure 5, Figure 6 and Figure 7. +From the figures, we can see that: i) On all three datasets, COCO- +SBRS outperforms all baseline algorithms in terms of Recall@20 on +sessions with all lengths. COCO-SBRS also has the best NDCG@20 +on Last.fm and Delicious, except for sessions of length five on De- +licious. This result shows that COCO-SBRS can better model the +causalities in SBRS and avoid the model learning spurious corre- +lations in the data. ii) There is a trend of decreasing performance +of COCO-SBRS when session lengths become longer. COCO-SBRS +performs better than all baselines when the the session length is +not greater than five. This shows that COCO-SBRS is not good at +extracting information from long sessions, which may be because +we adopt a simple attention neural network to model the ISCs for +sessions. Thus, we suggest that for those datasets containing a large +number of long sessions, it’s better to replace the attention-based +model in COCO-SBRS with a more powerful session encoder to +model ISCs in SBRSs. iii) On Reddit, the methods consider both +ISCs and OSCs, i.e., HGRU, II-RNN, SASRec, BERT4Rec, INSERT +and COCO-SBRS, perform better than those methods that consider +ISCs only. This is because the preferred topics of each user, i.e., the +items on Reddit, are stable, so users’ long-term static preferences, +i.e., OSCs, are the main causes for users to select the next topic. +Thus, it is more difficult to predict user-item interactions based on +the co-occurrence and sequential patterns of the topic, i.e., ISCs. +2 +3 +4 +5 +Session Length +0.0 +0.2 +0.4 +0.6 +0.8 +Recall@20 +Recall@20 on Last.fm +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +2 +3 +4 +5 +Session Length +0.0 +0.1 +0.2 +0.3 +0.4 +NDCG@20 +NDCG@20 on Last.fm +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +Figure 5: Recommendation Performance of All Methods on +Sessions with Different Lengths on Last.fm. +2 +3 +4 +5 +Session Length +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Recall@20 +Recall@20 on Delicious +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +2 +3 +4 +5 +Session Length +0.0 +0.1 +0.2 +0.3 +NDCG@20 +NDCG@20 on Delicious +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +Figure 6: Recommendation Performance of All Methods on +Sessions with Different Lengths on Delicious. +2 +3 +4 +5 +Session Length +0.0 +0.2 +0.4 +0.6 +Recall@20 +Recall@20 on Reddit +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +2 +3 +4 +5 +Session Length +0.0 +0.1 +0.2 +0.3 +NDCG@20 +NDCG@20 on Reddit +SKNN +GRU4Rec +STAMP +CSRM +SR-GNN +HGRU +II-RNN +SASRec +BERT4Rec +INSERT +COCO-SBRS +Figure 7: Recommendation Performance of All Methods on +Sessions with Different Lengths on Reddit. + diff --git a/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/load_file.txt b/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2846be8158c221b3cf3cd6ed6913497e8f3385f --- /dev/null +++ b/j9FQT4oBgHgl3EQfmDaC/content/tmp_files/load_file.txt @@ -0,0 +1,914 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf,len=913 +page_content='A Counterfactual Collaborative Session-based Recommender System Wenzhuo Song Northeast Normal University China wzsong@nenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='cn Shoujin Wang University of Technology Sydney Australia shoujin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='wang@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='au Yan Wang Macquarie University Australia yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='wang@mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='au Kunpeng Liu Portland State University United States kunpeng@pdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu Xueyan Liu∗ Jilin University China xueyanliu@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='cn Minghao Yin Northeast Normal University China ymh@nenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='cn ABSTRACT Most session-based recommender systems (SBRSs) focus on extract- ing information from the observed items in the current session of a user to predict a next item, ignoring the causes outside the session (called outer-session causes, OSCs) that influence the user’s selection of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, these causes widely exist in the real world, and few studies have investigated their role in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this work, we analyze the causalities and correlations of the OSCs in SBRSs from the perspective of causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We find that the OSCs are essentially the confounders in SBRSs, which leads to spu- rious correlations in the data used to train SBRS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To address this problem, we propose a novel SBRS framework named COCO- SBRS (COunterfactual COllaborative Session-Based Recommender Systems) to learn the causality between OSCs and user-item interac- tions in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO-SBRS first adopts a self-supervised approach to pre-train a recommendation model by designing pseudo-labels of causes for each user’s selection of the item in data to guide the train- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, COCO-SBRS adopts counterfactual inference to recommend items based on the outputs of the pre-trained recom- mendation model considering the causalities to alleviate the data sparsity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' As a result, COCO-SBRS can learn the causalities in data, preventing the model from learning spurious correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The experimental results of our extensive experiments conducted on three real-world datasets demonstrate the superiority of our proposed framework over ten representative SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='XXXXXXX KEYWORDS session-based recommendation, self-supervised learning, counter- factuals 1 INTRODUCTION Existing studies on Recommender Systems (RSs) mainly focus on modeling and predicting all user-item interactions to learn users’ static item preferences, ignoring the dynamic nature of user pref- erences in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To bridge this gap, session-based rec- ommender systems (SBRSs) have recently emerged and gained in- creasing attention [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Most existing SBRS models formalize the recommendation of next user-item interactions (next item for short) in sessions as a supervised learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For a current session of a user to be predicted (called target session), an SBRS model takes the items of the observed interactions in this session (called the session’s context) as input and outputs a list of items as predicted next items [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Most of these SBRSs imply a strong assumption that a user selects an item as the next item in a session only because the item correlates with the items in the same session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, they ignore the selection of the next items caused by factors other than items in the same session [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For example, on an e-commerce site, a user may select a product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', an item, because of his/her short-term intent or previous interest, a discount for this product, or a short-term item popularity trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In either case, the item selected is not related to other items in the same session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In contrast, in this work, we suggest that it is not appropriate to train SBRSs without considering the influence of causality of the factors outside the session context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Let us consider that a user selects the next items in a session based on two types of causes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', inner-session causes and outer-session causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Inner-session causes (ISCs for short) refer to the selection of the next items caused by the context of the same session, which may be due to the co-occurrence pattern and sequential pattern between items in the same session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In contrast, outer-session causes (OSCs for short) refer to the selec- tion of the next item caused by factors outside the target session, such as long-term user preferences, product popularity trends and discounts for products on e-commerce sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To facilitate an insight- ful analysis, we first present a graph in Figure 1 (a) showing the correlations between variables considered in existing SBRS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For example, when camera and fan are on discount at the same time in a supermarket, many users buy them together because they are cheaper than usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Suppose camera and fan are ISCs 𝑀 and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='13364v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='IR] 31 Jan 2023 Conference’17, July 2017, Washington, DC, USA W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Song, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' correlation causality spurious correlation camera fan camera fan camera lens ( c ) ( a ) discount ( b ) Figure 1: (a) Existing SBRSs make recommendations based on the co-occurrence-based correlation between camera and fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (b) However, the correlation between camera and fan is spurious since the purchase of fan is not because of the purchase of camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Instead, the co-purchase of camera and fan rely on the fact that they have discounted prices at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (c) Actually, there are causalities between items in SBRS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For example, a user buys a lens because he/she has bought a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' next item 𝐼 depicted in Figure 1 (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The SBRS models learn the co-occurrence pattern 𝑀 ↔ 𝐼 in the sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' If a user buys a camera, the models will recommend a fan to him/her even if the discount has ended, because there are a large number of ses- sions/transactions containing camera and fan together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, camera and fan are not related in terms of purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A user buys fan is not because he/she has bought camera, but because fan is on discount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Therefore, this co-occurrence correlation may result in the mis-modeling problem in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To further explain this problem, we present two causal graphs to describe the causalities in Figure 1 (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The causal graphs are directed acyclic graphs defined based on the structural causal model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We use the causal graph in Figure 1 (b) to represent the causalities in the above-mentioned example, where 𝑁 denotes the discount of a product, which is one of OSCs in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 𝑁 → 𝐼 and 𝑁 → 𝑀 denote a user buys a fan and a camera due to their discounted price respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Hence, the discounted price is the common cause of the purchase of camera and fan, and it is called a confounder1 in SBRSs [21, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Clearly, the confounder results in a spurious correlation between camera and fan as depicted in Figure 1 (b) since the purchase of fan is not because the purchase of camera even though they have been purchased successively in one session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A reliable SBRS should not make recommendations based on such spurious correlation since it cannot reflect the true casual relations between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Actually, there are some other reliable causality relations between items in SBRS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For example, as depicted in Figure 1 (c), the purchase of a lens is often because of the prior purchase of a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Such kind of relations should be the basis for making reliable recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' From above examples, we can conclude that the OSCs can be the confounders in SBRSs, and the relationship between items in session contexts and the next item in the data is a mixture of the causality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', camera and lens in Figure 1 (c), and the spurious correlations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', camera and fan in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Therefore, (Problem 1) without considering the 1Confounders are variables that affect both the treatment variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', camera and the outcome variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', fan and lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' causalities of the OSCs, the SBRSs will learn both the causality and the spurious correlations caused by the confounder and thus may generate incorrect recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' One possible way to solve this problem is to list all possible causes of the next item selection in sessions and then train a deep learning model in an end-to-end manner to learn the complex relationship between the next item and the causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, the true cause, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', which of ISCs and OSCs is the reason why a user selects a specific item, is a latent variable in the model, and they are not provided in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Thus, (Problem 2) it is difficult for deep learning models to identify the true causes of the item selections and learn the causalities shown in Figure 1 (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this paper, we address Problem 1 by proposing a novel SBRS framework considering both OSCs and ISCs for the next item recom- mendation in sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The proposed model is named COunterfactual COllaborative Session-Based Recommender System (COCO-SBRS, or COCO for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO is inspired by the ideas of counterfactu- als [45] and collaborative filtering for sessions [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, COCO first pre-trains a recommendation model to learn the causal- ities among ISCs, OSCs and user-item interactions in SBRSs, and then predicts the next item for a session based on some neighbor sessions with the same ISCs and OSCs as this session and the rec- ommendation model used to simulate the user’s selection in the neighbor sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To address Problem 2, in the pre-training phase, we adopt a self-supervised approach to train the recommendation model by designing pseudo-labels of causes for each user-item inter- action to guide the training of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To alleviate the problem of the lack of sessions with required ISCs and OSCs in the predic- tion phase, we adopt counterfactual inference to generate sessions using required ISCs and OSCs, and simulate users’ decision-making process with the pre-trained recommendation model to recommend items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In summary, the main contributions of this work are: (1) We propose an SBRS framework named counterfactual col- laborative session-based recommender system (COCO-SBRS or COCO) for effective next item recommendations in ses- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To the best of our knowledge, this is the first work in the literature to address the problem of spurious correlations caused by the confounder in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) We are the first to formulate the next item recommendation in sessions in the framework of counterfactual computing and collaborative filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we first develop a self- supervised approach to pre-train a recommendation model to learn causalities in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Then, we recommend items for a given session using the model, taking other sessions with the same ISCs and OSCs as this session as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (3) To evaluate the effectiveness of COCO, we conduct exten- sive experiments on three real-world datasets from various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We compare COCO with the representative and state-of-the-art SBRSs, and the experimental results show COCO’s superiority over baseline SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 Session-based Recommender Systems In this section, we introduce two groups of SBRSs: (1) SBRSs that consider inner-session causes only, and (2) SBRSs that consider both inner-session and outer-session causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A Counterfactual Collaborative Session-based Recommender System Conference’17, July 2017, Washington, DC, USA SBRSs that consider inner-session causes only make recom- mendations based on the session context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' These algorithms aim to learn users’ short-term preferences reflected by the items in the session context and the complex relationships between items in sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' According to the employed technology, these SBRSs can be classified into conventional approaches, latent representation approaches, and deep neural network approaches [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' KNN-based SBRSs and Markov chain-based SBRSs are the most popular conven- tional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' KNN-based SBRSs such as SKNN recommend items in sessions similar to the target session [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Markov chain- based SBRSs such as FPMC make recommendations by modeling the transition patterns of items in sessions [14, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The latent rep- resentation SBRSs utilize the technique of latent factors models or matrix factorization to make recommendations [4, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In recent years, deep learning-based SBRSs are becoming popular, and re- searchers have developed SBRSs based on various deep learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Recurrent neural network (RNN) based SBRSs model the sequential pattern of items in sessions [9, 10, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, they are based on a rigid assumption that adjacent items in each session are sequentially dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Attention-based methods relax this assumption by emphasizing those more informative items in sessions to reduce the interference of uninformative ones [3, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To model the high-order transition among items, the graph neural net- work (GNN) based SBRSs first represent sessions with graphs and then employ GNN models to make recommendations [23, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Deep learning-based SBRSs can model the complex patterns of items in sessions and therefore achieve better performance over other approaches in many recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, the above-mentioned SBRSs can only predict the next item based on the limited informa- tion in the session context and ignore the global information and factors outside the session influence users’ selection of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' SBRSs that consider both inner-session and outer-session causes aim to extract and fuse information from both the target session and other sessions to make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The most common information from other sessions considered in existing works includes global information such as item dependency and long-term item preferences of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Global information is essential for SBRSs because the lengths of sessions in the real world tend to be very short and contain limited information, and the global information can be used as a supplement to the session context [22, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' User preferences are outer-session information that has great potential to improve the performance of recommendation algorithms because, in many real-world scenarios, it is easy to obtain user behavioral data reflecting user preferences that belong to the same user of the target session [15, 27, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Unlike the above algorithms, this work aims to eliminate the spurious correlations caused by the confounder in SBRSs introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 Counterfactuals in Recommender Systems Counterfactual computing has been introduced into the research of recommendation systems in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Compared to the rapid development in other machine learning fields, the studies on counterfactual-based recommender systems (CFBRSs for short) are still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' According to the problem to be addressed, existing CFBRSs can be classified into algorithms for data bias, algorithms for data missing, and algorithms for other tasks [5, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Among the algorithms for data bias, researchers have proposed the MACR algorithm for popularity bias, and the CR algorithm for clickbait bias [2, 16, 20, 37, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The data missing problem stems from the large number of inactive users and items in RSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Existing works use counterfactual-based data generation to alleviate the problem of missing data in RSs [39, 43, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Other problems studied in CFBRSs include explainability, diversity and fairness in recommender sys- tems [7, 31, 32, 38, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Different from the existing CFBRSs, we study the counterfactual-based model for learning causality in session- based recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 3 PROBLEM STATEMENT In this work, we use 𝑈 = {𝑢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=',𝑢|𝑈 |} to denote the set of |𝑈 | users and 𝑉 = {𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝑣 |𝑉 |} to denote the set of |𝑉 | items in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A user 𝑢 ∈ 𝑈 has a sequence of user-item interactions, which can be represented using a list of items corresponding to each interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In SBRSs, the interactions of each user form a series of sessions, where each session is a list of the user’s interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', item list 𝑠 = {𝑣𝑠 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝑣𝑠 𝑡 } in a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The subscript of each item in 𝑠 denotes its order in 𝑠 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='t its occurring time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Given a target session 𝑠 to be predicted, the goal of an SBRS is to predict the next interactions in 𝑠, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', the next item 𝑣𝑠 𝑡+1, based on known information such as user preferences and session context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this work, we consider two causes for a user to select the next item in sessions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', ISCs and OSCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we formulate the SBRS model as a probabilistic classifier 𝑝(𝑣|𝑀, 𝑁) over a set of items in the dataset conditioning on the ISCs variable 𝑀 and the OSCs variable 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The prediction result of 𝑝(𝑣|𝑀, 𝑁) represents the probability distribution of the item which user will select as the next item for 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Finally, items with top-K probabilities are selected as the recommendation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 4 COUNTERFACTUAL COLLABORATIVE SESSION-BASED RECOMMENDER SYSTEM In this section, we introduce the proposed SBRS method named COunterfactual COllaborative Session-Based Recommender System (COCO-SBRS, or COCO for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO is inspired by counterfac- tuals [8, 45] and collaborative filtering for sessions [19, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO first pre-trains a recommendation model to learn the causalities among ISCs, OSCs and user-item interactions in SBRSs, and then predicts the next item for a session based on some neighbor sessions with the same ISCs and OSCs as this session and the recommen- dation model used to simulate the user’s selection in the neighbor sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, COCO has the following three steps: (i) Ab- duction: In this step, we design a generative process to describe the users’ decision-making process in SBRSs, and then implement the generative process with an attention-based neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we adopt a self-supervised approach to pre-train the neu- ral network using the sessions in the training set to determine its optimal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' After pre-training the BRM, the framework makes recommendations with the steps of action and prediction, the process of which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (ii) Action: For a target session 𝑠 of user 𝑢, we first sample a session 𝑠′ of user 𝑢′ from the dataset, then replace its ISCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝐼𝑆𝐶(𝑠′) with the ISCs of the target session, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝐼𝑆𝐶(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we use the modified session as the input to the model trained in the abduction step to simulate the Conference’17, July 2017, Washington, DC, USA W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Song, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' decision-making process of 𝑢′ to answer a counterfactual question [44]: “Given a target session 𝑠 of 𝑢, which item will be selected as the next item by another user𝑢′ based on the similar ISCs and OSCs as 𝑠?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We sample a few sessions of similar users to the user of the target session, and then perform the action step for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (iii) Prediction: In this step, we make recommendations by combining the outputs of all the models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', the answers to the counterfactual question, and the users’ similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 Abduction: Base Recommendation Model and its Self-Supervised based Training Generative Process for SBRSs: Based on the causal graphs in Figure 1 (b) and (c), we present a generative process to describe a user’s decision-making process given specific ISCs and OSCs as follows: Given a target session 𝑠 of user 𝑢: (1) generate 𝑂𝑆𝐶(𝑠) according to the user 𝑢 of 𝑠;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) generate 𝐼𝑆𝐶(𝑠) according to the items in 𝑠’s context 𝑐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (3) sample a cause 𝐶(𝑢,𝑐) from 𝑂𝑆𝐶(𝑠) and 𝐼𝑆𝐶(𝑠) according to 𝑢 and 𝑐 for an item 𝑣 to predict;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (4) generates the score of the item based on 𝐶(𝑢,𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we implement the generative process for SBRSs based on attention neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We call the model the base recommen- dation model (BRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' BRM’s architecture is depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Modeling OSCs in BRM: We consider two OSCs in SBRSs: static preference and dynamic preference [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (1) Static preference is a long-term user preference that does not change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this work, we use a fixed-length vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', user embedding e𝑢 ∈ R𝑑, to represent the static preference of a user 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) Dynamic preference is the short-term user preference that changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We use an attention-based neural network to learn the embedding of a user’s dynamic preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, given a user 𝑢’s session 𝑠, the embedding of dynamic preference h𝑁 (𝑠) for 𝑠 is calculated by: h𝑁 (𝑠) = 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(e𝑢, E(𝑅(𝑠)), E(𝑅(𝑠))), (1) where 𝑅(𝑠) = {𝑣𝑅 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝑣𝑅 |𝑅|} is a set containing the most recent in- teractions of 𝑢 before 𝑠, |𝑅(𝑠)| is a hyper-parameter and is set as 10 for all sessions in our experiments, E(𝑅(𝑠)) ∈ R|𝑅(𝑠) |×𝑑 denotes the matrix containing all the embeddings of items in 𝑅(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Given a query vector q′ ∈ R𝑑, a matrix of 𝜅 key vectors K′ ∈ R𝜅×𝑑 and a matrix of 𝜅 value vectors V′ ∈ R𝜅×𝑑, 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(q′, K′, V′) is an attention network defined as: 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(q, K′, V′) = ∑︁ 𝑖 𝛼(q, K𝑖) × V𝑖, (2) where 𝛼(q, K′) is a vector containing the attention scores between q and each vector in K′ calculated by: 𝛼(q, K𝑖) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥𝑖 (q · K𝑇 𝑖 / √ 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (3) Finally, we use h𝑁 (𝑠) as the embedding of the OSCs for 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Modeling ISCs in BRM: Given 𝑠’s context 𝑐, the representation of ISCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', h𝑀 (𝑠) for 𝑠 is calculated by: h𝑀 (𝑠) = 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑎𝑣𝑔(E(𝑐)), E(𝑐), E(𝑐)), (4) where E(𝑐) ∈ R|𝑐 |×𝑑 denotes a matrix containing the embeddings of all items in 𝑐, 𝑎𝑣𝑔(E(𝑐)) = 1 |𝑐 | � 𝑣∈𝑐 e𝑣 is the mean vector of the item embeddings in 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next-Item Prediction in BRM: In the generation process, the user selects one of the OSCs and ISCs as the cause for the next item selection in 𝑠 according to the user and the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We implement this process with an attention-based neural network to learn a soft weight of the two causes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝜆 ∈ [0, 1]: 𝜆 = 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(W𝑇 (h𝑀 (𝑠)||h𝑁 (𝑠)||e𝑣) + 𝑏), (5) where 𝑠𝑖𝑔𝑚𝑜𝑖𝑑(𝑧) = 1/(1 + 𝑒−𝑧), || denotes the concatenate oper- ation, 𝑣 ∈ 𝑉 is the item to be predicted, and W ∈ R𝑑 and 𝑏 ∈ R are parameters to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A large 𝜆 means that the OSCs have more influence than the ISCs on the next item selection, and a small 𝜆 means the ISCs have more influence than the OSCs for 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we incorporate the two causes by: h(𝑠) = 𝜆h𝑁 (𝑠) + (1 − 𝜆)h𝑀 (𝑠), (6) and predict the next item for 𝑠 by: 𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥(e(Ω)h(𝑠)𝑇 ), (7) where Ω is the set of items to be predicted, and e(Ω) ∈ R|Ω|×𝑑 is a matrix containing the embeddings of all items in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Training BRM with Cross Entropy Loss: By regarding the next item prediction in sessions as a multi-class classification task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' we can train the model using the Cross Entropy Loss: 𝓁1 = 1 |𝑆𝑏| ∑︁ 𝑠 ∈𝑆𝑏 ∑︁ 𝑣′∈𝑉𝑏 1[𝑣′ ∈ 𝑐] log𝑝(𝑣 = 𝑣′|𝑀 = 𝐼𝑆𝐶(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 𝑁 = 𝑂𝑆𝐶(𝑠)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (8) where 𝑐 is the context of 𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 𝑆𝑏 denotes the set of all sessions in the training batch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 𝑉𝑏 is the set of all items in the training batch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' and 1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] is defined as: 1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] = � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' if 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 is True,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' if 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 is False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (9) Improve BRM with Self-Supervised Learning: In the pro- posed generation process, an item is selected as the next item due to one of the ISCs and OSCs, which we call the true cause for the corresponding user-item interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, the true cause is a latent variable in the model since they are not provided in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To address this problem, in BRM, we use a linear model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', Equation (5), to learn the relationship between the latent variable 𝜆 and the observational variables 𝑀 and 𝑁 so that the model gets the ability to identify the true cause corresponding to each interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To improve the accuracy of true cause identification, we propose a self-supervised learning approach to guide the training of the parameters in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Our main idea is to construct a pseudo- label for the soft weights of two causes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we assume that: given each item of each session, (1) if an item appears in the interaction history of 𝑢, the user is more likely to select the item due to the OSCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝜆 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) if an item appears in the context of the target session, the user is more likely to select the item due to the ISCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', (1 − 𝜆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Based on these two assumptions, we obtain the definitions of 𝜆’s pseudo-labels: � 𝑦𝑁 (𝑠) = 1[𝑣 ∈ 𝑉𝑢], 𝑦𝑀 (𝑠) = 1[𝑣 ∈ 𝑐], (10) A Counterfactual Collaborative Session-based Recommender System Conference’17, July 2017, Washington, DC, USA : sessions similar to Action: Replace session context with BRM BRM BRM calculate similarity Action: Predict with BRMs target session Top-K List Item 1 Item 2 Item 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Prediction: Combine BRM outputs Top-K List Item 1 Item 3 Item 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Enhance with Boost Factor Inputs: interactions before session the context of the session + Output: replace weighted sum add to relevant items .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Figure 2: The steps of Action and Prediction of COCO-SBRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The BRMs are pre-trained with the sessions in the training set in the Abduction step, which is not shown in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' User Input Layer Embed- ding attention network attention network Cause Repre- sentation Cause Weights Recent Interactions Context of the Session Candidate Item OSCs ISCs Next Item Prediction Cause Modeling Output: score of the item .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Figure 3: Base Recommendation Model (BRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' where 𝑉𝑢 is the set of items the user of 𝑠 has interacted before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we consider the true cause prediction problem as a binary classification problem and present a self-supervised loss based on the Binary Cross Entropy Loss: 𝓁2 = 1 |𝑆𝑏| ∑︁ 𝑠 ∈𝑆𝑏 𝐵𝐶𝐸(𝜆,𝑦𝑁 (𝑠)) + 𝐵𝐶𝐸((1 − 𝜆),𝑦𝑀 (𝑠)), (11) where given a prediction score 𝑥 and the corresponding label 𝑦, Binary Cross Entropy Loss 𝐵𝐶𝐸(𝑥,𝑦) is defined as: 𝐵𝐶𝐸(𝑥,𝑦) = 𝑦 · log𝑥 + (1 − 𝑦) · log(1 − 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (12) Training BRM according to Equation (11) encourages the model to generate soft weights of the OSCs and ISCs that are consistent with pseudo-labels, improving the model’s ability to untangle the ISCs and OSCs, and identify the true causes of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Finally, the loss function for the training of BRM is defined as a trade-off of two loss functions: 𝓁 = 𝓁1 + 𝛽 ∗ 𝓁2, (13) where 𝛽 ∈ [0, +∞) is the trade-off hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 Action and Prediction: Counterfactual and Collaborative Next-Item Recommendation The key idea of collaborative filtering for sessions is predicting items in other sessions similar to the target session [6, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, when calculating session similarities, these methods only consider the items in the session context, and ignore outer-session causes such as static user preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In addition, these methods assume that every session in the dataset has similar sessions, which may not be true when the dataset is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' As a result, they may treat sessions of users who have a completely different preference as similar sessions, and thus make incorrect recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' To address the above problems, given a target session 𝑠 of a user 𝑢, we use the BRM model to simulate users’ decision-making process for the counterfactual question: “which next item will be selected by another user 𝑢′, who has similar preferences to 𝑢 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝑂𝑆𝐶(𝑠) in BRM) in the same session context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝐼𝑆𝐶(𝑠) in BRM)?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this way, we can ensure that the next item for each similar session is generated when the user has similar ISCs and OSCs to the target session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, the process contains two steps: Action: Given a target session 𝑠 of a user 𝑢, the action step aims to find a user𝑢′ who has a similar preference to𝑢 and then compute the probability distribution of the next item under the same ISCs of 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we first find a session 𝑠′ with similar OSCs to 𝑠 by calculating the similarity of the recent interaction sets2: 𝑠𝑖𝑚(𝑠,𝑠′) = |𝑅(𝑠) ∩ 𝑅(𝑠′)| |𝑅(𝑠) ∪ 𝑅(𝑠′)|, (14) where 𝑠′ ≠ 𝑠 is a session sampled from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A large 𝑠𝑖𝑚(𝑠,𝑠′) means that 𝑢′ has similar OSCs to 𝑢 when the session 𝑠′ occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, a larger value of 𝑠𝑖𝑚(𝑠,𝑠′) does not mean that the ISCs of 𝑠′ are similar to the ISCs of 𝑠, and direct use of BRM to predict for 2The most recent interaction set of a session, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', 𝑅(𝑠), can reflect the dynamic user preferences when the session occurs, which is the main part of the OSCs for 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Song, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Table 1: The detailed statistical information of the three datasets used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm Delicious Reddit #sessions 5915 45,772 1,122,150 #interactions 38,367 249,919 2,874,671 #users 1,101 1,752 19,878 #items 711 5,047 13,742 #interactions per user 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='85 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='65 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='62 #interactions per session 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='56 #sessions per user 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='37 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='13 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='45 𝑠 based on 𝑠′ will produce erroneous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We replace the 𝐼𝑆𝐶(𝑠′) with 𝐼𝑆𝐶(𝑠) as an action on the BRM’s prediction in Equation (7) to address this problem: 𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (15) In this way, the BRM can predict the next item as it simulates 𝑢′ in the context of session 𝑠, which does not exist in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Prediction: In the action step, we only consider one similar user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Based on the idea of collaborative filtering, in the prediction step, we consider that increasing the number of similar users could improve the recommendation performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we first select the most similar sessions of 𝑠 from the dataset to form a session set 𝜋(𝑠) according to the session similarities of Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, for each user 𝑢𝑖 of session 𝑠𝑖 ∈ 𝜋(𝑠), we replace the 𝐼𝑆𝐶(𝑠𝑖) in its corresponding BRM with 𝐼𝑆𝐶(𝑠) while keeping its 𝑂𝑆𝐶(𝑠𝑖) unchanged, and then we weighted sum the results of all BRMs of each session in 𝜋(𝑠) to obtain the result: 𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = 1 𝐶 ∑︁ 𝑠𝑖 ∈𝜋 (𝑠) 𝑠𝑖𝑚(𝑠,𝑠𝑖) × 𝑝(𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠𝑖)), (16) where 𝐶 is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 Enhancement with Boost Factor After obtaining the item distribution 𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀, 𝑁) for the next item recommendation, we find that emphasizing the recently seen items in the target session 𝑠 can further improve the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, for each item 𝑣𝑖 ∈ (𝑅(𝑠) ∩ 𝑐), we first modify the weights of recently seen items by adding a boost factor 𝜖 > 0 and then obtain the final result of COCO after enhancement with the boost factor : 𝑝′ 𝑐𝑜𝑐𝑜 (𝑣|𝑀 = 𝐼𝑆𝐶(𝑠), 𝑁 = 𝑂𝑆𝐶(𝑠)) = 1 𝐶′ (𝑝𝑐𝑜𝑐𝑜 (𝑣|𝑀, 𝑁) + 𝜖 ∗ 1[𝑣 ∈ (𝑅(𝑠) ∩ 𝑐)]), (17) where 1[𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛] is a vector containing the results of Equation (10) for all 𝑣 ∈ 𝑉 , and 𝐶′ is the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 5 EXPERIMENTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 Experimental Settings Baseline Algorithms: In the comparison experiments, we select (1) five SBRSs that consider inner-session causes only (SKNN, GRU4Rec, STAMP, CSRM and SRGNN) and (2) five SBRSs that consider both inner-session and outer-session causes (HGRU, II-RNN, SASRec, BERT4Rec and INSERT) as baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' These baselines are based on various techniques, including collaborative filtering, RNN, attention neural networks, memory neural networks and graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' SKNN: A collaborative filtering SBRS built on the idea of recommending the items in neighbor sessions which are similar to the target session [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' GRU4Rec: GRU4Rec employs an RNN to extract dynamic user preference in the session context for the next item rec- ommendation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' STAMP: A memory neural network-based SBRS uses atten- tion to capture users’ short-term preferences for the next item recommendations in sessions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' CSRM: A memory neural network-based SBRS uses atten- tion networks to learn the similarities between sessions and make recommendations based on the information extracted from neighbor sessions and their similarities [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' SR-GNN: A graph neural network-based SBRS first repre- sents the sessions with graphs and then employs a graph neural network to recommend items based on the item tran- sition patterns extracted from the graph [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The following five SBRSs consider both inter-session causes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', the information from the session context, and outer-session causes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', users preference: HGRU: HGRU employs a hierarchical RNN where one RNN models the sequential patterns in the session context, and the other RNN learns a user’s preference across his sessions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' II-RNN: II-RNN utilizes the information extracted from the most recent session to complement and initialize the RNN modeling the target session [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' SASRec: SASRec is a self-attention based sequential RS de- signed to model users’ interaction sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In this work, we concatenate all interactions of each user to form the user sequences [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' BERT4Rec: BERT4Rec is a deep bidirectional self-attention based sequential RS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' BERT4Rec adopts the Cloze ob- jective and predicts an item based on its context and the user’s historical interactions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' INSERT: INSERT is the state-of-the-art SBRS considering both user preference and item patterns in sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' It is de- signed for next item recommendations in short sessions based on few-shot learning and meta-learning [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Datasets: We conduct the experiments on the following three publicly available real-world datasets used in previous SBRS works: Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm3 used in [13] contains the logs of users’ music lis- tening behaviors in Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm online music service [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Delicious is a dataset used in [28] which contains the user tagging records in a social network-based bookmarking sys- tem named Delicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 3Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm and Delicious are from https://grouplens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='org/datasets/hetrec-2011/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A Counterfactual Collaborative Session-based Recommender System Conference’17, July 2017, Washington, DC, USA Table 2: Recommendation performance of all compared methods on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' R@5 and R@20 are short for Recall@5 and Recall@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' N@5 and N@20 are short for NDCG@5 and NDCG@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We adopt 5-fold cross-validation and report the av- erage value for each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For all evaluation metrics, higher numbers represent better method performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The bold and underlined numbers under each metric represent the best and the second-best performing method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' * means the improvement is significant at 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm Delicious Reddit R@5 N@5 R@20 N@20 R@5 N@5 R@20 N@20 R@5 N@5 R@20 N@20 SKNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='235 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='159 GRU4Rec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='542 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', discussion topics, in Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Data Preparation: We prepare the datasets based on the process in previous SBRS works [13, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For each dataset, we remove inactive users and items with the number of interactions less than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Then, we put two successive items in a user’s interaction history into one session if the interval between them is less than 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In contrast, if the interval is greater than 6 hours, they are put into two different sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Next, we remove the sessions containing only one item and the sessions with more than 20 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The reason for not considering sessions with one item is that these sessions cannot have the context together with a item to be predicted [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Removing long sessions is a common practice in SBRSs [13, 27], because there are few long sessions, so removing them will not affect the results of SBRSs, but keeping them will greatly increase the running time of many baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' After the pre-processing, we show the basic information of the three datasets in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Evaluation Protocol and Metrics: We use 5-fold cross-validation to obtain reliable experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, we randomly divide all sessions in the dataset into five equal parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We select one of the five parts as the test set and the remaining as the training set for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The experiments for each dataset will be conducted five times so that the test sets cover all the data, and the average experimental results of the five parts are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Note that in each fold, we ensure that the training set contains all users and all items to avoid abnormal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Besides, we randomly select half of the sessions in the test set to form the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We use commonly used ranking measures in previous works, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', recall and NDCG, to evaluate the performance of each SBRS [12, 27, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Specifically, for each test session, we iteratively select each item as the target item and the items before this item is the session’s context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For each algorithm, we sort all items based on their recommendation scores output by the model and select the top-K scored items to calculate recall@K and NDCG@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='com/datasets/colemaclean/subreddit-interactions To obtain the best performance for each algorithm, we first ini- tialize its hyper-parameters using the settings given in the paper for each algorithm and then fine-tune the most important hyper- parameters based on the performance of each algorithm on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We report the performance of the next item rec- ommendations in short sessions since some SBRSs will perform better (see Appendix A for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The main parameters of the baselines are set as follows: In KNN, the number of similar sessions is 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In CSRM, the number of memory slots is 256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In II-RNN, the dimension of embeddings is 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In INSERT, the dimension of the hidden state is 50, and the number of similar sessions is set to 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In both SASRec and BERT4Rec, we set the number of attention layers 𝐿 = 2 and the number of attention heads ℎ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In the pro- posed COCO-SBRS, the number of other sessions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', |𝜋|, is 10, and the trade-off parameter 𝛽 is set to 1 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We implement COCO-SBRS using PyTorch and GPU to accelerate the model’s training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The source code of COCO-SBRS is available in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='com/wzsong17/COCO-SBRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 Recommendation Performance Evaluation and Analysis In this section, we conduct comparison experiments to evaluate the performance of the proposed COCO-SBRS and all baseline algo- rithms to answer the question: “How does the proposed COCO- SBRS perform compared with the baseline SBRSs?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Table 2 presents the recommendation performance of all com- pared algorithms on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In general, the first five SBRSs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', SKNN, GRU4Rec, STAMP, CSRM and SR-GNN, do not perform well compared with other SBRSs, which consider the outer-session causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' SKNN does not consider the complex item patterns and user preferences when calculating session similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' STAMP uses attention neural networks to model sessions and is good at extract- ing information from sessions with uninformative items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' CSRM is a collaborative filtering-based SBRS, but its performance relies on the session similarities learned with attention networks and the Conference’17, July 2017, Washington, DC, USA W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Song, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' quality of session embedding obtained by the memory neural net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' GRU4Rec considers the sequential pattern among the items in sessions, while SR-GNN models the item transition patterns in sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Thus, their performance depends on the proportion of the corresponding patterns present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Different from the above five algorithms, HGRU, II-RNN, SASRec, BERT4Rec, and INSERT consider both inner-session causes and outer-session causes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', user preference across sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' HGRU considers the dynamic change of user preferences between succes- sive sessions of the same user and models it with a user-level RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In contrast, II-RNN considers that user preferences are constant be- tween successive sessions of the same user, so II-RNN directly uses the information of the most recent session to supplement the lim- ited information in the target session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' For SASRec and BERT4Rec, we concatenate the target session and all the historical interactions of the user as the interaction sequence, which allows the models to fully extract information from both the ISCs and OSCs based on their deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We can see that BERT4Rec can obtain good performance on Reddit due to many repeat items in user sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' INSERT is the state-of-the-art SBRS for next item recommendations in short sessions, which uses a specially designed session similarity calculation module to incorporate information from other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' From Table 2, we can see that INSERT performs better than the above methods w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' half of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' However, IN- SERT still needs to find similar sessions from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' It has difficulty finding similar sessions from the data when considering both inner-session causes and outer-session causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The proposed COCO-SBRS achieves a significant improvement in both Recall and NDCG compared to the baseline methods (except for the NDCG in Reddit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The reason is that the counterfactual computing framework alleviates the difficulty of finding similar sessions due to data sparsity while considering both inner-session causes and outer-session causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Besides, we pre-train BRM in the framework so that the model can better untangle and identify the true causes and model the causalities in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Finally, the performance of COCO is further enhanced by the boost factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 Ablation Analysis In this experiment, we test the performance of several simplified versions of COCO-SBRS to answer the question: “How does the proposed counterfactual computing framework benefit the next item recommendation in SBRS?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We design three simplified variants of COCO-SBRS: (1) BRM, the base recommendation model predicts the next item in sessions without the counterfactual computing framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) COCO w/o BF, which removes the boost factor by setting 𝛽 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (3) COCO w/o SSL, which removes the self-supervised loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Table 3 shows the performance of the three simplified variants and the full version of COCO-SBRS on Delicious and Lastfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The table shows that BRM performs the worst, showing the effective- ness of the proposed collaborative filtering-based counterfactual framework proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' It also shows that even consid- ering all the causes for the selection of the next item, it is difficult to model the causalities in SBRSs well using deep learning models only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The poor performance of COCO w/o BF and COCO w/o SSL compared to the full version COCO-SBRS indicates that both the Table 3: Recommendation performance of COCO-SBRS and its simplified variants on Delicious and Lastfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' dataset Variant R@5 N@5 R@20 N@20 Lastfm BRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='292 COCO w/o BF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='335 COCO w/o SSL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='346 COCO-SBRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='374 Delicious BRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='190 COCO w/o BF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='223 COCO w/o SSL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='225 COCO-SBRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='263 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='48 Recall@20 Recall@20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='24 NDCG@20 NDCG@20 010 50 100 150 200 | | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='48 Recall@20 Recall@20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='24 NDCG@20 NDCG@20 Figure 4: Sensitivity of 𝛽 and |𝜋| on Delicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' boost factor and the pre-training with self-supervised loss can ef- fectively improve the performance of the proposed COCO-SBRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The booster factor can improve the performance of COCO-SBRS by emphasizing the most recently seen items since, in real-world data, users often interact with preferred items repeatedly [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In addition, the self-supervised loss can help COCO-SBRS learn un- tangled representations of ISCs and OSCs and improve the ability to identify the true causes in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='4 Hyper-parameters Sensitivity Test We use two groups of experiments to test COCO’s hyper-parameter sensitivity: (1) The process of pre-training BRM in COCO-SBRS considers two loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', Equation (8) and Equation (11), so in the first group of experiments we test the sensitivity of the model to the balance of two losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' (2) In collaborative filtering-based models, an important hyper-parameter affecting the algorithms’ performance is the number of similar neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In COCO-SBRS, this parameter refers to the number of other sessions that answer the counterfactual question, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', |𝜋|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Thus, in the second group of experiments, we test the performance of COCO-SBRS under different |𝜋| while keeping other hyper-parameters unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The experimental results of the two groups of experiments are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' From Figure 4, we can see that the performance of COCO-SBRS increases as 𝛽 goes from 0 to 1 but gradually decreases after the 𝛽 is greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The experimental results indicate that the optimal trade-off weight for balancing the two losses is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Besides, the model performs best when |𝜋| = 50 but decreases as |𝜋| continues to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Note that for COCO-SBRS a large |𝜋| will result in more running time of the model, so in the previous experiments, A Counterfactual Collaborative Session-based Recommender System Conference’17, July 2017, Washington, DC, USA we set |𝜋| = 10 to balance the recommended performance and model efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 6 CONCLUSION The present study was designed to address the problem that the confounder in SBRSs can cause the SBRS models to learn spurious correlations in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' This work proposes a counterfactual-based framework named COCO-SBRS for next item recommendation in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO-SBRS first adopts a self-supervised approach to pre- train a recommendation model to learn the causalities in SBRSs, and then make recommendations for a session based on some neighbor sessions with the same causes as this session and the recommen- dation model used to simulate the user’s selection in sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We conduct extensive experiments on three real-world datasets, and the results show that the proposed COCO-SBRS can eliminate the influence of spurious correlations caused by the confounder in SBRSs and make accurate next item recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A limitation of this study is that COCO only considers user preference and item co-occurrence when pre-training the base recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' In the future, we will explore more inner-session causes and outer- session causes such as social influence and their impacts on SBRSs for better recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' ACKNOWLEDGMENTS This 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Song, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' A RECOMMENDATION PERFORMANCE OF ALL METHODS ON SESSIONS WITH DIFFERENT LENGTHS In this section, we test the performance of all compared methods on sessions with different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The length of a session refers to the number of interactions contained in the session [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' The shorter the session length, the less information contained in the context of the session, and the lower the probability that the user select an item because of the ISCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', the information in the session context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Therefore, the performance of each algorithm on sessions with different lengths can reflect their ability to learn the causality and the correlation among ISCs, OSCs and the interactions in SBRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' We conduct comparison experiments on three datasets, and the results are depicted in Figure 5, Figure 6 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' From the figures, we can see that: i) On all three datasets, COCO- SBRS outperforms all baseline algorithms in terms of Recall@20 on sessions with all lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO-SBRS also has the best NDCG@20 on Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm and Delicious, except for sessions of length five on De- licious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' This result shows that COCO-SBRS can better model the causalities in SBRS and avoid the model learning spurious corre- lations in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' ii) There is a trend of decreasing performance of COCO-SBRS when session lengths become longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' COCO-SBRS performs better than all baselines when the the session length is not greater than five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' This shows that COCO-SBRS is not good at extracting information from long sessions, which may be because we adopt a simple attention neural network to model the ISCs for sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Thus, we suggest that for those datasets containing a large number of long sessions, it’s better to replace the attention-based model in COCO-SBRS with a more powerful session encoder to model ISCs in SBRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' iii) On Reddit, the methods consider both ISCs and OSCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', HGRU, II-RNN, SASRec, BERT4Rec, INSERT and COCO-SBRS, perform better than those methods that consider ISCs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' This is because the preferred topics of each user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', the items on Reddit, are stable, so users’ long-term static preferences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', OSCs, are the main causes for users to select the next topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' Thus, it is more difficult to predict user-item interactions based on the co-occurrence and sequential patterns of the topic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=', ISCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='8 Recall@20 Recall@20 on Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='4 NDCG@20 NDCG@20 on Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS Figure 5: Recommendation Performance of All Methods on Sessions with Different Lengths on Last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='6 Recall@20 Recall@20 on Delicious SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 NDCG@20 NDCG@20 on Delicious SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS Figure 6: Recommendation Performance of All Methods on Sessions with Different Lengths on Delicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content=' 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='6 Recall@20 Recall@20 on Reddit SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS 2 3 4 5 Session Length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FQT4oBgHgl3EQfmDaC/content/2301.13364v1.pdf'} +page_content='3 NDCG@20 NDCG@20 on Reddit SKNN GRU4Rec STAMP CSRM SR-GNN HGRU II-RNN SASRec BERT4Rec INSERT COCO-SBRS Figure 7: Recommendation Performance of All Methods on Sessions with Different Lengths on Reddit.' 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diff --git a/odFMT4oBgHgl3EQf7DEn/content/tmp_files/2301.12462v1.pdf.txt b/odFMT4oBgHgl3EQf7DEn/content/tmp_files/2301.12462v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3f293e66a1b56bee4a0730e66036efb5bdb24d1 --- /dev/null +++ b/odFMT4oBgHgl3EQf7DEn/content/tmp_files/2301.12462v1.pdf.txt @@ -0,0 +1,1044 @@ +arXiv:2301.12462v1 [cs.GT] 29 Jan 2023 +Combinatorial Pen Testing (or Consumer Surplus of +Deferred-Acceptance Auctions) +Aadityan Ganesh +Jason Hartline +January 31, 2023 +Abstract +Pen testing is the problem of selecting high capacity resources when the only way to mea- +sure the capacity of a resource expends its capacity. We have a set of n pens with unknown +amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in +them. We are allowed to gather more information by writing with them, but this uses up ink +that was previously in the pens. Algorithms are evaluated against the standard benchmark, i.e, +the optimal pen testing algorithm, and the omniscient benchmark, i.e, the optimal selection if +the quantity of ink in the pens are known. +We identify optimal and near optimal pen testing algorithms by drawing analogues to auc- +tion theoretic frameworks of deferred-acceptance auctions and virtual values. Our framework +allows the conversion of any near optimal deferred-acceptance mechanism into a pen testing +algorithm with an additional overhead of at most (1+o(1)) ln n in the approximation factor of +the omniscient benchmark. We use this framework to give pen testing algorithms for various +combinatorial constraints like matroid, knapsack and general downward-closed constraints and +also for online environments. +1 +Introduction +Pen testing is the problem of selecting high capacity resources when the only way to measure +the capacity of the resource expends its capacity (Qiao and Valiant, 2023). We show that any +ascending-price auction can be converted into an equivalent pen testing algorithm. We apply the +auction theoretic framework of virtual values to identify optimal pen testing algorithms. This +connection allows many existing results from auction theory to be applied to pen testing and gives +optimal and near optimal pen testing algorithms in combinatorial and online environments. +The pen testing problem of Qiao and Valiant (2023) is the following. We have a set of pens with +varying amounts of remaining ink and we want to choose a pen with the largest amount of ink left. +We have access to the distribution of ink levels in these pens, but we are only allowed to gather +more information by writing with the pens to test them. While writing gives information about +1 + +whether there is still ink left, it uses up ink that was previously in the pen. Pens that are expended +due to testing can be discarded without incurring any penalty. We compare the performance of our +algorithms against two benchmarks– the optimal pen testing algorithm (the standard benchmark) +and the optimal algorithm that knows the amount of ink in each pen (the omniscient benchmark). +The combinatorial pen testing problem generalizes the pen testing problem to selecting a feasible +subset of pens, e.g., according to a matroid or knapsack constraint, to maximize the total remaining +ink in the chosen subset. +Deferred-acceptance auctions (Milgrom and Segal, 2014) capture a wide range of ascending-price +mechanisms. These are mechanisms that greedily reject the least promising agent by increasing +the price for getting allocated at each stage. For instance, to auction off one good, the deferred- +acceptance mechanism keeps increasing the price, rejecting agents whenever their value for the +good falls below the price until exactly one agent remains in the auction. The surplus of the +mechanism is the sum of the values of all the allocated agents and the consumer surplus is the +surplus minus the payments made by the agents. We provide a black box approach to convert any +surplus optimal (or near optimal) deferred-acceptance auction into an optimal (or near optimal) +pen testing algorithm with equivalent performance guarantees. In this construction, pens and their +ink levels are analogous to agents and their values. While the standard benchmark is similar +to comparing the underlying deferred-acceptance auction against the consumer surplus optimal +auction, the omniscient benchmark compares it against the optimal surplus. These problems are +related by the following theorem. +Theorem 1. Consider a combinatorial pen testing environment with n pens, the analogous auction +environment, and a deferred acceptance mechanism DA for the auction environment. Let γ(n) +denote the standard approximation of DA, i.e., the worst-case ratio of the expected optimal surplus +to the expected surplus of DA for the auction environment. Let ζ(n) denote the optimal omniscient +approximation, i.e., the worst-case ratio of the omniscient benchmark to the expected optimal +consumer surplus for the auction environment. Then, there is a pen testing algorithm that is a γ(n)- +approximation to the standard benchmark and π(n) = γ(n) ζ(n)-approximation to the omniscient +benchmark. +Theorem 1 gives a reduction framework for designing near optimal pen testing algorithms under +arbitrary feasibility constraints. This approach improves bounds from previous literature for all the +environments that we discuss; however, it does not always yield a tight bound. For the online IID +environment (described below), we identify an algorithm with omniscient approximation π(n) that +is a constant factor less than γ(n) ζ(n). +The first step of the framework is to obtain bounds for the optimal omniscient approximation +ζ(n) under various feasibility constraints. We show ζ(n) ≤ (1 + o(1)) ln n in any combinatorial +environment. For the special case of k-identical goods, Hartline and Roughgarden (2008) prove +ζ(n) ≤ +2 +ln 2(1 + o(1)) ln n +k; our analysis gives a better bound when ln k = o(1) ln n. +The second step of the framework combines our bounds on the optimal omniscient approximation +ζ(n) with various γ(n)-approximate deferred-acceptance mechanisms from the literature. Surplus +2 + +optimal deferred-acceptance auctions are known for k-identical goods and more general matroid +feasibility constraints (Milgrom and Segal, 2014). We give a 2-approximate deferred-acceptance +mechanism for knapsack constraints. Beyond matroids and knapsacks, Feldman et al. (2022) give +a logarithmic approximate deferred-acceptance auction for any downward-closed feasibility con- +straint. The selection of any pen testing algorithm can be padded by adding (possibly expended) +pens to always output a maximal feasible set. This inherent downward-closure of the pen testing +problem allows the auction of Feldman et al. (2022) to be extended to all combinatorial constraints. +Online posted-price mechanisms are a special case of deferred-acceptance auctions. Online envi- +ronments are ones where an irrevocable decision regarding choosing a pen needs to be taken before +testing the next one. We consider the settings where the order of testing pens is adversarially cho- +sen (the oblivious version, cf. prophet inequalities, Chawla et al., 2010) and where the algorithm +can choose the order (the sequential version, cf. correlation gap, Yan, 2011). With our analysis +of the optimal omniscient approximation ζ and known bounds on the standard approximation γ +of online pricing mechanisms, the reduction framework yields better performance guarantees than +the bounds by Qiao and Valiant (2023) for these environments. For the special case where all ink +levels are drawn IID, the bound on the omniscient approximation from the reduction framework is +loose; instead, we give a pen testing algorithm that is within an additive 1 of the lower bound of +Qiao and Valiant (2023). +These results are summarized in Table 1. +2 +The Pen Testing Problem and Ascending-Price Auctions +In this section, we formally define the combinatorial pen testing problem, generalized from Qiao and Valiant +(2023), and explore its connections to ascending-price mechanisms. +Definition 1. A pen testing instance is described by a set N = {1, 2, . . . , n} of pens with unknown +ink levels v1, . . . , vn, each vi drawn independently from the distribution Fi, and a subset P of the +powerset of the pens denoting feasible subsets of the pens. The residual ink level ui of pen i is +initiated to vi. We can test the pens before making a decision. Test (i, θi) is done by writing with +pen i for time θi. We receive a binary signal at the end of the test: +1. If ui ≥ θi: The test succeeds. The pen now has an (unobservable) ink level ui ←− ui − θi. +2. If ui < θi: The test fails. The pen now has no ink left, i.e, ui ←− 0. +We can choose to run multiple tests on the pens. We need to output some feasible subset P ∈ P of +pens, maximizing +� +i∈P ui. +The goal is to optimize the total amount of ink remaining in the set of chosen pens. We compare +the performance of the pen testing algorithm against the following benchmarks: +3 + +Environment +ζ(n) +(1+o(1)) +γ(n) +π(n) +(1+o(1)) +Select k +2 +ln 2 · ln n +k +* +ln n +1** +min{ 2 +ln 2 ln n +k, ln n} +Matroids +ln n +1** +ln n +Knapsack +ln n +2 +2 · ln n +General Downward-Closed +ln n +O(log n)† +O(log2 n) +Select 1 Online Oblivious +ln n +2†† +O(log n) ‡ +2 · ln n +Select 1 Online Sequential +ln n +e +e−1 +‡‡ +O(log n) ‡ +e +e−1 · ln n +Select 1 Online IID +ln n +e +e−1 +‡‡ +e ln n ‡ +ln n +Matroid Online Oblivious +ln n +2§ +2 · ln n +Matroid Online Sequential +ln n +e +e−1 +‡‡ +e +e−1 · ln n +Table 1: Selected upper bounds. The bounds for optimal omniscient approximation ζ and standard +approximation π are normalized by a (1 + o(1)) factor. Prior work: * Hartline and Roughgarden, +2008; +** +Milgrom and Segal, +2014, +Bikhchandani et al., 2011; +† +Feldman et al., +2022; +†† Samuel-Cahn, 1984, Chawla et al., 2010; § Kleinberg and Weinberg, 2012; ‡ Qiao and Valiant, +2023; ‡‡ Chawla et al., 2010, Yan, 2011. Note that Hartline and Roughgarden (2008) give a better +bound for Select k when k = ω(log n). All bounds in the table also hold for the underlying auction +environment. Pen testing algorithms with the same guarantees as downward-closed constraints can +be achieved for arbitrary combinatorial constraints. +• The standard benchmark is the expected performance of the optimal pen testing algorithm. +The standard approximation of a pen testing algorithm is the ratio of this benchmark to its +performance, denoted γ(n). +• The omniscient benchmark is the expected performance of the optimal algorithm that knows +the quantity of ink in each pen. The omniscient approximation of a pen testing algorithm is +the ratio of this benchmark to its performance, denoted π(n). +Compare a pen testing instance with n pens and a feasibility constraint P with an n-agent auction +under the same feasibility constraint. The original levels of ink correspond to the value each agent +gets upon being allocated. The ink spent through testing is analogous to the prices in the auction. +Thus, maximizing total residual ink is equivalent to optimizing consumer surplus, i.e, the sum of +the values of the winning agents minus the sum of all payments. The two benchmarks correspond +to the optimal consumer surplus and the optimal surplus, i.e, the total value of the winning agents. +Note that auctions generally give the auctioneer the added advantage of the ability to solicit bids +from agents. In pen testing algorithms, we only get to learn whether the ink left in the pen is more +than some threshold and in doing so, we irrevocably expend ink up to the threshold. This is similar +to ascending-price auctions, where the auctioneer irreversibly increases the price faced by each +agent, and in doing so, only learns whether the agent has a value at least the price. +Milgrom and Segal (2014) describe the wide class of ascending-price auctions called deferred- +acceptance mechanisms. +4 + +Definition 2 (Milgrom and Segal, 2014; Deferred-Acceptance Auctions). A deferred-acceptance +auction is held across multiple stages. For each stage t, the auction maintains a set of active +bidders At satisfying A1 ⊇ A2 ⊇ · · · ⊇ At, where the initial active set A1 is the set of all bidders. +The auction is characterized by a (possibly randomized) pricing rule ⃗p, mapping the history of the +auction at stage t to (discriminatory) prices for each agent satisfying pi(Ht) ≥ pi(Ht−1) for all +agents i and for all histories Ht at stage t arising out of stage t − 1 history Ht−1, i.e, the prices +are monotonously non-decreasing for all agents over time. Agents can opt to drop out once the +prices are updated in stage t. At+1 is the set of agents in At that did not drop out in stage t. +The mechanism terminates at stage t when At becomes feasible. The agents in At are charged +according to ⃗p(Ht). +The definition from Milgrom and Segal (2014) restricts deferred-acceptance mechanisms to be +deterministic, which we relax for the purpose of designing pen testing algorithms. From the dis- +cussion above, note that any deferred-acceptance mechanism can be converted into a pen testing +algorithm, by setting thresholds equal to the prices recommended by the mechanism. The con- +sumer surplus of the mechanism corresponds to the residual ink in the pens chosen by the pen +testing algorithm. +Now that we have established the consumer surplus is a quantity of interest for pen testing, we +review consumer surplus optimization through virtual values in the next section. +2.1 +Consumer Surplus Optimization and Virtual Valuations +Let us begin in a single-agent environment with one good. Let the agent’s value be drawn from +the distribution F. The quantile q of an agent with value v ∼ F is the measure with respect to F +of stronger values, i.e, q = 1 − F(v). For q ∈ [0, 1], let v(q) be the inverse demand function of +F satisfying F(v(q)) = 1 − q. In other words, Prˆv∼F(ˆv > v(q)) = q. Throughout the paper, we +assume v(1) = 0 and +� 0 +0 v(t)dt = 0. It can be shown that these assumptions can be made without +loss of generality. +Let V (q) = +� q +0 v(t)dt be the price-posting surplus curve. Notice that V (q) is the expected surplus +from posting a price v(q) to the agent, thereby allocating the good with probability q. By our +assumption on F, V (0) = 0. Let U(q) = V (q) − qv(q) be the price-posting consumer surplus +curve. Similar to the price-posting surplus curve, U(q) is the consumer-surplus from posting a +price v(q). Note that 0 ≤ U(0) ≤ V (0) = 0, and thus, U(0) = 0. +Let u(q) = U′(q) = −qv′(q) be the marginal price-posting consumer surplus curve. Note that v is +monotonously non-increasing, and hence its derivative v′ is non-positive. Thus, u(q) ≥ 0 and U +is monotone non-decreasing. A quick note on derivatives: whenever v′(q) is well defined, u(q) is +well defined. At other points, u(q) can be calculated using any sub-derivative of v. At points q of +discontinuity of v, v′(q) = −∞ and thus u(q) = ∞. +Theorem 2 (Myerson, 1981). In a Bayesian incentive-compatible mechanism with allocation rule +5 + +y and payment rule p (over quantiles), the expected consumer surplus of an agent satisfies +Eq∼U[0,1][v(q)y(q) − p(q)] = Eq∼U[0,1][u(q)y(q)] = Eq∼U[0,1][−U(q) y′(q)] + U(0)y(0) +Thus, the expected consumer surplus equals the expected virtual surplus with marginal consumer +surplus as virtual values. Myerson (1981) states that an allocation rule can be implemented as a +truthful auction if and only if the allocation rule is monotone non-increasing in quantile space. +Thus, the consumer surplus optimal mechanism optimizes for virtual surplus subject to the alloca- +tion rule being monotonously non-increasing. +When the marginal price-posting consumer surplus curve is non-increasing (and hence U is con- +cave), optimizing for virtual surplus automatically ensures monotonicity of the allocation rule. We +will call these distributions with non-increasing marginal price-posting consumer surplus curves +consumer surplus regular. +However, u is not monotone non-increasing for common distributions like the uniform and normal +distribution. In fact, u is monotone non-decreasing for these distributions. In such a case, the +allocation that pointwise optimizes for virtual surplus might not satisfy monotonicity. Myerson +(1981) prescribes an ironing procedure to optimize for virtual surplus subject to monotonicity. +1. Construct the concave hull U of the price-posting consumer surplus curve U +2. Define u(q) = U +′(q) +3. Find the virtual surplus optimal mechanism using u as virtual surplus +Since U is concave, u is non-increasing, and hence optimizing for the ironed virtual surplus is +easier. Throughout the paper, we will follow the convention of attaching a bar on top of a curve to +describe the ironed curve. +In a multi-agent environment, the interim allocation rule for an agent is the single-agent allocation +rule that arises in expectation over the quantiles of all other agents. This captures the perspective +of the agent after knowing its value, but before learning the values of the other agents. +Theorem 3. Let y be the interim allocation rule for some agent, satisfying +d +dqy(q) = 0 for all q +such that U(q) ̸= U(q). Then +Eq[u(q)y(q)] = Eq[u(q)y(q)] +In other words, if the allocation rule does not differentiate between agents in an ironed region, +then the ironed virtual surplus can be used to compute the consumer surplus instead of the actual +virtual surplus. +Theorem 4 (Alaei et al., 2012, 2019). Let Y be some allocation rule with interim allocation rule +yi monotonously non-increasing for each agent i. Then, there exists a mechanism Y with interim +6 + +allocation rule yi for agent i such that the expected consumer surplus for agent i equals +Eq∼U[0,1][ui(q) yi(q)] = Eq∼U[0,1][ui(q) yi(q)] +In other words, the expected consumer surplus of Y is the consumer surplus of the original mech- +anism if the virtual values were given by ui for agent i instead of ui +2.2 +Near-Optimal Deferred-Acceptance Mechanisms for Consumer Surplus +In this section, we prescribe a recipe similar to Feng et al. (2023) to convert any approximately op- +timal deferred-acceptance mechanism for surplus into an approximately optimal deferred-acceptance +mechanism for consumer surplus. +Definition 3 (Virtual-Pricing Transformation of a Deferred-Acceptance Mechanism). Let DAV +be a deferred-acceptance mechanism designed to optimize surplus. Let vi be the inverse demand +function and ui be the virtual value function (for consumer surplus) for agent i. The virtual- +pricing transformation on DAV , denoted by DAU, implements DAV in ironed virtual price space, +i.e, whenever DAV posts a price ˆvi to agent i, DAU posts a price vi(θi) satisfying +θi = sup{θ : ui(θ) ≥ ˆvi} +We will make some preliminary observations about the transformation: +1. Consider two runs of the transformed mechanism DAU with some agent having two different +values, both corresponding to the same ironed virtual value (which can happen if both these +values are within the same ironed interval). From the definition of the transform, the agent +faces the smallest price needed to differentiate itself from virtual values smaller than the +threshold set by the mechanism. Consequently, the mechanism does not post prices from the +middle of an ironed interval. Thus, in both these runs of the mechanism, the agent behaves +identically. Since the mechanism does not discriminate between values with the same ironed +virtual value, the consumer surplus of the mechanism equals the ironed virtual surplus. +2. If the original mechanism DAV is a γ-approximation to the optimal surplus, in expectation +over all product distributions, then the transformed mechanism DAU is a γ-approximation +to the optimal ironed virtual surplus, and hence, to the optimal consumer surplus. +3. Finally, it is straightforward to see that the transformed mechanism DAU posts non-decreasing +prices to each agent. Hence, DAU is also a deferred-acceptance mechanism. +Recall that the optimal omniscient approximation ζ(n) is the ratio between the optimal surplus and +the optimal consumer surplus. The proof of Theorem 1 is now straightforward. +Proof of Theorem 1. Let OPTU denote the expected optimal consumer surplus, OPTV denote the +expected optimal surplus, and DAU denote the expected consumer surplus of the deferred accep- +7 + +tance mechanism of Definition 3. Then, +DAU ≥ γ(n) × OPTU ≥ γ(n) ζ(n) × OPTV +where the inequalities follow from the definition of standard approximation γ(n) and optimal om- +niscient approximation ζ(n), respectively. The equivalent pen testing algorithm yields the neces- +sary approximation ratios against the two benchmarks. +In the next section, we analyze the optimal omniscient approximation ζ(n) for general combinato- +rial environments. +3 +Consumer Surplus versus Surplus +Our approximation procedure follows two steps. In Section 3.1, we approximate the consumer +surplus to the optimal surplus in a single-agent environment, where the ex-ante probability of +allocation is constrained to be at most q. We get a 1 − ln q approximation in this setting. We then +move from the single-agent environment to multi-agent environments using an approach similar +to Feng et al. (2023). However, the single-agent approximation diverges to ∞ when q −→ 0, and +does not help in approximating the consumer surplus against the optimal surplus for these small +quantiles. Section 3.2 applies an approach of Hartline and Taggart (2019), that shows it is sufficient +to have a good approximation for consumer surplus only in the larger quantiles (q = ω( 1 +n)), to get +around this challenge and to give a (1 + o(1)) ln n approximation for n-agent environments with +general feasibility constraints. +3.1 +The Single-Agent Problem +Recall that V, U and U are the price-posting surplus curve, price-posting consumer surplus curve +and the ironed consumer surplus curve respectively. In this section, we establish a “closeness +property” like in Feng et al. (2023). +Theorem 5. For an ex-ante allocation probability q ∈ [0, 1], the ratio of the optimal surplus V (q) +to the optimal consumer surplus U(q) is at most 1 − ln q. +We will begin by proving Theorem 5 for consumer surplus regular distributions, and then extend +the result to all distributions. +Lemma 1. For any consumer surplus regular distribution and an ex-ante allocation probability +q ∈ [0, 1], the ratio of the optimal surplus V (q) to the optimal consumer surplus U(q) is at most +1 − ln q. +Proof. Conditional on generating a surplus V (q), we want to compute the distribution that mini- +8 + +mizes consumer surplus U(q). +U(q) = +� q +0 +u(t)dt. +V (q) = U(q) + qv(q) += +� q +0 +u(t)dt − q +� 1 +q +v′(t)dt += +� q +0 +u(t)dt + q +� 1 +q +u(t) +t dt. +(1) +We substitute u(t) = −tv′(t) in the last equation. +Simultaneously, we enforce u(t) ≥ 0, and u(t) is non-increasing (since the distribution is consumer +surplus regular), v(1) = 0 and V (0) = 0. +Finding the minimum consumer surplus distribtuion reduces to the following program. +min +� q +0 +u(t)dt +subject to +� q +0 +u(t)dt + q +� 1 +q +u(t) +t dt = V (q) +u(t) ≥ 0, u(t) is monotonously non-increasing +v(1) = 0, V (0) = 0 +The constraints in the last line can be enforced after solving for u. +Monotonicity dictates u(t) ≥ u(q) for t ≤ q. Hence, minimizing +� q +0 u(t)dt would correspond to +setting u(t) = u(q) for t ≤ q. Similarly, u(t) ≤ u(q) for t ≥ q. Minimizing +� q +0 u(t)dt would +mean minimizing V (q) − q +� 1 +q +u(t) +t dt which is achieved by setting u(t) = u(q) for t ≥ q. Let +u(0) = u(t) = u(1) = u. For a constant marginal price-posting consumer surplus curve, +U(q) = +� q +0 +u(t)dt = q u +V (q) = +� q +0 +u(t)dt + q +� 1 +q +u(t) +t dt = [q − q ln q] u +Thus, for any consumer surplus regular distribution, V (q) +U(q) ≤ [1 − ln q]. The constant marginal +consumer surplus curve that achieves this ratio corresponds to the exponential distribution. +Lemma 2. Let the consumer surplus curves U, ˆU satisfy ˆU(0) = 0 and ˆU(t) ≥ U(t) for all +t ∈ [0, 1]. Then, for the corresponding surplus curves, ˆV (q) ≥ V (q). +9 + +Proof. We rewrite equation (1) as follow. +V (q) = +� q +0 +u(t)dt + q +� 1 +q +u(t) +t dt += U(q) + q +�U(t) +t +�t=1 +t=q + q +� 1 +q +U(t) +t2 dt += qU(1) + q +� 1 +q +U(t) +t2 dt +(2) +The second equality is obtained through integrating by parts. Increasing U pointwise clearly results +in an increase in V . +Proof of Theorem 5. The optimal consumer surplus for an ex-ante allocation probability q equals +U(q). +Consider the distribution with price-posting consumer surplus curve U. Let V be the price-posting +surplus curve corresponding to U. From Lemma 2 and Lemma 1, +V (q) +U(q) ≤ V (q) +U(q) ≤ 1 − ln q +The second inequality holds from Lemma 1, since V is the price-posting surplus curve of a con- +sumer surplus regular distribution (we made U concave through ironing). +See Appendix A for a brief discussion on the surplus generated by the single-agent optimal con- +sumer surplus auction. +3.2 +Multi-Agent Environments +Consider the interim allocation rule yi for agent i in the surplus optimal mechanism OPTV . With- +out loss of generality, assume yi(1) = 0. The expected surplus for agent i equals +Eq∈U[0,1][yi(q) vi(q)] = +� +yi(q) Vi(q) +�q=1 +q=0 + Eq∈U[0,1][−y′ +i(q) V (q)] = Eq∈U[0,1][−y′ +i(q) V (q)] +This equality comes from integrating by parts (yi(q) × Vi(q) vanishes at both, q = 0 and 1, since +Vi(0) = 0 and yi(1) = 0). Suppose there exists a constant α such that Vi(q) ≤ α U i(q) for all +q ∈ [0, 1], then, +Eq∈U[0,1][−y′ +i(q) V (q)] ≤ Eq∈U[0,1][−y′ +i(q) α U(q)] = α × Eq∈U[0,1][yi(q) ui(q)] +Note that the left hand side of the above equation cannot be related to the consumer surplus yet, +since d +dqyi need not be equal to 0 whenever U(q) ̸= U(q). However, Theorem 4 shows the existence +of a mechanism with expected consumer surplus of agent i equal to Eq∈U[0,1][yi(q) ui(q)]. Thus, +10 + +the expected consumer surplus of this mechanism is an α-approximation to the optimal surplus. +The consumer surplus optimal mechanism OPTU will only do better. +Unfortunately, as q −→ 0, the bound from Theorem 5 of Vi(q) +Ui(q) ≤ (1 − ln q) diverges to ∞. Thus, +we need to show that the loss from ignoring the small (strong) quantiles is not much. Suppose we +find a mechanism M1 with interim allocation ˆyi for agent i, near optimal surplus, and y′ +i(q) = 0 +for q ∈ [0, ǫ]. Then, y′ +i(q) × V (q) = y′ +i(q) × U(q) = 0 in this region. For q ∈ [ǫ, 1], we know +V (q) ≤ (1 − ln ǫ) × U(q). Thus, +surplus(M1) ≤ (1 − ln ǫ) × OPTU +Hartline and Taggart (2019) give the construction of such a mechanism M1. +Definition 4 (Hartline and Taggart, 2019; ǫ-buffering rule). Given an allocation rule ⃗y = (y1, y2, . . . , yn) +and a quantile ǫ ∈ [0, 1], the ǫ-buffering rule for ⃗y runs y with quantiles transformed on each agent +as follows: +• Top inflate: for any qi ∈ [0, ǫ], return 0 +• For any qi ∈ [ǫ, 1 − ǫ], return qi−ǫ +1−2ǫ +• Bottom deflate: for any qi ∈ [1 − ǫ, 1], return 1 +In essence, the top inflate branch makes the ǫ-buffering rule treat all agents with a small quantile +as if they had quantile 0 and thus, y′ +i(q) = 0 for q ∈ [0, ǫ] (the bottom deflate branch is unnecessary +for the analysis of monotone payoff curves like surplus V and consumer surplus U; see Remark 1). +Theorem 6 (Hartline and Taggart, 2019). Let M be the ǫ-buffering rule of the surplus optimal +mechanism ⃗y. Then, surplus(⃗y) ≤ +1 +(1− +ǫ +1−ǫ ) (1−ǫ) (1−2nǫ) × surplus(⃗y). +Theorem 7. Consider an n-agent environment with an arbitrary feasibility constraint. The optimal +omniscient approximation is at most +1−ln ǫ +(1− +ǫ +1−ǫ ) (1−ǫ) (1−2nǫ). +Proof. The proof follows by combining Theorem 6 with the discussion above. Let M1 be the +ǫ-buffering rule of the surplus optimal mechanism OPTV . Then, +OPTV ≤ +1 +(1 − +ǫ +1−ǫ) (1 − ǫ) (1 − 2nǫ) × surplus(M1) ≤ +1 − ln ǫ +(1 − +ǫ +1−ǫ) (1 − ǫ) (1 − 2nǫ) × OPTU +Corollary 1. In n-agent environments with an arbitrary feasibility constraint, the optimal omni- +scient approximation ζ(n) ≤ (1 + o(1)) ln n. +11 + +Proof. Setting ǫ = +1 +n ln n in Theorem 7, we get +ζ(n) ≤ +1 − ln +1 +n ln n +(1 − +1 +n ln n +1− +1 +n ln n ) (1 − +1 +n ln n) (1 − 2n · +1 +n ln n) += +1 + ln n + ln ln n +(1 − +1 +n ln n−1) (1 − +1 +n ln n) (1 − +2 +ln n) += n ln n − 1 +n ln n − 2 · +n ln n +n ln n − 1 · ln n − 2 +ln n +· 1 + ln n + ln ln n +ln n +× ln n += (1 + o(1)) ln n +Remark 1. The ǫ-buffering rule of Hartline and Taggart (2019) can be implemented without a bot- +tom deflate branch. This would strengthen the approximation guarantee to +1 +(1−ǫ) (1−nǫ). However, +substituting ǫ = +1 +n ln n will not result in a bound tighter than (1 + o(1)) ln n. +Remark 2. For the special case of k-unit auctions, Hartline and Roughgarden (2008) show ζ(n) ≤ +2 +ln 2(ln n +k +ln 2) = +2 +ln 2(1+o(1)− ln k +ln n) ln n. Our bound performs better whenever ln k +ln n = o(1), such +as k = 1. +Remark 3. The bound from Corollary 1 matches the lower bound of Qiao and Valiant (2023). In +the single-item environment with values of agents drawn IID from the exponential distribution, they +show that the optimal surplus is Hn = (1 + o(1)) ln n times the optimal consumer surplus. +4 +Pen Testing Corollaries from Deferred-Acceptance Mecha- +nisms +There are many environments in which deferred-acceptance mechanisms are known to be good. +By Theorem 1, these imply good pen testing algorithms (up to an additional ζ(n) factor in ap- +proximation to the omniscient benchmark). In this section, we discuss a few notable examples. A +summary of these results was given previously in Table 1. +Deferred-acceptance mechanisms that achieve the ex-post surplus optimal outcome are known +for various feasibility constraints. While the simple auction that uniformly increases the price +until exactly k bidders remain active is surplus optimal in the k-identical goods environment, +Milgrom and Segal (2014) and Bikhchandani et al. (2011) generalize the mechanism for matroid +feasibility constraints. We get the following corollary from the above auctions. +Corollary 2. For a pen testing environment with a matroid feasibility constraint, there exists a pen +testing algorithm with an omniscient approximation ratio (1 + o(1)) ln n. +D¨utting et al. (2014) initiated the study of prior-free deferred-acceptance approximation mecha- +nisms in environments where deferred-acceptance mechanisms are known to be suboptimal. Their +12 + +bounds can be improved in settings like ours where prior distributions of the agents’ values are +known. For general downward-closed constraints with a prior distribution on values, Feldman et al. +(2022) give a O(log log m) approximation to the optimal surplus, where m is the number of maxi- +mal feasible sets. Note that m ≤ 2n (every subset of agents might be feasible) and hence, log log m +is at most log n. Thus, we have a poly-logarithmic approximate pen testing algorithm for any +downward-closed feasibility environment. +Note, however, that the pen testing model is inherently downward-closed. Specifically, suppose +an algorithm wanted to select a set P ⊆ P, but P is not feasible while P is. Since all pens have +non-negative residual ink, even the ones that failed their tests, there is no loss in selecting P instead +of the set P. Thus, near optimal pen testing algorithms for downward-closed constraints can be +extended to give near optimal algorithms for general combinatorial constraints, giving the same +performance guarantee as the downward-closed environment. +Corollary 3. For any combinatorial pen testing environment, there exists a pen testing algorithm +with an omniscient approximation ratio O(log2 n). +Remark 4. Note that this reduction cannot be used to design deferred-acceptance mechanisms +for general feasibility constraints. Allocating to agents that have dropped out of the auction can +make the mechanism non-truthful (i.e, staying active till the price reaches the agent’s value and +dropping out subsequently might not be in the agent’s best interests). +The bound for downward-closed and general combinatorial environments can be improved upon +in special cases; for example, in Appendix B we give a natural deferred-acceptance mechanism for +knapsack constraints that is a 2-approximation. For this knapsack problem, the agents have sizes +along with values and feasible subsets are precisely those with a total size at most the knapsack +capacity. +Corollary 4. For a pen testing environment with a knapsack feasibility constraint, there exists a +pen testing algorithm with an omniscient approximation ratio 2 (1 + o(1)) lnn. +Online pricing mechanisms are a special case of deferred-acceptance mechanisms; thus, Theorem 1 +converts online pricing mechanisms with good surplus into good online pen testing algorithms. For +online pen testing problems, every pen can be tested exactly once and must either be selected or +discarded immediately after testing and before testing another pen. The algorithm might have +the ability to determine the order of testing pens (the sequential version) or the order might be +adversarially determined (the oblivious version). +For the sequential pricing problem, Chawla et al. (2010) gave an +e +e−1-approximation mechanism +for the single-item environment. Yan (2011) connected the sequential pricing problem to the cor- +relation gap and generalized the +e +e−1-approximation to matroid environments. Both of these papers +considered optimizing revenue, but the results can be easily adapted to optimize surplus. +Corollary 5. In the online sequential pen testing problem with a matroid feasibility constraint, +there exists a pen testing algorithm that achieves an omniscient approximation ratio +e +e−1 · (1 + +13 + +o(1)) ln n. +Hajiaghayi et al. (2007) and Chawla et al. (2010) adapt the prophet inequality of Samuel-Cahn +(1984) to give a non-adaptive 2-approximation for the oblivious price posting problem with 1 +good. Kleinberg and Weinberg (2012) extend this 2-approximation to matroid environments and +any arrival order of agents but with adaptive prices. +Corollary 6. In the online oblivious pen testing problem to select a single pen, there exists a non- +adaptive pen testing algorithm that achieves an omniscient approximation ratio 2(1 + o(1)) ln n. +Corollary 7. In the online oblivious pen testing problem with a matroid feasibility constraint, there +exists a pen testing algorithm that achieves an omniscient approximation ratio 2(1 + o(1)) ln n. +The bounds of Corollary 5 and Corollary 6 improve on the online pen testing bounds of Qiao and Valiant +(2023). +5 +The Online IID Environment +Consider the special case of the online pen testing problem where the ink levels are drawn inde- +pendently from the same distribution. We move away from the reduction framework of Theorem 1 +to obtain a better bound for this environment. +Theorem 8. In online single-item environments with IID agents, there exists a price-posting strat- +egy with an omniscient approximation at most (1 + o(1)) ln n. Equivalently, in the online pen +testing problem with IID ink levels, there exists an algorithm that achieves π(n) ≤ (1 + o(1)) ln n. +First, note that the above theorem is tight. In Remark 3, we saw an instance (originally from +Qiao and Valiant, 2023) with n IID agents whose values are drawn from the exponential distribu- +tion with consumer surplus only an Hn approximation to the optimal surplus. The optimal online +consumer surplus cannot do better. Hence, the (1 + o(1)) ln n approximation ratio in Theorem 8 is +tight. +Before proving Theorem 8, we briefly discuss the guarantee obtained through the reduction frame- +work of Theorem 1. For the online environment with IID agents, it is known that γ(n) = +e +e−1 +(Chawla et al., 2010). Thus, the framework would give us an +e +e−1 · (1 + o(1)) ln n bound. The +direct analysis below gives a tight Hn + 1 = (1 + o(1)) ln n bound up to an additive factor 1. +Proof of Theorem 8. Similar to the proof of Theorem 5, we will phrase our program as an opti- +mization problem and compute the optimal solution. +The surplus optimal auction always allocates the good to the agent with the highest value (smallest +quantile). Let vmax be the random variable denoting the value of the winner. +Pr(vmax ≤ v(t)) = (1 − t)n +14 + +Thus, the expected surplus of the surplus optimal mechanism equals +� 1 +0 +v(t) × n(1 − t)n−1dt = +� 1 +0 +� 1 +t +u(r) +r dr × n(1 − t)n−1dt += +� 1 +0 +u(t) +t dt − +� 1 +0 +u(t) +t (1 − t)ndt += +� 1 +0 +1 − (1 − t)n +t +u(t)dt += U(1) + +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t2 +U(t)dt +The first equality follows by integrating from u(t) = −tv′(t) and v(1) = 0. We integrate by parts +for the second and fourth equation. +Next, we look at the consumer surplus through price-posting. For a price v(q), we allocate the good +if at least one agent has a quantile in [0, q]. Conditioned on selling the good, we get an expected +consumer surplus of U(q) +q . Thus, the expected consumer surplus equals +1 − (1 − q)n +q +× U(q) +As before, we first show Theorem 8 for consumer surplus regular distributions. Without loss +of generality, assume that the maximum achievable consumer surplus through anonymous price +posting is at most 1. We want to find the consumer surplus regular distribution with the largest +surplus. +max U(1) + +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t2 +U(t)dt +subject to +1 − (1 − q)n +q +× U(q) ≤ 1 for all q ∈ [0, 1] +U is concave +First, observe that C(q) = +q +1−(1−q)n is convex (see Lemma 4 in Appendix C). We want to fit a +concave curve U under the convex curve C. Let Cq be the tangent to C at q. C ˆq is a feasible +solution for U for all ˆq ∈ [0, 1]. Further, given that U touches the curve C at ˆq, U(t) ≤ C ˆq(t) +for all t ∈ [0, 1] (from the concavity of U). Also, it is clear that the objective increases with a +pointwise increase in U. Thus, we conclude that the optimal solution to the program is achieved at +U = C ˆq at some ˆq ∈ [0, 1]. In particular, the worst-case ratio occurs when U is a straight-line. +Without loss of generality, we can normalize the slope to 1. Let U(q) = q + a. The surplus equals +1 + a + +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t +dt + a × +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t2 +dt = Hn + an +15 + +See Lemma 5 in Appendix C for the proof of the above equality. +By posting a price v(q), we generate a consumer surplus +1 − (1 − q)n +q +× U(q) = (1 + (1 − q) + · · · + (1 − q)n−1) × (q + a) +If a ≥ +1 +n, the ratio between the surplus and the consumer surplus is at most Hn + 1 by setting +q = 0. If a < 1 +n, the surplus is less than Hn + 1, and by setting q = 1, the consumer surplus is +at least 1. Thus, the worst-case ratio between surplus and consumer surplus is at most Hn + 1 = +(1 + o(1)) ln n. +We now show the result for all distributions. We rewrite the optimal surplus as follows. +� 1 +0 +v(t) × n(1 − t)n−1 = +� +V (t) × n(1 − t)n−1�t=1 +t=0+ +� 1 +0 +V (t) × n(n − 1)(1 − t)n−2dt +The equality follows through integration by parts. From the above expression, it can be concluded +that the optimal surplus increases with a pointwise increase in V . +Let V be the price-posting surplus curve of the distribution with a price-posting consumer surplus +curve U. U is pointwise larger than U, and hence, from Lemma 2, V is pointwise larger than V . +Hence, the optimal surplus is larger in the distribution with price-posting consumer surplus curve +U. However, from Theorem 3, the consumer surplus from posting prices is identical to both the +distributions. Thus, the consumer surplus irregular distribution has a better ratio between surplus +and consumer surplus than the consumer surplus regular distribution. Thus, the ratio between +optimal surplus to optimal consumer surplus is at most (1 + o(1)) ln n for all distributions. +6 +Conclusion +In this section, we discuss the various advantages and disadvantages of using the reduction frame- +work of Theorem 1. Connecting the pen testing problem to auction theory and building an easy-to- +use reduction framework enable us to design near optimal pen testing algorithms for any general +feasibility constraint. We are able to construct algorithms that are only a constant factor away +from the lower bound described in Qiao and Valiant (2023) for various feasibility constraints like +matroids (γ(n) = 1) and knapsack (γ(n) = 2). However, there is a scope for improvement on the +following two fronts: +1. obtaining tight approximation guarantees, and +2. obtaining approximation guarantees under other models of access to the prior distributions. +Tight Approximation Guarantees: The approximation ratio of any pen testing algorithm is at +least the optimal omniscient approximation ζ(n). This follows since the optimal consumer sur- +plus in the underlying auction environment is an upper bound on the performance of the optimal +16 + +pen testing algorithm. The framework matches this bound up to a multiplicative factor of the ap- +proximation γ(n) of deferred-acceptance mechanisms. Reducing this gap for various feasibility +environments stands as an open problem. +For instance, consider the online environments discussed in the paper, where ζ(n) = (1+o(1)) lnn. +Our framework yielded an omniscient approximation 2(1 + o(1)) ln n in the oblivious version +(γ(n) = 2) and +e +e−1 · (1 + o(1)) ln n in the sequential version (γ(n) = +e +e−1). We conjecture that the +approximation guarantees obtained by applying the reduction framework are not tight, and can be +strengthened to (1 + o(1)) ln n in both these environments, like in the IID setting (Section 5). +Other Models of Access to Distributions: Our framework crucially makes use of virtual val- +ues, which in turn need complete knowledge of the distribution of ink levels. Thus, tailoring this +approach to models where the algorithm gets partial access to the priors is challenging. +Qiao and Valiant (2023) study two such models. They study the online oblivious problem where +the algorithm gets access to just one sample from the distribution of each pen. +They give a +O(log n)-approximation to the omniscient benchmark in this setting. They also look at the online +secretary setting, where the pens arrive according to a permutation chosen uniformly at random. +The quantity of ink in the pens are adversarially determined. The algorithm is told the quantity +of ink in the pen with the largest ink level. They give a O(log n)-approximation in this environ- +ment. Further, when the algorithm is given access to the ink levels in all the n pens (with random +arrival order), they tighten their bound to O( +log n +log log n). It is an open question to identify reduction +frameworks for these different models of access to the distributions. +References +Alaei, S., Fu, H., Haghpanah, N., Hartline, J. D., and Malekian, A. (2012). Bayesian optimal +auctions via multi-to single-agent reduction. In 13th ACM Conference on Electronic Commerce, +EC’12, page 17. +Alaei, S., Fu, H., Haghpanah, N., Hartline, J. D., and Malekian, A. (2019). Efficient computation +of optimal auctions via reduced forms. Mathematics of Operations Research, 44(3):1058–1086. +Bikhchandani, S., De Vries, S., Schummer, J., and Vohra, R. V. (2011). An ascending vickrey +auction for selling bases of a matroid. Operations research, 59(2):400–413. +Chawla, S., Hartline, J. D., Malec, D. L., and Sivan, B. (2010). Multi-parameter mechanism design +and sequential posted pricing. In Proceedings of the forty-second ACM symposium on Theory +of computing, pages 311–320. +Dantzig, G. B. (1957). Discrete-variable extremum problems. Operations Research, 5(2):266–277. +D¨utting, P., Gkatzelis, V., and Roughgarden, T. (2014). The performance of deferred-acceptance +auctions. In Proceedings of the fifteenth ACM conference on Economics and computation, pages +187–204. +17 + +Feldman, M., Gkatzelis, V., Gravin, N., and Schoepflin, D. (2022). Bayesian and randomized clock +auctions. In Proceedings of the 23rd ACM Conference on Economics and Computation, pages +820–845. +Feng, Y., Hartline, J., and Li, Y. (2023). Simple mechanisms for agents with non-linear utilities. +(to appear in) Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms. +Hajiaghayi, M. T., Kleinberg, R., and Sandholm, T. (2007). Automated online mechanism design +and prophet inequalities. In AAAI, volume 7, pages 58–65. +Hartline, J. and Taggart, S. (2019). Sample complexity for non-truthful mechanisms. In Proceed- +ings of the 2019 ACM Conference on Economics and Computation, pages 399–416. +Hartline, J. D. and Roughgarden, T. (2008). Optimal mechanism design and money burning. In +Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 75–84. +Kleinberg, R. and Weinberg, S. M. (2012). Matroid prophet inequalities. In Proceedings of the +forty-fourth annual ACM symposium on Theory of computing, pages 123–136. +Milgrom, P. and Segal, I. (2014). Deferred-acceptance auctions and radio spectrum reallocation. +In Proceedings of the fifteenth ACM conference on Economics and computation, pages 185–186. +Myerson, R. B. (1981). Optimal auction design. Mathematics of operations research, 6(1):58–73. +Qiao, M. and Valiant, G. (2023). Online pen testing. 14th Innovations in Theoretical Computer +Science Conference, (ITCS) 2023, January 10-13, 2023, MIT, Cambridge, USA. +Samuel-Cahn, E. (1984). Comparison of threshold stop rules and maximum for independent non- +negative random variables. Ann. Probab., 12(4):1213–1216. +Yan, Q. (2011). Mechanism design via correlation gap. In Proceedings of the twenty-second +annual ACM-SIAM symposium on Discrete Algorithms, pages 710–719. SIAM. +A +A Discussion on the Surplus of the Single-Agent Optimal +Consumer Surplus Auction +In Section 3.1, we compare the optimal consumer surplus with an ex-ante allocation constraint q to +the optimal surplus. We now take a look at the surplus generated by the consumer surplus optimal +auction. +Corollary 8. For an ex-ante allocation probability q, the ratio of the optimal surplus to the surplus +from the consumer surplus maximizing auction is at most 1 − ln q. +The corollary follows since the surplus of the consumer surplus maximizing auction is only larger +than the consumer surplus. +The above corollary is tight. This follows immediately from the observations below: +18 + +1. For a distribution with a convex price-posting consumer surplus curve, the consumer surplus +optimal mechanism is obtained by giving away the good for free with probability q. For +such distributions, the consumer surplus of the optimal mechanism equals the surplus of the +mechanism. +2. The price-posting consumer surplus curve of the exponential distribution is a straight line. A +distribution obtained by adding a small convex distortion to this curve has the same ironed +price-posting consumer surplus curve as the exponential distribution and close to the same +price-posting surplus curve as the exponential distribution. +The ratio of surplus in the surplus optimal mechanism to the consumer surplus optimal mechanism +is close to 1 − ln q for distributions described in bullet 2. +B +Deferred-Acceptance Mechanism for Knapsack Constraints +Consider an environment with a knapsack feasibility constraint. Each agent i has a value vi drawn +independently from distribution Fi and a size si. The goal is to chose a set S of agents maximizing +surplus such the total size � +i∈S si of chosen agents is at most C. +We rely on the following folklore 2-approximation algorithm for the knapsack problem. +Algorithm 1. Run the alternative with a better expected surplus: +1. Bang-per-buck (Dantzig, 1957): Sort agents in decreasing order of vi +si. Pick agents in this +order until the knapsack is full. +2. Max: Pick the agent with the largest value. +Milgrom and Segal (2014) give deferred-acceptance implementations for both, bang-per-buck and +max. Thus, the above is a deferred-acceptance mechanism. +Lemma 3. Algorithm 1 is a 2-approximation to the optimal surplus. +Proof. Let OPTV be the random variable denoting the optimal surplus. Fix some realization of +values v1, v2, . . . , vn. Without loss of generality, assume v1 +s1 ≥ v2 +s2 ≥ · · · ≥ vn +sn. Let bang-per-buck +pick agents 1 through t for this realization of values. +Clearly, �t+1 +i=1 vi ≥ OPTV . This is because the optimal algorithm that can pack fractional agents +in the knapsack would greedily pick the first t agents and pick some fraction of agent t+1. Thus, if +vmax = max1≤i≤n vi, �t +i=1 vi+vmax ≥ OPTV. The surplus generated jointly by the two alternatives +pointwise dominates the optimal surplus and hence dominates the optimal surplus in expectation. +Evi∼Fi[surplus(bang-per-buck)] + Evi∼Fi[surplus(max)] ≥ Evi∼Fi[OPTV] +We get the larger of the two terms in the left hand side, which must be at least 1 +2Evi∼Fi[OPTV]. +19 + +C +Lemmas for Theorem 8 +Lemma 4. +q +1−(1−q)n is convex in q. +Proof. +q +1 − (1 − q)n = +1 +�n−1 +j=0(1 − q)j +d +dq +1 +�n−1 +j=0(1 − q)j = +�n−2 +j=0(j + 1)(1 − q)j +�n−1 +j=0(1 − q)j +· +1 +�n−1 +j=0(1 − q)j +The second term in the product is increasing in q. We will show the first term is also increasing in +q. The first term can be rewritten as +�n−2 +j=0(j + 1)(1 − q)j +�n−1 +j=0(1 − q)j += +�n−1 +j=0[(1 − q)j − (1 − q)n−1] +�n−1 +j=0(1 − q)j += 1 − n +(1 − q)n−1 +1 + (1 − q) + · · · + (1 − q)n−1 +Further rewriting this, it becomes apparent that this is indeed increasing in q. +�n−2 +j=0(j + 1)(1 − q)j +�n−1 +j=0(1 − q)j += 1 − n +1 +1 +(1−q)n−1 + · · · + +1 +(1−q)0 +. +Lemma 5. +1 + a + +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t +dt + a × +� 1 +0 +1 − (1 − t)n − nt(1 − t)n−1 +t2 +dt = Hn + an +Proof. The key is to rewrite 1−(1−t)n−nt(1−t)n−1 +t +. +1 − (1 − t)n − nt(1 − t)n−1 +t += +n−1 +� +j=0 +(1 − t)j − n(1 − t)n−1 += +n−1 +� +j=0 +[(1 − t)j − (1 − t)n−1] += t × +n−2 +� +j=0 +(j + 1) × (1 − t)j +Thus, the expression reduces to +1 + a + +n−2 +� +j=0 +� 1 +0 +(j + 1) × t(1 − t)jdt + a × +n−2 +� +j=0 +� 1 +0 +(j + 1) × (1 − t)jdt = Hn + an. +20 + diff --git a/odFMT4oBgHgl3EQf7DEn/content/tmp_files/load_file.txt b/odFMT4oBgHgl3EQf7DEn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3cf8912a4151a2e178995f68c05b3be908e83965 --- /dev/null +++ b/odFMT4oBgHgl3EQf7DEn/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf,len=592 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='12462v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='GT] 29 Jan 2023 Combinatorial Pen Testing (or Consumer Surplus of Deferred-Acceptance Auctions) Aadityan Ganesh Jason Hartline January 31, 2023 Abstract Pen testing is the problem of selecting high capacity resources when the only way to mea- sure the capacity of a resource expends its capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We have a set of n pens with unknown amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We are allowed to gather more information by writing with them, but this uses up ink that was previously in the pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Algorithms are evaluated against the standard benchmark, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, the optimal pen testing algorithm, and the omniscient benchmark, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, the optimal selection if the quantity of ink in the pens are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We identify optimal and near optimal pen testing algorithms by drawing analogues to auc- tion theoretic frameworks of deferred-acceptance auctions and virtual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Our framework allows the conversion of any near optimal deferred-acceptance mechanism into a pen testing algorithm with an additional overhead of at most (1+o(1)) ln n in the approximation factor of the omniscient benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We use this framework to give pen testing algorithms for various combinatorial constraints like matroid, knapsack and general downward-closed constraints and also for online environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1 Introduction Pen testing is the problem of selecting high capacity resources when the only way to measure the capacity of the resource expends its capacity (Qiao and Valiant, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We show that any ascending-price auction can be converted into an equivalent pen testing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We apply the auction theoretic framework of virtual values to identify optimal pen testing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This connection allows many existing results from auction theory to be applied to pen testing and gives optimal and near optimal pen testing algorithms in combinatorial and online environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The pen testing problem of Qiao and Valiant (2023) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We have a set of pens with varying amounts of remaining ink and we want to choose a pen with the largest amount of ink left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We have access to the distribution of ink levels in these pens, but we are only allowed to gather more information by writing with the pens to test them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' While writing gives information about 1 whether there is still ink left, it uses up ink that was previously in the pen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Pens that are expended due to testing can be discarded without incurring any penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We compare the performance of our algorithms against two benchmarks– the optimal pen testing algorithm (the standard benchmark) and the optimal algorithm that knows the amount of ink in each pen (the omniscient benchmark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The combinatorial pen testing problem generalizes the pen testing problem to selecting a feasible subset of pens, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', according to a matroid or knapsack constraint, to maximize the total remaining ink in the chosen subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Deferred-acceptance auctions (Milgrom and Segal, 2014) capture a wide range of ascending-price mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' These are mechanisms that greedily reject the least promising agent by increasing the price for getting allocated at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For instance, to auction off one good, the deferred- acceptance mechanism keeps increasing the price, rejecting agents whenever their value for the good falls below the price until exactly one agent remains in the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The surplus of the mechanism is the sum of the values of all the allocated agents and the consumer surplus is the surplus minus the payments made by the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We provide a black box approach to convert any surplus optimal (or near optimal) deferred-acceptance auction into an optimal (or near optimal) pen testing algorithm with equivalent performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In this construction, pens and their ink levels are analogous to agents and their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' While the standard benchmark is similar to comparing the underlying deferred-acceptance auction against the consumer surplus optimal auction, the omniscient benchmark compares it against the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' These problems are related by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Consider a combinatorial pen testing environment with n pens, the analogous auction environment, and a deferred acceptance mechanism DA for the auction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let γ(n) denote the standard approximation of DA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', the worst-case ratio of the expected optimal surplus to the expected surplus of DA for the auction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let ζ(n) denote the optimal omniscient approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', the worst-case ratio of the omniscient benchmark to the expected optimal consumer surplus for the auction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, there is a pen testing algorithm that is a γ(n)- approximation to the standard benchmark and π(n) = γ(n) ζ(n)-approximation to the omniscient benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 1 gives a reduction framework for designing near optimal pen testing algorithms under arbitrary feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This approach improves bounds from previous literature for all the environments that we discuss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' however, it does not always yield a tight bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the online IID environment (described below), we identify an algorithm with omniscient approximation π(n) that is a constant factor less than γ(n) ζ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The first step of the framework is to obtain bounds for the optimal omniscient approximation ζ(n) under various feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We show ζ(n) ≤ (1 + o(1)) ln n in any combinatorial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the special case of k-identical goods, Hartline and Roughgarden (2008) prove ζ(n) ≤ 2 ln 2(1 + o(1)) ln n k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' our analysis gives a better bound when ln k = o(1) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The second step of the framework combines our bounds on the optimal omniscient approximation ζ(n) with various γ(n)-approximate deferred-acceptance mechanisms from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Surplus 2 optimal deferred-acceptance auctions are known for k-identical goods and more general matroid feasibility constraints (Milgrom and Segal, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We give a 2-approximate deferred-acceptance mechanism for knapsack constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Beyond matroids and knapsacks, Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2022) give a logarithmic approximate deferred-acceptance auction for any downward-closed feasibility con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The selection of any pen testing algorithm can be padded by adding (possibly expended) pens to always output a maximal feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This inherent downward-closure of the pen testing problem allows the auction of Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2022) to be extended to all combinatorial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Online posted-price mechanisms are a special case of deferred-acceptance auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Online envi- ronments are ones where an irrevocable decision regarding choosing a pen needs to be taken before testing the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We consider the settings where the order of testing pens is adversarially cho- sen (the oblivious version, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' prophet inequalities, Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2010) and where the algorithm can choose the order (the sequential version, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' correlation gap, Yan, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' With our analysis of the optimal omniscient approximation ζ and known bounds on the standard approximation γ of online pricing mechanisms, the reduction framework yields better performance guarantees than the bounds by Qiao and Valiant (2023) for these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the special case where all ink levels are drawn IID, the bound on the omniscient approximation from the reduction framework is loose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' instead, we give a pen testing algorithm that is within an additive 1 of the lower bound of Qiao and Valiant (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' These results are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2 The Pen Testing Problem and Ascending-Price Auctions In this section, we formally define the combinatorial pen testing problem, generalized from Qiao and Valiant (2023), and explore its connections to ascending-price mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A pen testing instance is described by a set N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' , n} of pens with unknown ink levels v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' , vn, each vi drawn independently from the distribution Fi, and a subset P of the powerset of the pens denoting feasible subsets of the pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The residual ink level ui of pen i is initiated to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We can test the pens before making a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Test (i, θi) is done by writing with pen i for time θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We receive a binary signal at the end of the test: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' If ui ≥ θi: The test succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The pen now has an (unobservable) ink level ui ←− ui − θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' If ui < θi: The test fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The pen now has no ink left, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, ui ←− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We can choose to run multiple tests on the pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We need to output some feasible subset P ∈ P of pens, maximizing � i∈P ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The goal is to optimize the total amount of ink remaining in the set of chosen pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We compare the performance of the pen testing algorithm against the following benchmarks: 3 Environment ζ(n) (1+o(1)) γ(n) π(n) (1+o(1)) Select k 2 ln 2 · ln n k ln n 1** min{ 2 ln 2 ln n k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ln n} Matroids ln n 1** ln n Knapsack ln n 2 2 · ln n General Downward-Closed ln n O(log n)† O(log2 n) Select 1 Online Oblivious ln n 2†† O(log n) ‡ 2 · ln n Select 1 Online Sequential ln n e e−1 ‡‡ O(log n) ‡ e e−1 · ln n Select 1 Online IID ln n e e−1 ‡‡ e ln n ‡ ln n Matroid Online Oblivious ln n 2§ 2 · ln n Matroid Online Sequential ln n e e−1 ‡‡ e e−1 · ln n Table 1: Selected upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The bounds for optimal omniscient approximation ζ and standard approximation π are normalized by a (1 + o(1)) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Prior work: * Hartline and Roughgarden, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ** Milgrom and Segal, 2014, Bikhchandani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' † Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' †† Samuel-Cahn, 1984, Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' § Kleinberg and Weinberg, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ‡ Qiao and Valiant, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ‡‡ Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2010, Yan, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that Hartline and Roughgarden (2008) give a better bound for Select k when k = ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' All bounds in the table also hold for the underlying auction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Pen testing algorithms with the same guarantees as downward-closed constraints can be achieved for arbitrary combinatorial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The standard benchmark is the expected performance of the optimal pen testing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The standard approximation of a pen testing algorithm is the ratio of this benchmark to its performance, denoted γ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The omniscient benchmark is the expected performance of the optimal algorithm that knows the quantity of ink in each pen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The omniscient approximation of a pen testing algorithm is the ratio of this benchmark to its performance, denoted π(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Compare a pen testing instance with n pens and a feasibility constraint P with an n-agent auction under the same feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The original levels of ink correspond to the value each agent gets upon being allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The ink spent through testing is analogous to the prices in the auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, maximizing total residual ink is equivalent to optimizing consumer surplus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, the sum of the values of the winning agents minus the sum of all payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The two benchmarks correspond to the optimal consumer surplus and the optimal surplus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, the total value of the winning agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that auctions generally give the auctioneer the added advantage of the ability to solicit bids from agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In pen testing algorithms, we only get to learn whether the ink left in the pen is more than some threshold and in doing so, we irrevocably expend ink up to the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This is similar to ascending-price auctions, where the auctioneer irreversibly increases the price faced by each agent, and in doing so, only learns whether the agent has a value at least the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Milgrom and Segal (2014) describe the wide class of ascending-price auctions called deferred- acceptance mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 4 Definition 2 (Milgrom and Segal, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Deferred-Acceptance Auctions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A deferred-acceptance auction is held across multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For each stage t, the auction maintains a set of active bidders At satisfying A1 ⊇ A2 ⊇ · · · ⊇ At, where the initial active set A1 is the set of all bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The auction is characterized by a (possibly randomized) pricing rule ⃗p, mapping the history of the auction at stage t to (discriminatory) prices for each agent satisfying pi(Ht) ≥ pi(Ht−1) for all agents i and for all histories Ht at stage t arising out of stage t − 1 history Ht−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, the prices are monotonously non-decreasing for all agents over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Agents can opt to drop out once the prices are updated in stage t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' At+1 is the set of agents in At that did not drop out in stage t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The mechanism terminates at stage t when At becomes feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The agents in At are charged according to ⃗p(Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The definition from Milgrom and Segal (2014) restricts deferred-acceptance mechanisms to be deterministic, which we relax for the purpose of designing pen testing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' From the dis- cussion above, note that any deferred-acceptance mechanism can be converted into a pen testing algorithm, by setting thresholds equal to the prices recommended by the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The con- sumer surplus of the mechanism corresponds to the residual ink in the pens chosen by the pen testing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Now that we have established the consumer surplus is a quantity of interest for pen testing, we review consumer surplus optimization through virtual values in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='1 Consumer Surplus Optimization and Virtual Valuations Let us begin in a single-agent environment with one good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let the agent’s value be drawn from the distribution F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The quantile q of an agent with value v ∼ F is the measure with respect to F of stronger values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, q = 1 − F(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For q ∈ [0, 1], let v(q) be the inverse demand function of F satisfying F(v(q)) = 1 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In other words, Prˆv∼F(ˆv > v(q)) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Throughout the paper, we assume v(1) = 0 and � 0 0 v(t)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' It can be shown that these assumptions can be made without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let V (q) = � q 0 v(t)dt be the price-posting surplus curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Notice that V (q) is the expected surplus from posting a price v(q) to the agent, thereby allocating the good with probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' By our assumption on F, V (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let U(q) = V (q) − qv(q) be the price-posting consumer surplus curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Similar to the price-posting surplus curve, U(q) is the consumer-surplus from posting a price v(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that 0 ≤ U(0) ≤ V (0) = 0, and thus, U(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let u(q) = U′(q) = −qv′(q) be the marginal price-posting consumer surplus curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that v is monotonously non-increasing, and hence its derivative v′ is non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, u(q) ≥ 0 and U is monotone non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A quick note on derivatives: whenever v′(q) is well defined, u(q) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' At other points, u(q) can be calculated using any sub-derivative of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' At points q of discontinuity of v, v′(q) = −∞ and thus u(q) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 2 (Myerson, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In a Bayesian incentive-compatible mechanism with allocation rule 5 y and payment rule p (over quantiles), the expected consumer surplus of an agent satisfies Eq∼U[0,1][v(q)y(q) − p(q)] = Eq∼U[0,1][u(q)y(q)] = Eq∼U[0,1][−U(q) y′(q)] + U(0)y(0) Thus, the expected consumer surplus equals the expected virtual surplus with marginal consumer surplus as virtual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Myerson (1981) states that an allocation rule can be implemented as a truthful auction if and only if the allocation rule is monotone non-increasing in quantile space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the consumer surplus optimal mechanism optimizes for virtual surplus subject to the alloca- tion rule being monotonously non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' When the marginal price-posting consumer surplus curve is non-increasing (and hence U is con- cave), optimizing for virtual surplus automatically ensures monotonicity of the allocation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We will call these distributions with non-increasing marginal price-posting consumer surplus curves consumer surplus regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, u is not monotone non-increasing for common distributions like the uniform and normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In fact, u is monotone non-decreasing for these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In such a case, the allocation that pointwise optimizes for virtual surplus might not satisfy monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Myerson (1981) prescribes an ironing procedure to optimize for virtual surplus subject to monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Construct the concave hull U of the price-posting consumer surplus curve U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Define u(q) = U ′(q) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Find the virtual surplus optimal mechanism using u as virtual surplus Since U is concave, u is non-increasing, and hence optimizing for the ironed virtual surplus is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Throughout the paper, we will follow the convention of attaching a bar on top of a curve to describe the ironed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In a multi-agent environment, the interim allocation rule for an agent is the single-agent allocation rule that arises in expectation over the quantiles of all other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This captures the perspective of the agent after knowing its value, but before learning the values of the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let y be the interim allocation rule for some agent, satisfying d dqy(q) = 0 for all q such that U(q) ̸= U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then Eq[u(q)y(q)] = Eq[u(q)y(q)] In other words, if the allocation rule does not differentiate between agents in an ironed region, then the ironed virtual surplus can be used to compute the consumer surplus instead of the actual virtual surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 4 (Alaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2012, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let Y be some allocation rule with interim allocation rule yi monotonously non-increasing for each agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, there exists a mechanism Y with interim 6 allocation rule yi for agent i such that the expected consumer surplus for agent i equals Eq∼U[0,1][ui(q) yi(q)] = Eq∼U[0,1][ui(q) yi(q)] In other words, the expected consumer surplus of Y is the consumer surplus of the original mech- anism if the virtual values were given by ui for agent i instead of ui 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='2 Near-Optimal Deferred-Acceptance Mechanisms for Consumer Surplus In this section, we prescribe a recipe similar to Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2023) to convert any approximately op- timal deferred-acceptance mechanism for surplus into an approximately optimal deferred-acceptance mechanism for consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Definition 3 (Virtual-Pricing Transformation of a Deferred-Acceptance Mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let DAV be a deferred-acceptance mechanism designed to optimize surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let vi be the inverse demand function and ui be the virtual value function (for consumer surplus) for agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The virtual- pricing transformation on DAV , denoted by DAU, implements DAV in ironed virtual price space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, whenever DAV posts a price ˆvi to agent i, DAU posts a price vi(θi) satisfying θi = sup{θ : ui(θ) ≥ ˆvi} We will make some preliminary observations about the transformation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Consider two runs of the transformed mechanism DAU with some agent having two different values, both corresponding to the same ironed virtual value (which can happen if both these values are within the same ironed interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' From the definition of the transform, the agent faces the smallest price needed to differentiate itself from virtual values smaller than the threshold set by the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Consequently, the mechanism does not post prices from the middle of an ironed interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, in both these runs of the mechanism, the agent behaves identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Since the mechanism does not discriminate between values with the same ironed virtual value, the consumer surplus of the mechanism equals the ironed virtual surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' If the original mechanism DAV is a γ-approximation to the optimal surplus, in expectation over all product distributions, then the transformed mechanism DAU is a γ-approximation to the optimal ironed virtual surplus, and hence, to the optimal consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Finally, it is straightforward to see that the transformed mechanism DAU posts non-decreasing prices to each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Hence, DAU is also a deferred-acceptance mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Recall that the optimal omniscient approximation ζ(n) is the ratio between the optimal surplus and the optimal consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The proof of Theorem 1 is now straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let OPTU denote the expected optimal consumer surplus, OPTV denote the expected optimal surplus, and DAU denote the expected consumer surplus of the deferred accep- 7 tance mechanism of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, DAU ≥ γ(n) × OPTU ≥ γ(n) ζ(n) × OPTV where the inequalities follow from the definition of standard approximation γ(n) and optimal om- niscient approximation ζ(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The equivalent pen testing algorithm yields the neces- sary approximation ratios against the two benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In the next section, we analyze the optimal omniscient approximation ζ(n) for general combinato- rial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 3 Consumer Surplus versus Surplus Our approximation procedure follows two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='1, we approximate the consumer surplus to the optimal surplus in a single-agent environment, where the ex-ante probability of allocation is constrained to be at most q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We get a 1 − ln q approximation in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We then move from the single-agent environment to multi-agent environments using an approach similar to Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, the single-agent approximation diverges to ∞ when q −→ 0, and does not help in approximating the consumer surplus against the optimal surplus for these small quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='2 applies an approach of Hartline and Taggart (2019), that shows it is sufficient to have a good approximation for consumer surplus only in the larger quantiles (q = ω( 1 n)), to get around this challenge and to give a (1 + o(1)) ln n approximation for n-agent environments with general feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='1 The Single-Agent Problem Recall that V, U and U are the price-posting surplus curve, price-posting consumer surplus curve and the ironed consumer surplus curve respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In this section, we establish a “closeness property” like in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For an ex-ante allocation probability q ∈ [0, 1], the ratio of the optimal surplus V (q) to the optimal consumer surplus U(q) is at most 1 − ln q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We will begin by proving Theorem 5 for consumer surplus regular distributions, and then extend the result to all distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For any consumer surplus regular distribution and an ex-ante allocation probability q ∈ [0, 1], the ratio of the optimal surplus V (q) to the optimal consumer surplus U(q) is at most 1 − ln q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Conditional on generating a surplus V (q), we want to compute the distribution that mini- 8 mizes consumer surplus U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' U(q) = � q 0 u(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' V (q) = U(q) + qv(q) = � q 0 u(t)dt − q � 1 q v′(t)dt = � q 0 u(t)dt + q � 1 q u(t) t dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (1) We substitute u(t) = −tv′(t) in the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Simultaneously, we enforce u(t) ≥ 0, and u(t) is non-increasing (since the distribution is consumer surplus regular), v(1) = 0 and V (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Finding the minimum consumer surplus distribtuion reduces to the following program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' min � q 0 u(t)dt subject to � q 0 u(t)dt + q � 1 q u(t) t dt = V (q) u(t) ≥ 0, u(t) is monotonously non-increasing v(1) = 0, V (0) = 0 The constraints in the last line can be enforced after solving for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Monotonicity dictates u(t) ≥ u(q) for t ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Hence, minimizing � q 0 u(t)dt would correspond to setting u(t) = u(q) for t ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Similarly, u(t) ≤ u(q) for t ≥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Minimizing � q 0 u(t)dt would mean minimizing V (q) − q � 1 q u(t) t dt which is achieved by setting u(t) = u(q) for t ≥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let u(0) = u(t) = u(1) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For a constant marginal price-posting consumer surplus curve, U(q) = � q 0 u(t)dt = q u V (q) = � q 0 u(t)dt + q � 1 q u(t) t dt = [q − q ln q] u Thus, for any consumer surplus regular distribution, V (q) U(q) ≤ [1 − ln q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The constant marginal consumer surplus curve that achieves this ratio corresponds to the exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let the consumer surplus curves U, ˆU satisfy ˆU(0) = 0 and ˆU(t) ≥ U(t) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, for the corresponding surplus curves, ˆV (q) ≥ V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We rewrite equation (1) as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' V (q) = � q 0 u(t)dt + q � 1 q u(t) t dt = U(q) + q �U(t) t �t=1 t=q + q � 1 q U(t) t2 dt = qU(1) + q � 1 q U(t) t2 dt (2) The second equality is obtained through integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Increasing U pointwise clearly results in an increase in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The optimal consumer surplus for an ex-ante allocation probability q equals U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Consider the distribution with price-posting consumer surplus curve U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let V be the price-posting surplus curve corresponding to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' From Lemma 2 and Lemma 1, V (q) U(q) ≤ V (q) U(q) ≤ 1 − ln q The second inequality holds from Lemma 1, since V is the price-posting surplus curve of a con- sumer surplus regular distribution (we made U concave through ironing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' See Appendix A for a brief discussion on the surplus generated by the single-agent optimal con- sumer surplus auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='2 Multi-Agent Environments Consider the interim allocation rule yi for agent i in the surplus optimal mechanism OPTV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' With- out loss of generality, assume yi(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The expected surplus for agent i equals Eq∈U[0,1][yi(q) vi(q)] = � yi(q) Vi(q) �q=1 q=0 + Eq∈U[0,1][−y′ i(q) V (q)] = Eq∈U[0,1][−y′ i(q) V (q)] This equality comes from integrating by parts (yi(q) × Vi(q) vanishes at both, q = 0 and 1, since Vi(0) = 0 and yi(1) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Suppose there exists a constant α such that Vi(q) ≤ α U i(q) for all q ∈ [0, 1], then, Eq∈U[0,1][−y′ i(q) V (q)] ≤ Eq∈U[0,1][−y′ i(q) α U(q)] = α × Eq∈U[0,1][yi(q) ui(q)] Note that the left hand side of the above equation cannot be related to the consumer surplus yet, since d dqyi need not be equal to 0 whenever U(q) ̸= U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, Theorem 4 shows the existence of a mechanism with expected consumer surplus of agent i equal to Eq∈U[0,1][yi(q) ui(q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, 10 the expected consumer surplus of this mechanism is an α-approximation to the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The consumer surplus optimal mechanism OPTU will only do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Unfortunately, as q −→ 0, the bound from Theorem 5 of Vi(q) Ui(q) ≤ (1 − ln q) diverges to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, we need to show that the loss from ignoring the small (strong) quantiles is not much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Suppose we find a mechanism M1 with interim allocation ˆyi for agent i, near optimal surplus, and y′ i(q) = 0 for q ∈ [0, ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, y′ i(q) × V (q) = y′ i(q) × U(q) = 0 in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For q ∈ [ǫ, 1], we know V (q) ≤ (1 − ln ǫ) × U(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, surplus(M1) ≤ (1 − ln ǫ) × OPTU Hartline and Taggart (2019) give the construction of such a mechanism M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Definition 4 (Hartline and Taggart, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ǫ-buffering rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Given an allocation rule ⃗y = (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' yn) and a quantile ǫ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' the ǫ-buffering rule for ⃗y runs y with quantiles transformed on each agent as follows: Top inflate: for any qi ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ǫ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' return 0 For any qi ∈ [ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1 − ǫ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' return qi−ǫ 1−2ǫ Bottom deflate: for any qi ∈ [1 − ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' return 1 In essence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' the top inflate branch makes the ǫ-buffering rule treat all agents with a small quantile as if they had quantile 0 and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' y′ i(q) = 0 for q ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' ǫ] (the bottom deflate branch is unnecessary for the analysis of monotone payoff curves like surplus V and consumer surplus U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' see Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 6 (Hartline and Taggart, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let M be the ǫ-buffering rule of the surplus optimal mechanism ⃗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, surplus(⃗y) ≤ 1 (1− ǫ 1−ǫ ) (1−ǫ) (1−2nǫ) × surplus(⃗y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Consider an n-agent environment with an arbitrary feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The optimal omniscient approximation is at most 1−ln ǫ (1− ǫ 1−ǫ ) (1−ǫ) (1−2nǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The proof follows by combining Theorem 6 with the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let M1 be the ǫ-buffering rule of the surplus optimal mechanism OPTV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Then, OPTV ≤ 1 (1 − ǫ 1−ǫ) (1 − ǫ) (1 − 2nǫ) × surplus(M1) ≤ 1 − ln ǫ (1 − ǫ 1−ǫ) (1 − ǫ) (1 − 2nǫ) × OPTU Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In n-agent environments with an arbitrary feasibility constraint, the optimal omni- scient approximation ζ(n) ≤ (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Setting ǫ = 1 n ln n in Theorem 7, we get ζ(n) ≤ 1 − ln 1 n ln n (1 − 1 n ln n 1− 1 n ln n ) (1 − 1 n ln n) (1 − 2n · 1 n ln n) = 1 + ln n + ln ln n (1 − 1 n ln n−1) (1 − 1 n ln n) (1 − 2 ln n) = n ln n − 1 n ln n − 2 · n ln n n ln n − 1 · ln n − 2 ln n 1 + ln n + ln ln n ln n × ln n = (1 + o(1)) ln n Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The ǫ-buffering rule of Hartline and Taggart (2019) can be implemented without a bot- tom deflate branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This would strengthen the approximation guarantee to 1 (1−ǫ) (1−nǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, substituting ǫ = 1 n ln n will not result in a bound tighter than (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the special case of k-unit auctions, Hartline and Roughgarden (2008) show ζ(n) ≤ 2 ln 2(ln n k +ln 2) = 2 ln 2(1+o(1)− ln k ln n) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Our bound performs better whenever ln k ln n = o(1), such as k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The bound from Corollary 1 matches the lower bound of Qiao and Valiant (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In the single-item environment with values of agents drawn IID from the exponential distribution, they show that the optimal surplus is Hn = (1 + o(1)) ln n times the optimal consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 4 Pen Testing Corollaries from Deferred-Acceptance Mecha- nisms There are many environments in which deferred-acceptance mechanisms are known to be good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' By Theorem 1, these imply good pen testing algorithms (up to an additional ζ(n) factor in ap- proximation to the omniscient benchmark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In this section, we discuss a few notable examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A summary of these results was given previously in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Deferred-acceptance mechanisms that achieve the ex-post surplus optimal outcome are known for various feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' While the simple auction that uniformly increases the price until exactly k bidders remain active is surplus optimal in the k-identical goods environment, Milgrom and Segal (2014) and Bikhchandani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2011) generalize the mechanism for matroid feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We get the following corollary from the above auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For a pen testing environment with a matroid feasibility constraint, there exists a pen testing algorithm with an omniscient approximation ratio (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' D¨utting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2014) initiated the study of prior-free deferred-acceptance approximation mecha- nisms in environments where deferred-acceptance mechanisms are known to be suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Their 12 bounds can be improved in settings like ours where prior distributions of the agents’ values are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For general downward-closed constraints with a prior distribution on values, Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2022) give a O(log log m) approximation to the optimal surplus, where m is the number of maxi- mal feasible sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that m ≤ 2n (every subset of agents might be feasible) and hence, log log m is at most log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, we have a poly-logarithmic approximate pen testing algorithm for any downward-closed feasibility environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note, however, that the pen testing model is inherently downward-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Specifically, suppose an algorithm wanted to select a set P ⊆ P, but P is not feasible while P is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Since all pens have non-negative residual ink, even the ones that failed their tests, there is no loss in selecting P instead of the set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, near optimal pen testing algorithms for downward-closed constraints can be extended to give near optimal algorithms for general combinatorial constraints, giving the same performance guarantee as the downward-closed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For any combinatorial pen testing environment, there exists a pen testing algorithm with an omniscient approximation ratio O(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Note that this reduction cannot be used to design deferred-acceptance mechanisms for general feasibility constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Allocating to agents that have dropped out of the auction can make the mechanism non-truthful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='e, staying active till the price reaches the agent’s value and dropping out subsequently might not be in the agent’s best interests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The bound for downward-closed and general combinatorial environments can be improved upon in special cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' for example, in Appendix B we give a natural deferred-acceptance mechanism for knapsack constraints that is a 2-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For this knapsack problem, the agents have sizes along with values and feasible subsets are precisely those with a total size at most the knapsack capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For a pen testing environment with a knapsack feasibility constraint, there exists a pen testing algorithm with an omniscient approximation ratio 2 (1 + o(1)) lnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Online pricing mechanisms are a special case of deferred-acceptance mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' thus, Theorem 1 converts online pricing mechanisms with good surplus into good online pen testing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For online pen testing problems, every pen can be tested exactly once and must either be selected or discarded immediately after testing and before testing another pen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The algorithm might have the ability to determine the order of testing pens (the sequential version) or the order might be adversarially determined (the oblivious version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the sequential pricing problem, Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2010) gave an e e−1-approximation mechanism for the single-item environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Yan (2011) connected the sequential pricing problem to the cor- relation gap and generalized the e e−1-approximation to matroid environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Both of these papers considered optimizing revenue, but the results can be easily adapted to optimize surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In the online sequential pen testing problem with a matroid feasibility constraint, there exists a pen testing algorithm that achieves an omniscient approximation ratio e e−1 · (1 + 13 o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Hajiaghayi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2007) and Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2010) adapt the prophet inequality of Samuel-Cahn (1984) to give a non-adaptive 2-approximation for the oblivious price posting problem with 1 good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Kleinberg and Weinberg (2012) extend this 2-approximation to matroid environments and any arrival order of agents but with adaptive prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In the online oblivious pen testing problem to select a single pen, there exists a non- adaptive pen testing algorithm that achieves an omniscient approximation ratio 2(1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In the online oblivious pen testing problem with a matroid feasibility constraint, there exists a pen testing algorithm that achieves an omniscient approximation ratio 2(1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The bounds of Corollary 5 and Corollary 6 improve on the online pen testing bounds of Qiao and Valiant (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 5 The Online IID Environment Consider the special case of the online pen testing problem where the ink levels are drawn inde- pendently from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We move away from the reduction framework of Theorem 1 to obtain a better bound for this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In online single-item environments with IID agents, there exists a price-posting strat- egy with an omniscient approximation at most (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Equivalently, in the online pen testing problem with IID ink levels, there exists an algorithm that achieves π(n) ≤ (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' First, note that the above theorem is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In Remark 3, we saw an instance (originally from Qiao and Valiant, 2023) with n IID agents whose values are drawn from the exponential distribu- tion with consumer surplus only an Hn approximation to the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The optimal online consumer surplus cannot do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Hence, the (1 + o(1)) ln n approximation ratio in Theorem 8 is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Before proving Theorem 8, we briefly discuss the guarantee obtained through the reduction frame- work of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For the online environment with IID agents, it is known that γ(n) = e e−1 (Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the framework would give us an e e−1 · (1 + o(1)) ln n bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The direct analysis below gives a tight Hn + 1 = (1 + o(1)) ln n bound up to an additive factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Similar to the proof of Theorem 5, we will phrase our program as an opti- mization problem and compute the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The surplus optimal auction always allocates the good to the agent with the highest value (smallest quantile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let vmax be the random variable denoting the value of the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Pr(vmax ≤ v(t)) = (1 − t)n 14 Thus, the expected surplus of the surplus optimal mechanism equals � 1 0 v(t) × n(1 − t)n−1dt = � 1 0 � 1 t u(r) r dr × n(1 − t)n−1dt = � 1 0 u(t) t dt − � 1 0 u(t) t (1 − t)ndt = � 1 0 1 − (1 − t)n t u(t)dt = U(1) + � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t2 U(t)dt The first equality follows by integrating from u(t) = −tv′(t) and v(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We integrate by parts for the second and fourth equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Next, we look at the consumer surplus through price-posting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For a price v(q), we allocate the good if at least one agent has a quantile in [0, q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Conditioned on selling the good, we get an expected consumer surplus of U(q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the expected consumer surplus equals 1 − (1 − q)n q × U(q) As before, we first show Theorem 8 for consumer surplus regular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Without loss of generality, assume that the maximum achievable consumer surplus through anonymous price posting is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We want to find the consumer surplus regular distribution with the largest surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' max U(1) + � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t2 U(t)dt subject to 1 − (1 − q)n q × U(q) ≤ 1 for all q ∈ [0, 1] U is concave First, observe that C(q) = q 1−(1−q)n is convex (see Lemma 4 in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We want to fit a concave curve U under the convex curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let Cq be the tangent to C at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' C ˆq is a feasible solution for U for all ˆq ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Further, given that U touches the curve C at ˆq, U(t) ≤ C ˆq(t) for all t ∈ [0, 1] (from the concavity of U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Also, it is clear that the objective increases with a pointwise increase in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, we conclude that the optimal solution to the program is achieved at U = C ˆq at some ˆq ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In particular, the worst-case ratio occurs when U is a straight-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Without loss of generality, we can normalize the slope to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let U(q) = q + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The surplus equals 1 + a + � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t dt + a × � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t2 dt = Hn + an 15 See Lemma 5 in Appendix C for the proof of the above equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' By posting a price v(q), we generate a consumer surplus 1 − (1 − q)n q × U(q) = (1 + (1 − q) + · · · + (1 − q)n−1) × (q + a) If a ≥ 1 n, the ratio between the surplus and the consumer surplus is at most Hn + 1 by setting q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' If a < 1 n, the surplus is less than Hn + 1, and by setting q = 1, the consumer surplus is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the worst-case ratio between surplus and consumer surplus is at most Hn + 1 = (1 + o(1)) ln n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We now show the result for all distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We rewrite the optimal surplus as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' � 1 0 v(t) × n(1 − t)n−1 = � V (t) × n(1 − t)n−1�t=1 t=0+ � 1 0 V (t) × n(n − 1)(1 − t)n−2dt The equality follows through integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' From the above expression, it can be concluded that the optimal surplus increases with a pointwise increase in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let V be the price-posting surplus curve of the distribution with a price-posting consumer surplus curve U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' U is pointwise larger than U, and hence, from Lemma 2, V is pointwise larger than V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Hence, the optimal surplus is larger in the distribution with price-posting consumer surplus curve U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, from Theorem 3, the consumer surplus from posting prices is identical to both the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the consumer surplus irregular distribution has a better ratio between surplus and consumer surplus than the consumer surplus regular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the ratio between optimal surplus to optimal consumer surplus is at most (1 + o(1)) ln n for all distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 6 Conclusion In this section, we discuss the various advantages and disadvantages of using the reduction frame- work of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Connecting the pen testing problem to auction theory and building an easy-to- use reduction framework enable us to design near optimal pen testing algorithms for any general feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We are able to construct algorithms that are only a constant factor away from the lower bound described in Qiao and Valiant (2023) for various feasibility constraints like matroids (γ(n) = 1) and knapsack (γ(n) = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' However, there is a scope for improvement on the following two fronts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' obtaining tight approximation guarantees, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' obtaining approximation guarantees under other models of access to the prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Tight Approximation Guarantees: The approximation ratio of any pen testing algorithm is at least the optimal omniscient approximation ζ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This follows since the optimal consumer sur- plus in the underlying auction environment is an upper bound on the performance of the optimal 16 pen testing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The framework matches this bound up to a multiplicative factor of the ap- proximation γ(n) of deferred-acceptance mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Reducing this gap for various feasibility environments stands as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For instance, consider the online environments discussed in the paper, where ζ(n) = (1+o(1)) lnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Our framework yielded an omniscient approximation 2(1 + o(1)) ln n in the oblivious version (γ(n) = 2) and e e−1 · (1 + o(1)) ln n in the sequential version (γ(n) = e e−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We conjecture that the approximation guarantees obtained by applying the reduction framework are not tight, and can be strengthened to (1 + o(1)) ln n in both these environments, like in the IID setting (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Other Models of Access to Distributions: Our framework crucially makes use of virtual val- ues, which in turn need complete knowledge of the distribution of ink levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, tailoring this approach to models where the algorithm gets partial access to the priors is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Qiao and Valiant (2023) study two such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' They study the online oblivious problem where the algorithm gets access to just one sample from the distribution of each pen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' They give a O(log n)-approximation to the omniscient benchmark in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' They also look at the online secretary setting, where the pens arrive according to a permutation chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The quantity of ink in the pens are adversarially determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The algorithm is told the quantity of ink in the pen with the largest ink level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' They give a O(log n)-approximation in this environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Further, when the algorithm is given access to the ink levels in all the n pens (with random arrival order), they tighten 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Optimal auction design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Mathematics of operations research, 6(1):58–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Qiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' and Valiant, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Online pen testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 14th Innovations in Theoretical Computer Science Conference, (ITCS) 2023, January 10-13, 2023, MIT, Cambridge, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Samuel-Cahn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Comparison of threshold stop rules and maximum for independent non- negative random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=', 12(4):1213–1216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Yan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Mechanism design via correlation gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 710–719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A A Discussion on the Surplus of the Single-Agent Optimal Consumer Surplus Auction In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content='1, we compare the optimal consumer surplus with an ex-ante allocation constraint q to the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We now take a look at the surplus generated by the consumer surplus optimal auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For an ex-ante allocation probability q, the ratio of the optimal surplus to the surplus from the consumer surplus maximizing auction is at most 1 − ln q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The corollary follows since the surplus of the consumer surplus maximizing auction is only larger than the consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The above corollary is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This follows immediately from the observations below: 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For a distribution with a convex price-posting consumer surplus curve, the consumer surplus optimal mechanism is obtained by giving away the good for free with probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' For such distributions, the consumer surplus of the optimal mechanism equals the surplus of the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The price-posting consumer surplus curve of the exponential distribution is a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' A distribution obtained by adding a small convex distortion to this curve has the same ironed price-posting consumer surplus curve as the exponential distribution and close to the same price-posting surplus curve as the exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The ratio of surplus in the surplus optimal mechanism to the consumer surplus optimal mechanism is close to 1 − ln q for distributions described in bullet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' B Deferred-Acceptance Mechanism for Knapsack Constraints Consider an environment with a knapsack feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Each agent i has a value vi drawn independently from distribution Fi and a size si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The goal is to chose a set S of agents maximizing surplus such the total size � i∈S si of chosen agents is at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We rely on the following folklore 2-approximation algorithm for the knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Run the alternative with a better expected surplus: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Bang-per-buck (Dantzig, 1957): Sort agents in decreasing order of vi si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Pick agents in this order until the knapsack is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Max: Pick the agent with the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Milgrom and Segal (2014) give deferred-acceptance implementations for both, bang-per-buck and max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, the above is a deferred-acceptance mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Algorithm 1 is a 2-approximation to the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let OPTV be the random variable denoting the optimal surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Fix some realization of values v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Without loss of generality, assume v1 s1 ≥ v2 s2 ≥ · · · ≥ vn sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Let bang-per-buck pick agents 1 through t for this realization of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Clearly, �t+1 i=1 vi ≥ OPTV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' This is because the optimal algorithm that can pack fractional agents in the knapsack would greedily pick the first t agents and pick some fraction of agent t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Thus, if vmax = max1≤i≤n vi, �t i=1 vi+vmax ≥ OPTV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The surplus generated jointly by the two alternatives pointwise dominates the optimal surplus and hence dominates the optimal surplus in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Evi∼Fi[surplus(bang-per-buck)] + Evi∼Fi[surplus(max)] ≥ Evi∼Fi[OPTV] We get the larger of the two terms in the left hand side, which must be at least 1 2Evi∼Fi[OPTV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 19 C Lemmas for Theorem 8 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' q 1−(1−q)n is convex in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' q 1 − (1 − q)n = 1 �n−1 j=0(1 − q)j d dq 1 �n−1 j=0(1 − q)j = �n−2 j=0(j + 1)(1 − q)j �n−1 j=0(1 − q)j 1 �n−1 j=0(1 − q)j The second term in the product is increasing in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' We will show the first term is also increasing in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The first term can be rewritten as �n−2 j=0(j + 1)(1 − q)j �n−1 j=0(1 − q)j = �n−1 j=0[(1 − q)j − (1 − q)n−1] �n−1 j=0(1 − q)j = 1 − n (1 − q)n−1 1 + (1 − q) + · · · + (1 − q)n−1 Further rewriting this, it becomes apparent that this is indeed increasing in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' �n−2 j=0(j + 1)(1 − q)j �n−1 j=0(1 − q)j = 1 − n 1 1 (1−q)n−1 + · · · + 1 (1−q)0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1 + a + � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t dt + a × � 1 0 1 − (1 − t)n − nt(1 − t)n−1 t2 dt = Hn + an Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' The key is to rewrite 1−(1−t)n−nt(1−t)n−1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 1 − (1 − t)n − nt(1 − t)n−1 t = n−1 � j=0 (1 − t)j − n(1 − t)n−1 = n−1 � j=0 [(1 − t)j − (1 − t)n−1] = t × n−2 � j=0 (j + 1) × (1 − t)j Thus, the expression reduces to 1 + a + n−2 � j=0 � 1 0 (j + 1) × t(1 − t)jdt + a × n−2 � j=0 � 1 0 (j + 1) × (1 − t)jdt = Hn + an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFMT4oBgHgl3EQf7DEn/content/2301.12462v1.pdf'} diff --git a/pNE1T4oBgHgl3EQfPAOE/content/2301.03022v1.pdf b/pNE1T4oBgHgl3EQfPAOE/content/2301.03022v1.pdf new 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b/pNE1T4oBgHgl3EQfPAOE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6664cd72392b7081c04402800c5486f8204706e317346497fbbd75ebb369c7e4 +size 330257 diff --git a/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/2301.01169v1.pdf.txt b/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/2301.01169v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f03c299c8ee1134fed03ae118c9bd692c11bb06 --- /dev/null +++ b/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/2301.01169v1.pdf.txt @@ -0,0 +1,1414 @@ +Noname manuscript No. +(will be inserted by the editor) +Charge superradiance on charged BTZ black holes +Sebastian Konewko · Elizabeth Winstanley +Abstract We study superradiance for a charged scalar field +subject to Robin (mixed) boundary conditions on a charged +BTZ black hole background. Scalar field modes having a +real frequency do not exhibit superradiance, independent of +the boundary conditions applied. For scalar field modes with +a complex frequency, irrespective of the boundary condi- +tions, no charge superradiance occurs if the black hole is +static. We demonstrate the existence of superradiant modes +with complex frequencies for a charged and rotating BTZ +black hole. Most of the superradiant modes we find satisfy +Robin boundary conditions, but there are also superradiant +modes with complex frequencies satisfying Dirichlet and +Neumann boundary conditions. We explore the effect of the +black hole and scalar field charge on the outgoing energy +flux of these superradiant modes. +1 Introduction +In the scattering of waves incident on a black hole, super- +radiance occurs if the reflected wave has greater amplitude +than the incident wave [1]. For example, low-frequency bos- +onic waves scattered by a rotating Kerr black hole exhibit su- +perradiance [2–4]. This phenomenon is the wave analogue +of the Penrose process for particles [5], with the wave ex- +tracting rotational energy from the black hole. An analogous +effect (charge superradiance) occurs for charged scalar field +waves scattered by a charged Reissner-Nordstr¨om black hole +S. Konewko +Independent Researcher +E-mail: sebastian.konewko@gmail.com +E. Winstanley +Consortium for Fundamental Physics, School of Mathematics and +Statistics, The University of Sheffield, Hicks Building, Hounsfield +Road, Sheffield. S3 7RH United Kingdom +E-mail: E.Winstanley@sheffield.ac.uk +[6–10]. In this case it is electrostatic rather than rotational +energy which is extracted from the black hole by the wave. +If one considers asymptotically anti-de Sitter (adS) rather +than asymptotically flat black holes, superradiance persists. +The adS boundary acts like a mirror, reflecting the amplified +wave back onto the black hole. For sufficiently small rotat- +ing and/or charged asymptotically-adS black holes in four +or more space-time dimensions, this process can lead to a +superradiant instability (see, for example, [11–34] for a se- +lection of papers from the extensive literature on this topic). +The situation for asymptotically-adS black holes in three +space-time dimensions is subtly different from that in four +or more space-time dimensions. In particular, one can con- +sider the famous BTZ metric [35–37]. Since null infinity is a +time-like hypersurface in adS, boundary conditions have to +be applied to an incident wave. For a neutral scalar field, ap- +plying the simplest boundary conditions, namely Dirichlet +boundary conditions, superradiance is absent [38]. Super- +radiance also does not occur for a fermion field vanishing +on the boundary [39]. However, one can impose more gen- +eral (Neumann or Robin) boundary conditions on the scalar +field [40], and, for at least some Robin boundary conditions, +superradiant modes exist [41]. The impact of Robin bound- +ary conditions on superradiance has also been considered on +four-dimensional Kerr-adS black holes [42]. +In this paper we examine whether superradiance exists +for a charged scalar field subject to Robin (mixed) boundary +conditions on a charged analogue of the BTZ black hole. We +begin, in Section 2, by reviewing the black hole metric and +separable solutions of the charged scalar field equation on +this spacetime background, paying particular attention to the +boundary conditions satisfied by the field far from the black +hole. In Section 3 we study the possibility of superradiance +using two approaches: firstly a Wronskian condition which +is valid for waves having real frequency, and secondly, fol- +lowing [41], considering the energy flux down the horizon +of an ingoing wave. In particular, an ingoing superradiant +arXiv:2301.01169v1 [hep-th] 3 Jan 2023 + +2 +Sebastian Konewko, Elizabeth Winstanley +wave will have an outgoing energy flux. Using the Wron- +skian, we find that there is no superradiance for field modes +having real frequency. If the frequency is complex, by con- +sidering the energy flux, we show that charge superradiance +is absent if the black hole is nonrotating. This leaves open +the possibility of superradiance for charged, rotating black +holes, which is studied in Section 4. Using a simple numeri- +cal method, valid for frequencies in the superradiant regime, +we find charged superradiant modes. We explore the effect +of increasing either the black hole or scalar field charge on +the energy fluxes of these superradiant modes. Finally our +conclusions are presented in Section 5. +2 Charged scalar field on a charged BTZ black hole +2.1 Charged BTZ black holes +The neutral BTZ black hole [35–37] is a solution of the +three-dimensional Einstein equations with a negative cos- +mological constant Λ = −ℓ−2, having metric +ds2 = −N0(r)dt2 +N0(r)−1 dr2 +r2 � +dϕ +Nϕ +0 (r)dt +�2 (1a) +where +N0(r) = r2 +ℓ2 −M + J2 +4r2 , +Nϕ +0 (r) = − J +2r2 , +(1b) +with M the mass and J the angular momentum of the black +hole. +In the static case (J = 0), the black hole acquires an elec- +tric charge Q by introducing the electromagnetic potential +Aµ = A0δt +µ, where +A0 = −Qln +� r +r0 +� +, +(2a) +and an arbitrary length scale r0 has been introduced to render +the argument of the logarithm dimensionless. The metric for +a static charged BTZ black hole is then [35–37] +ds2 = −N(r)dt2 +N(r)−1 dr2 +r2 dϕ2 +(2b) +and the lapse function takes the form +N(r) = r2 +ℓ2 −M − Q2 +2 ln +� r +r0 +� +. +(2c) +The generalization of the rotating BTZ black hole to in- +clude an electric charge is far from straightforward [37, 43]. +Various three-dimensional, charged, rotating black holes are +presented in [44–48]. In this paper we consider the follow- +ing charged generalization of the BTZ metric [48]: +ds2 = −N(r) +r2 +R(r)2 dt2 +N(r)−1 dr2 ++R(r)2 [dϕ +Nϕ(r)dt]2 +(3a) +where N(r) is the same as in the static case (2c) and the other +functions appearing in the metric are: +R(r)2 = r2 + Ω 2ℓ2 +1−Ω 2 +� +M + Q2 +2 ln +� r +r0 +�� +, +Nϕ(r) = − +Ωℓ +(1−Ω 2)R(r)2 +� +M + Q2 +2 ln +� r +r0 +�� +, +(3b) +where M, Q and Ω ∈ [0,1) are constants. When Ω > 0, the +electromagnetic potential Aµ acquires a nonzero magnetic +part and is given by: +Aµdxµ = − +Q +√ +1−Ω 2 [dt −Ωℓdϕ]ln +� r +r0 +� +. +(3c) +The mass � +M, angular momentum �J and charge �Q of the black +hole are given in terms of the parameters M, Q, and Ω as +follows [48]: +� +M = +1 +1−Ω 2 +� +M +� +1+Ω 2� +− 1 +2Q2Ω 2 +� +, +�J = +2Ω +1−Ω 2 +� +M − 1 +4Q2 +� +, +�Q = +Q +√ +1−Ω 2 . +(4) +Unlike the neutral BTZ black hole, the metric (3a) cannot +be obtained by identifying points in three-dimensional adS +space-time. In particular, the scalar curvature R is not con- +stant: +R = Q2 +2r2 − 6 +ℓ2 . +(5) +In the limit Ω → 0, the metric and gauge field potential +(3) reduce to those in the static case (2). If we set Q = 0, +the metric (3a) does not reduce to the original rotating BTZ +metric (1a) in (t,r,ϕ) coordinates. However, using R as the +radial coordinate, when Q = 0 the metric (3a) becomes +ds2 = −�N(R)dt2 + �N(R)−1 dR2 +R2 [dϕ +Nϕ(r)dt]2 (6a) +where we have defined the function +�N(R) = N(r) r2 +R2 = R2 +ℓ2 − M +� +1+Ω 2� +1−Ω 2 ++ +M2Ω 2ℓ2 +(1−Ω 2)2 R2 . (6b) +We therefore have a metric of the form (1a) with mass � +M = +M +� +1+Ω 2� +/ +� +1−Ω 2�2 and angular momentum +�J = 2MΩℓ/ +� +1−Ω 2� +, in accordance with (4). +The horizons of the black hole are located at those val- +ues of the radial coordinate r for which N(r) vanishes. If +M < Q2 [1−2ln(Qℓ/2r0) ]/2 there is a naked singularity +at r = 0; we do not consider this possibility further. For +M > Q2 [1−2ln(Qℓ/2r0)]/2 there is an event horizon at +r = rh, the largest zero of N(r) and an inner horizon at the +smaller positive zero of N(r). These two horizons coincide + +Charge superradiance on charged BTZ black holes +3 +when M = Q2 [1−2ln(Qℓ/2r0)]/2 and in this case we have +an extremal black hole [49]. In this paper we focus on the +case where the black hole is nonextremal. +By making a gauge transformation of the form +Aµ → Aµ +∂µχ, +χ = +Q +√ +1−Ω 2 (t −Ωℓϕ)ln +�rh +r0 +� +, +(7) +we may set r0 = rh without loss of generality. We then find, +by considering the zeros of (2c), that +M = r2 +h +ℓ2 . +(8) +At the horizon, we have R(rh)2 = r2 +h/ +� +1−Ω 2� +and Nϕ(rh) = +−Ω/ℓ, so that Ω/ℓ is the angular speed with which the event +horizon rotates. +In our analysis of superradiance, we will be interested +in the flux of energy down the event horizon of the black +hole. For this analysis, we will require suitable coordinates +which are regular across the horizon. We will employ ingo- +ing Eddington-Finkelstein (EF) coordinates. First we define +an ingoing null coordinate v by +dv = dt + 1 +r +R(r) +N(r) dr, +(9a) +and a new angular coordinate �ϕv by +d �ϕv = dϕ − R(r) +r +Nϕ(r) +N(r) dr. +(9b) +Then the coordinates (v,r, �ϕv) are ingoing EF coordinates, in +terms of which the metric (3a) becomes +ds2 = −N(r) +r2 +R(r)2 dv2 + 2r +R(r) dvdr ++R(r)2 [d �ϕv +Nϕ(r)dv]2 . +(10) +The metric (10) is regular when r = rh and N(r) = 0, as +required. Near the horizon, the ingoing EF coordinates take +the form +v = t + +r∗ +√ +1−Ω 2 , +�ϕv = ϕ − +Ωr∗ +ℓ +√ +1−Ω 2 +(11) +where r∗ is the usual tortoise coordinate, defined by +dr∗ +dr = +1 +N(r). +(12) +2.2 Charged scalar field +We consider a scalar field Φ with charge q and mass m prop- +agating on the rotating charged black hole (3a), and satisfy- +ing the charged scalar field equation +� +DµDµ −m2� +Φ = 0, +(13) +where Dµ = ∇µ − iqAµ is the covariant derivative. We as- +sume that the scalar field is minimally coupled to the ge- +ometry. The stress-energy tensor for the charged scalar field +is +Tµν = ℜ +�� +DµΦ +�∗ DνΦ − 1 +2gµνgρσ � +DρΦ +�∗ DσΦ +−1 +2m2gµνΦ∗Φ +� +, +(14) +where ℜ is the real part and a star is used to denote complex +conjugation. +Mode solutions of the scalar field equation (13) take the +form +Φωk(t,r,ϕ) = 1 +√r e−iωt eikϕ Xωk(r), +(15) +where ω is the frequency of the wave (which may be com- +plex) and k ∈ Z is the azimuthal quantum number. In terms +of the tortoise coordinate r∗ (12), the radial function Xωk(r) +satisfies the equation +� d2 +dr2∗ ++Vωk(r) +� +Xωk(r) = 0 +(16a) +where the potential Vωk(r) takes the form +Vωk(r) = +�ω −kΩℓ−1 +√ +1−Ω 2 −qQln +� r +rh +��2 +−m2N(r) ++ N(r)2 +4r2 +− N′(r)N(r) +2r +− +� +ωΩ −kℓ−1�2 ℓ2N(r) +(1−Ω 2)r2 +. +(16b) +As r → rh and the event horizon is approached, we have +r∗ → −∞ and +Vωk(r) → �ω2 +(17) +where we have defined +�ω = ω −kΩℓ−1 +√ +1−Ω 2 . +(18) +Therefore, near the horizon, the radial function Xωk(r) takes +the form +Xωk(r) ∼ Aωkei �ωr∗ +Bωke−i �ωr∗ +(19) +where Aωk and Bωk are complex constants. The frequency of +the wave has effectively been shifted due to the rotation of + +4 +Sebastian Konewko, Elizabeth Winstanley +the black hole. The fact that �ω does not depend on the charge +stems from our choice of gauge, in that the electromagnetic +gauge potential (3c) vanishes at the horizon since we have +taken r0 = rh. +Far from the black hole, as r → ∞, the leading-order be- +haviour of the potential (16b) is, in general, +Vωk(r) ∼ − +� +m2 + 3 +4ℓ2 +� r2 +ℓ2 . +(20) +This leading-order behaviour is the same as for the neutral +scalar field, and does not depend on the frequency ω or the +azimuthal quantum number k. In this regime the tortoise co- +ordinate has the following form: +r∗ ∼ −ℓ2 +r , +(21) +yielding the equidimensional differential equation +� d2 +dr2∗ +− µ2 +r2∗ +� +Xωk(r) = 0 +(22) +where µ2 is a constant given by +µ2 = m2ℓ2 + 3 +4. +(23) +Let us assume for the moment that µ2 ̸= 0. The solutions of +(22) are Xωk ∼ rp +∗ ∼ r−p, where +p = 1 +2 +� +1± +� +1+4µ2 +� +. +(24) +For 4µ2 > −1, the values of p are real and +Xωk(r) ∼ Cωkr− 1 +2 (1+√ +1+4µ2) +Dωkr− 1 +2 (1−√ +1+4µ2) +(25) +for complex constants Cωk, Dωk. The second term gives a ra- +dial function which is not square integrable at infinity when +4µ2 > 0 and therefore we set Dωk = 0 in this case. In this +situation there is no choice of boundary conditions which +can be imposed on the scalar field at infinity. +For −1 < 4µ2 < 0, both solutions in (25) are square- +integrable, resulting in some freedom in the choice of bound- +ary conditions at infinity [50, 51]. The solution with Dωk = +0 satisfies Dirichlet boundary conditions, while, following +[50], we define Neumann boundary conditions to be such +that Cωk = 0. If both Cωℓ and Dωℓ are nonzero, then we +have Robin (mixed) boundary conditions. In this situation +we write the solution (25) in the form [41, 50] +Xωk(r) ∼ Eωk +� +r− 1 +2 (1+√ +1+4µ2) cosζ +r− 1 +2 (1−√ +1+4µ2) sinζ +� +(26) +where Eωk is a complex constant and the real angle ζ (which +we term the “Robin parameter”) can be taken to lie in the +interval 0 ≤ ζ < π (we could equally well take ζ ∈ (− π +2 , π +2 ]). +Setting ζ = 0 yields Dirichlet boundary conditions, while +ζ = π +2 corresponds to Neumann boundary conditions. +When 4µ2 = −1, we have +Xωk(r) ∼ Cωkr− 1 +2 +Dωkr− 1 +2 ln +� r +rh +� +. +(27) +Both solutions are square-integrable in this case, so again +we have a choice of boundary conditions. For 4µ2 < −1, the +exponent p (24) is complex and Xωk(r) is oscillatory. Once +again both linearly independent solutions of the radial equa- +tion are square-integrable at infinity. However, these values +of µ2 violate the Breitenlohner-Freedman bound [52, 53] +and therefore we do not consider them further in this work. +The above discussion of the boundary conditions at in- +finity is valid only when µ2 ̸= 0, in which case the behaviour +of the charged scalar field at infinity is identical to that for +a neutral scalar field. In the special case µ2 = 0 the leading +order behaviour of the potential (16b) is no longer (20), but +instead we have, as r → ∞, +Vωk(r) ∼ q2Q2 +� +ln +� r +rh +��2 +. +(28) +In this case it is not possible to solve the asymptotic form of +the radial equation exactly in terms of elementary functions. +However, it is possible to perform an asymptotic expansion +for the radial function Xωk(r) in this case. The first couple +of terms in this asymptotic expansion are: +Xωℓ(r) ∼ Cωk +� +1 +r − q2Q2ℓ4 +6r3 +� +ln +� r +rh +��2 ++... +� ++Dωk +� +1− q2Q2ℓ4 +2r2 +� +ln +� r +rh +��2 ++... +� +. +(29) +The second solution gives a mode which is not square in- +tegrable at infinity, so we set Dωk = 0 in this case. The be- +haviour at infinity of a massless and conformally coupled +charged scalar field is thus rather different from that seen in +the neutral case. +3 Criterion for superradiance +We now explore whether superradiance occurs for a charged +scalar field, examining separately the cases where the fre- +quency ω is real or complex. +3.1 Wronskian condition +We first consider the situation in which the frequency ω is +real. In this case, the potential Vωk(r) (16b) is also real and +therefore the Wronskian +Wωk = X∗ +ωk +dXωk +dr∗ +−Xωk +dX∗ +ωk +dr∗ +(30) + +Charge superradiance on charged BTZ black holes +5 +is a constant. Near the horizon, using (19) we find +Wωk = 2i �ω +� +|Aωk|2 −|Bωk|2� +. +(31) +The value of Wωk as r → ∞ depends on the form of the radial +function Xωk(r) in this regime. Consider first the solution +(25) valid when 4µ2 > −1. In this case we have +Wωk = 2i +ℓ2 ℑ(C∗ +ωkDωk) +� +1+4µ2, +(32) +where ℑ denotes the imaginary part. Therefore, if we spec- +ify Dirichlet boundary conditions (for which Dωk = 0) or +Neumann boundary conditions (for which Cωk = 0), equat- +ing (31, 32) gives that |Aωk|2 = |Bωk|2. This means that the +amplitudes of the ingoing and outgoing waves at the hori- +zon are equal and there is no superradiance, generalizing the +result of [38] to the charged case. +When −1 < 4µ2 < 0, Dirichlet and Neumann bound- +ary conditions are not the only possibility, we can also im- +pose Robin boundary conditions for which both Cωk and +Dωk are nonzero. Using the parameterization (26), we have +Cωk = Eωk cosζ and Dωk = Eωk sinζ and the Wronskian +(32) becomes +Wωk = 2i +ℓ2 ℑ +� +|Eωk|2 cosζ sinζ +�� +1+4µ2 = 0. +(33) +We therefore deduce that there are no superradiant modes +having real frequency, even when Robin boundary condi- +tions are applied. +There are two special cases which need to be considered +separately. First, when 4µ2 = −1, the radial function Xωk(r) +has the form (27) as r → ∞, whence +Wωk = 2i +ℓ2 ℑ(C∗ +ωkDωk). +(34) +Our conclusions are however unchanged: (34) vanishes for +Dirichlet, Neumann and Robin boundary conditions and there +is no superradiance. +Finally we have the case 4µ2 = 0, for which the radial +function takes the form (29) as r → ∞, with Dωk = 0 to +ensure square integrability. In this situation the Wronskian +tends to zero as r → ∞, so that amplitudes of the ingoing +and outgoing waves at the horizon are again equal and there +is no superradiance. +The inclusion of a scalar field charge has made no dif- +ference to the analysis of superradiance using the Wron- +skian. In particular, we find that there is no superradiance +for modes having real frequency, irrespective of the bound- +ary conditions. This is in accordance with the results of [41] +for the neutral scalar field. In that case there are superradi- +ant modes when Robin boundary conditions are applied, but +these modes have complex frequencies. +3.2 Energy flux down the horizon +In this subsection, we take an alternative approach to inves- +tigate whether there are superradiant modes having complex +frequency ω. Following [41], we consider the energy flux +down the event horizon due to an ingoing mode. For the re- +mainder of this paper the frequency ω will be complex. +Consider an ingoing mode for which +Xωk(r) ∼ e−i �ωr∗as r∗ → −∞. +(35) +In terms of the ingoing EF coordinates (9), the scalar field +mode (15) takes the form +φωk ∼ Bωk +√rh +exp[−i(ωt + �ωr∗ −kϕ)] += Bωk +√rh +exp[−i(ωv−k�ϕv)] +(36) +where Bωk is a complex constant. In terms of the Killing +vectors ξ = ∂v and χ = ∂v +Ω/ℓ∂�ϕv, where Ω/ℓ is the an- +gular speed of the event horizon, the flux of energy down the +black hole is [41] +FE = +� 2π +0 +d �ϕv rhχµT µ +ν ξ ν = 2πrh T r +t |r=rh +� +1−Ω 2, +(37) +where the stress-energy tensor for the charged scalar field +is given by (14). Evaluating the required components of the +stress-energy tensor (14) gives +FE +F += ℜ(ω) +� +ℜ(ω)− kΩ +ℓ +� ++ℑ(ω)2, +(38) +where +F = 2πrh |Bωk|2 e2vℑ(ω). +(39) +Thus the flux of energy down the horizon due to an ingoing +mode will be positive unless +ℜ(ω) +� +ℜ(ω)− kΩ +ℓ +� ++ℑ(ω)2 < 0. +(40) +This is exactly the same condition for superradiance as in the +neutral case [41]. In particular, for the nonrotating charged +black hole we have +FE,Ω=0 +F += ℜ(ω)2 +ℑ(ω)2 ≥ 0 +(41) +and there is no superradiance for a charged scalar field on a +static, charged BTZ black hole, irrespective of the boundary +conditions applied to the field. For the rotating black hole, +the presence of charge will affect the frequencies ω of the +modes, so in the next section we investigate whether there +are charged scalar field modes for which (40) is satisfied. + +6 +Sebastian Konewko, Elizabeth Winstanley +4 Superradiance on charged rotating BTZ black holes +We now demonstrate the existence of superradiant modes +satisfying the condition (40), using a numerical method. We +restrict attention to the regime −1 < 4µ2 < 0, for which +there is a choice of boundary conditions that can be applied +to the scalar field at infinity. We briefly outline our numer- +ical method before discussing a selection of results for the +flux of energy (38) from superradiant modes. +4.1 Numerical method +We seek to solve the radial equation (16) to find complex +frequencies ω for which the radial function Xωℓ(r) satisfies +ingoing boundary conditions (35) at the horizon and Robin +boundary conditions (26) at infinity. In other words, we are +seeking quasi-normal modes (QNMs). There are many meth- +ods in the literature for the accurate computation of QNM +frequencies (see, for example [54–57] for reviews and [58– +65] for a selection of references concerning QNMs on BTZ +black holes). However, the form of the potential (16b) (in +particular, the presence of the nonanalytic ln +� +r +rh +� +term) hin- +ders implementing these in our situation. Our aims in this +section are rather less ambitious than the high-precision com- +putation of QNM frequencies. Instead, we are looking for +numerical evidence for the existence of superradiant modes, +and some qualitative information about the energy flux (38) +for these modes. With this in mind, we employ a rather naive +direct integration method, which is sufficiently accurate for +our purposes for modes lying in the region (40) for which su- +perradiance is possible. Our computations are implemented +in MATHEMATICA. +Given a complex frequency ω, we impose ingoing bound- +ary conditions (35) on the radial function Xωk(r) at r = rh + +ε, where ε ≪ 1. For r ≫ rh, the function Xωk(r) takes the +form (25) for complex constants Cωk, Dωℓ. We rewrite the +radial equation (16) in terms of the radial coordinate r and a +new dependent variable Yωk(r) = r +1 +2 (1−√ +1+4µ2)Xωk(r). We +numerically integrate this new radial equation from r = rh + +ε to r = rmax, where rmax ≫ rh. The value of Dωk can be +found as the limit of Yωk(r) as r → ∞. The value of Cωk is +found from the limit of r1+√ +1+4µ2Y ′ +ωk(r) as r → ∞. +For a general frequency ω, the constants Cωk and Dωk +thus found will be complex. To apply Robin boundary con- +ditions (26), we require the ratio Dωk/Cωk to be real. Ac- +cording to (40), superradiant modes can exist only for fre- +quencies have a real part ℜ(ω) satisfying 0 < ℜ(ω) < kΩ/ℓ. +We therefore consider only frequencies whose real parts lie +in this interval. For fixed ℜ(ω), we use MATHEMATICA’s in- +built root-finding command FindRoot to find the value of +ℑ(ω) for which the imaginary part of Dωk/Cωk vanishes. +We then determine the parameter ζ (26) governing the Robin +boundary conditions from +ζ = arctan +�Dωk +Cωk +� +, +(42) +and the energy flux FE/F using (38). We mostly take a +branch of the arctan function such that ζ ∈ [0,π); however +in part of our analysis it will be helpful to consider instead +ζ ∈ (− π +2 , π +2 ]. +This direct integration method has a number of draw- +backs. First, we require the numerical integration of the ra- +dial equation to very high precision in order to extract the +constants Cωk and Dωk to a reasonable accuracy. Second, +we find that the method yields satisfactory results only when +either the scalar field charge vanishes (q = 0) or for reason- +ably large values of at least one of the charges Q, q. In or- +der to obtain good results for a wider range of values of the +charges, and for nonsuperradiant modes, a more sophisti- +cated method would be needed to find the QNM frequen- +cies. However, our method is sufficiently accurate to give a +selection of superradiant modes which enables us to quali- +tatively explore the effect of black hole and/or scalar field +charge on superradiance. +4.2 Numerical results +In Figures 1 and 2 we present our numerical results demon- +strating the existence of superradiant modes for a charged +scalar field on a charged, rotating BTZ black hole. To aid +comparison with the results for a neutral scalar field in [41], +we set µ2 = −0.65, M = 16 (which corresponds to Figure +2 in [41]) and consider only modes with azimuthal quan- +tum number k = 1. In all our plots we show the energy flux +FE/F (38) as a function of the Robin parameter ζ (42). A +negative energy flux corresponds to a superradiant mode. +We begin by setting the scalar field charge q = 0, see +Figure 1(a), where we have also fixed the rotation parame- +ter Ω = 0.6 and varied the black hole charge parameter Q. +When Q = 0, we reproduce the results in [41], which pro- +vides verification of our numerical method. QNM for Q ̸= 0 +and q = 0 were studied in [49], although the focus in that +work was the mode frequencies rather than the energy flux, +as is the case here. As the black hole charge parameter Q in- +creases, we find that the energy flux FE/F in superradiant +modes decreases in magnitude, and that superradiant modes +exist for smaller values of the Robin parameter ζ. Superra- +diant effects are small in this situation: there is only a nar- +row interval of values of ζ for which there are superradiant +modes, and the resulting fluxes of energy have small magni- +tudes. We also note that all values of the Robin parameter ζ +for the superradiant modes when q = 0 are greater than π/2, +the value corresponding to Neumann boundary conditions. +We now examine the effect of the scalar field charge on +the superradiant energy flux. In Figure 1(b) we set the scalar + +Charge superradiance on charged BTZ black holes +7 +0.46 +0.48 +0.50 +0.52 +0.54 +0.56 +0.58 +0.60 +ζ/π +−0.010 +−0.005 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +FE/F +Energy Flux (q=0) +Q = 0.0 +Q = 0.1 +Q = 0.2 +Q = 0.5 +Q = 1.0 +Q = 2.0 +Q = 3.0 +Q = 4.0 +Q = 5.0 +(a) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ζ/π +−0.08 +−0.07 +−0.06 +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0.00 +FE/F +Energy Flux (q=1) +Q = 1.0 +Q = 1.5 +Q = 2.0 +Q = 2.5 +Q = 3.0 +Q = 4.0 +Q = 5.0 +(b) +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +ζ/π +−0.08 +−0.07 +−0.06 +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0.00 +FE/F +Energy Flux (Q=2) +q = 0.7 +q = 1.0 +q = 1.2 +q = 1.5 +q = 1.7 +(c) +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +ζ/π +−0.14 +−0.12 +−0.10 +−0.08 +−0.06 +−0.04 +−0.02 +0.00 +FE/F +Energy Flux (Q=3, q=1) +Ω = 0.2 +Ω = 0.4 +Ω = 0.6 +Ω = 0.8 +(d) +Fig. 1: Energy flux FE/F (38) for superradiant charged scalar field modes as a function of the Robin parameter ζ (42). A +negative energy flux corresponds to superradiance. We have fixed µ2 = −0.65, M = 16 and k = 1. In plots (a), (b) and (c) +the rotation parameter Ω = 0.6, and in (d) a selection of values of Ω are considered. The values of the black hole charge +parameter Q and scalar field charge q are as given in the legends. +field charge q = 1 and consider a selection of values of the +black hole charge parameter Q, again for fixed rotation pa- +rameter Ω = 0.6. For fixed Q, superradiant modes exist only +in a narrow interval of values of the Robin parameter ζ, with +the width of this interval decreasing as Q increases. As Q +varies, the possible values of ζ for which there are super- +radiant modes is much broader than in the case q = 0, and +we find superradiant modes with boundary conditions close +to Dirichlet (ζ = 0) when Q is large. The magnitude of the +energy flux FE/F for the superradiant modes with q = 1 in +Figure 1(b) is roughly an order of magnitude greater than +those in Figure 1(a) for q = 0, indicating a significant en- +hancement in superradiance due to the scalar field charge. +The values of the Robin parameter ζ for the superradiant +modes in Figure 1(b) mostly lie between the Dirichlet value +ζ = 0 and that for Neumann boundary conditions ζ = π +2 . We +see that there is a superradiant mode with ζ = π +2 (Neumann +boundary conditions) for Q = 1. This does not contradict our +analysis in Section 3.1, as this mode will have a complex +frequency. +In Figure 1(c), with the rotation parameter again set to +be Ω = 0.6, we fix the black hole charge parameter Q = 2 +and vary the scalar field charge q. The interval of values of +the Robin parameter ζ for which superradiant modes exists +shrinks as the scalar field charge q increases, and moves to +smaller values of ζ. At the same time, the maximum magni- + +8 +Sebastian Konewko, Elizabeth Winstanley +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ζ/π +−0.08 +−0.07 +−0.06 +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0.00 +0.01 +FE/F +Energy Flux (Q=5) +q = 1.8 +q = 1.9 +q = 2.0 +q = 2.1 +q = 2.2 +q = 2.5 +(a) +−0.020 +−0.015 +−0.010 +−0.005 +0.000 +0.005 +0.010 +ζ/π +−0.08 +−0.07 +−0.06 +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0.00 +0.01 +FE/F +Energy Flux (Q=5) +q = 1.2 +q = 1.3 +q = 1.5 +q = 1.7 +(b) +Fig. 2: Energy flux FE/F (38) for superradiant charged scalar field modes as a function of the Robin parameter ζ (42). A +negative energy flux corresponds to superradiance. We have set µ2 = −0.65, M = 16, Ω = 0.6 and k = 1. The black hole +charge parameter is fixed to be Q = 5, and a selection of values of the scalar field charge q are considered. +tude of the superradiant energy flux FE/F increases as q in- +creases. Combining the results in Figures 1(b,c), we deduce +that increasing either the black hole or scalar field charge +gives a narrower interval of values of ζ yielding superradi- +ant modes, with that interval being closer to Dirichlet bound- +ary conditions. The maximum magnitude of the superradiant +energy flux generally increases as either q or Q increases. +So far we have studied superradiant modes with the ro- +tation parameter Ω fixed. In Figure 1(d) we fix the black +hole charge parameter Q = 3 and scalar field charge q = 1, +and consider a selection of values of Ω. Increasing the ro- +tation parameter results in large increases in both the width +of the interval of values of ζ for which there are superradi- +ant modes, and the maximum magnitude of the superradiant +energy flux. These effects are significantly larger than those +resulting from changing either the scalar field or black hole +charges. We deduce from this that the most important factor +influencing superradiance is the rotation of the black hole. +We close our discussion of superradiant modes by ex- +ploring, in Figure 2, some results for a large value of the +black hole charge parameter, namely Q = 5, again with the +rotation parameter Ω = 0.6. Here we find behaviour which +is qualitatively different from that shown in Figure 1. For +larger values of the scalar field charge q ≥ 2.2 (see the left- +hand plot), we find a narrow interval of values of the Robin +parameter ζ which yield superradiant modes, and further- +more these values of ζ lie close to the Dirichlet value ζ = 0, +similarly to the results in Figure 1(c) for Q = 2. However, as +q decreases (again in the left-hand-plot), the interval of val- +ues of ζ for which there are superradiant modes widens sig- +nificantly, and comprises the majority of the interval 0 < ζ < +π. In particular, for q = 2.0, 2.1 and 2.2 we find superradiant +modes for which ζ = π +2 , corresponding to Neumann bound- +ary conditions. For 1.8 < q < 2.2, the left-hand plot shows +that the value of the Robin parameter ζ at which the en- +ergy flux has its maximum magnitude shifts from a location +close to ζ = 0 to a location close to ζ = π. On decreasing +q further, for fixed q we find two “branches” of superradiant +modes, one in a neighbourhood of ζ = π and one in a neigh- +bourhood of ζ = 0. These superradiant modes are depicted +in the right-hand plot in Figure 2, where we have chosen a +branch of the arctan function in (42) for which − π +2 < ζ < π +2 +instead of 0 < ζ < π as used elsewhere. In the right-hand +plot, we can see that for q = 1.5 and 1.7, there are super- +radiant modes for which ζ = 0, corresponding to Dirichlet +boundary conditions. +We therefore find that, unlike the situation for a neutral +scalar field, for a charged scalar field on a charged BTZ +black hole background, at least for a small subset of the +(Q,q)-parameter space, there are superradiant modes with +complex frequencies satisfying either Dirichlet or Neumann +boundary conditions at infinity. +5 Conclusions +In this paper we have explored the effect of black hole and +scalar field charge on superradiance on three-dimensional +BTZ black holes. We considered separable mode solutions +of the charged scalar field equation on the charged general- +ization of the rotating BTZ black hole metric [48]. Working +in the frequency domain, we find, as in the neutral scalar + +Charge superradiance on charged BTZ black holes +9 +field case, that modes with real frequency do not exhibit su- +perradiance. For modes with complex frequencies, follow- +ing [41], we define superradiance as occurring if the ingoing +flux of energy due to an ingoing scalar field mode is nega- +tive (in other words, if an ingoing mode results in an out- +going flux of energy it is said to be superradiant). We find +that it is necessary for the black hole to be rotating in order +for superradiance to occur. Therefore, there is no charge su- +perradiance for nonrotating BTZ black holes, unlike the sit- +uation for four-dimensional, Reissner-Nordstr¨om-adS black +holes. Superradiant modes lie in a region of the complex fre- +quency plane satisfying the inequality (40); however only a +small proportion of modes in this region are superradiant. +We use a simple numerical method, applicable to modes +in the superradiant regime, to demonstrate the existence of +superradiant charged scalar field modes when the black hole +charge is nonzero. We have not performed an exhaustive +search of the parameter space, but instead considered a sam- +ple of black holes. The presence of black hole and scalar +field charges results in a flux of outgoing energy which is +about an order of magnitude larger than in the uncharged +case. However, the dominant parameter affecting the mag- +nitude of the outgoing energy flux is the black hole rotation +rather than the charges. +We have also examined the range of boundary condi- +tions satisfied by the superradiant modes at infinity. These +boundary conditions are labelled by the Robin parameter +ζ. For most fixed values of the black hole and scalar field +charge, we find that superradiant modes correspond to val- +ues of ζ lying in a narrow interval. For a large black hole +charge parameter Q = 5, we have found some values of the +scalar field charge q ∼ 2 where the interval of values of ζ +is considerably wider than in the generic case. We have also +found some values of Q and q for which there are superra- +diant modes satisfying either Dirichlet or Neumann bound- +ary conditions, which are absent in the neutral scalar field +case. Superradiant modes satisfying Dirichlet boundary con- +ditions have also been found for charged scalar perturbations +of a Coulomb-like adS black hole in nonlinear electrody- +namics in three dimensions [66]. +Our numerical method has limited us to exploring a com- +paratively small region of the parameter space. In particular, +we find reliable numerical results only when at least one of +the scalar field charge q or black hole charge parameter Q is +comparatively large. We have also fixed the black hole mass +parameter M and azimuthal quantum number k, as well as +the scalar field mass m. Furthermore, we have restricted our +attention to a charged scalar field minimally coupled to the +spacetime curvature. With a more sophisticated numerical +method, it would be interesting to probe the parameter space +more widely. +In this paper we have studied a classical charged scalar +field. A natural extension of our work would be to con- +sider a quantum charged scalar field. The study of a mass- +less, conformally coupled quantum scalar field on a neu- +tral BTZ black hole is comparatively straightforward due to +the construction of the BTZ metric by identifying points in +adS space-time [35, 36]. In particular, when either Dirichlet +or Neumann boundary conditions are applied, the maximal +symmetry of adS can be exploited to enable the computation +of the renormalized expectation value of the stress-energy +tensor using the method of images [67, 68], see also [69– +71]. This method is not applicable when Robin boundary +conditions are applied as these break the maximal symmetry +of the underlying adS geometry [72–74]. The ground state +Green’s function for a neutral scalar field with Robin bound- +ary conditions applied is constructed in [50] using a mode +sum decomposition. +It would be interesting to explore what effect the su- +perradiant modes we have found in this paper have on the +definition of quantum states for a charged scalar field on a +charged BTZ black hole. On four-dimensional asymptoti- +cally flat black holes, the presence of superradiant modes +introduces subtleties in the construction of quantum states, +both in the rotating [75–79] and charged scenarios [80], and +one might anticipate similar challenges on a BTZ black hole. +Neutral scalar field modes on a neutral BTZ black hole are +given by hypergeometric functions which simplifies the anal- +ysis [41, 50]. For a charged scalar field on a charged BTZ +background there appears to be no simple closed-form ex- +pression for the modes, which will complicate the construc- +tion. We therefore postpone further consideration of the quan- +tum charged scalar field to future work. +Acknowledgements We thank Sam Dolan for helpful discussions re- +garding the numerical computation of QNM frequencies. The work of +E.W. is supported by the Lancaster-Manchester-Sheffield Consortium +for Fundamental Physics under STFC grant ST/T001038/1. This re- +search has also received funding from the European Union’s Horizon +2020 research and innovation program under the H2020-MSCA-RISE- +2017 Grant No. FunFiCO-777740. +Conflict of interest +The authors declare that they have no conflict of interest. +References +1. R. Brito, V. Cardoso, P. Pani, Lect. Notes Phys. 906, +pp.1 (2015). doi: 10.1007/978-3-319-19000-6 +2. C.W. Misner, Phys. Rev. Lett. 28, 994 (1972). +doi: +10.1103/PhysRevLett.28.994 +3. W.H. Press, S.A. Teukolsky, Nature 238, 211 (1972). +doi: 10.1038/238211a0 +4. S. 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Rev. +D 106(12), 125013 (2022). +doi: 10.1103/Phys- +RevD.106.125013 + diff --git a/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/load_file.txt b/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8069e86508184601eed11b1fb366ae0d9137914b --- /dev/null +++ b/pdAzT4oBgHgl3EQfOvsC/content/tmp_files/load_file.txt @@ -0,0 +1,1176 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf,len=1175 +page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (will be inserted by the editor) Charge superradiance on charged BTZ black holes Sebastian Konewko · Elizabeth Winstanley Abstract We study superradiance for a charged scalar field subject to Robin (mixed) boundary conditions on a charged BTZ black hole background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Scalar field modes having a real frequency do not exhibit superradiance, independent of the boundary conditions applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For scalar field modes with a complex frequency, irrespective of the boundary condi- tions, no charge superradiance occurs if the black hole is static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We demonstrate the existence of superradiant modes with complex frequencies for a charged and rotating BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Most of the superradiant modes we find satisfy Robin boundary conditions, but there are also superradiant modes with complex frequencies satisfying Dirichlet and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We explore the effect of the black hole and scalar field charge on the outgoing energy flux of these superradiant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 1 Introduction In the scattering of waves incident on a black hole, super- radiance occurs if the reflected wave has greater amplitude than the incident wave [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For example, low-frequency bos- onic waves scattered by a rotating Kerr black hole exhibit su- perradiance [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This phenomenon is the wave analogue of the Penrose process for particles [5], with the wave ex- tracting rotational energy from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' An analogous effect (charge superradiance) occurs for charged scalar field waves scattered by a charged Reissner-Nordstr¨om black hole S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Konewko Independent Researcher E-mail: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='konewko@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='com E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Winstanley Consortium for Fundamental Physics, School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' S3 7RH United Kingdom E-mail: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='Winstanley@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='uk [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this case it is electrostatic rather than rotational energy which is extracted from the black hole by the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' If one considers asymptotically anti-de Sitter (adS) rather than asymptotically flat black holes, superradiance persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The adS boundary acts like a mirror, reflecting the amplified wave back onto the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For sufficiently small rotat- ing and/or charged asymptotically-adS black holes in four or more space-time dimensions, this process can lead to a superradiant instability (see, for example, [11–34] for a se- lection of papers from the extensive literature on this topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The situation for asymptotically-adS black holes in three space-time dimensions is subtly different from that in four or more space-time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, one can con- sider the famous BTZ metric [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Since null infinity is a time-like hypersurface in adS, boundary conditions have to be applied to an incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For a neutral scalar field, ap- plying the simplest boundary conditions, namely Dirichlet boundary conditions, superradiance is absent [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Super- radiance also does not occur for a fermion field vanishing on the boundary [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, one can impose more gen- eral (Neumann or Robin) boundary conditions on the scalar field [40], and, for at least some Robin boundary conditions, superradiant modes exist [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The impact of Robin bound- ary conditions on superradiance has also been considered on four-dimensional Kerr-adS black holes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this paper we examine whether superradiance exists for a charged scalar field subject to Robin (mixed) boundary conditions on a charged analogue of the BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We begin, in Section 2, by reviewing the black hole metric and separable solutions of the charged scalar field equation on this spacetime background, paying particular attention to the boundary conditions satisfied by the field far from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In Section 3 we study the possibility of superradiance using two approaches: firstly a Wronskian condition which is valid for waves having real frequency, and secondly, fol- lowing [41], considering the energy flux down the horizon of an ingoing wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, an ingoing superradiant arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01169v1 [hep-th] 3 Jan 2023 2 Sebastian Konewko, Elizabeth Winstanley wave will have an outgoing energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Using the Wron- skian, we find that there is no superradiance for field modes having real frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' If the frequency is complex, by con- sidering the energy flux, we show that charge superradiance is absent if the black hole is nonrotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This leaves open the possibility of superradiance for charged, rotating black holes, which is studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Using a simple numeri- cal method, valid for frequencies in the superradiant regime, we find charged superradiant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We explore the effect of increasing either the black hole or scalar field charge on the energy fluxes of these superradiant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Finally our conclusions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 2 Charged scalar field on a charged BTZ black hole 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 Charged BTZ black holes The neutral BTZ black hole [35–37] is a solution of the three-dimensional Einstein equations with a negative cos- mological constant Λ = −ℓ−2, having metric ds2 = −N0(r)dt2 +N0(r)−1 dr2 +r2 � dϕ +Nϕ 0 (r)dt �2 (1a) where N0(r) = r2 ℓ2 −M + J2 4r2 , Nϕ 0 (r) = − J 2r2 , (1b) with M the mass and J the angular momentum of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In the static case (J = 0), the black hole acquires an elec- tric charge Q by introducing the electromagnetic potential Aµ = A0δt µ, where A0 = −Qln � r r0 � , (2a) and an arbitrary length scale r0 has been introduced to render the argument of the logarithm dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The metric for a static charged BTZ black hole is then [35–37] ds2 = −N(r)dt2 +N(r)−1 dr2 +r2 dϕ2 (2b) and the lapse function takes the form N(r) = r2 ℓ2 −M − Q2 2 ln � r r0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (2c) The generalization of the rotating BTZ black hole to in- clude an electric charge is far from straightforward [37, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Various three-dimensional, charged, rotating black holes are presented in [44–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this paper we consider the follow- ing charged generalization of the BTZ metric [48]: ds2 = −N(r) r2 R(r)2 dt2 +N(r)−1 dr2 +R(r)2 [dϕ +Nϕ(r)dt]2 (3a) where N(r) is the same as in the static case (2c) and the other functions appearing in the metric are: R(r)2 = r2 + Ω 2ℓ2 1−Ω 2 � M + Q2 2 ln � r r0 �� , Nϕ(r) = − Ωℓ (1−Ω 2)R(r)2 � M + Q2 2 ln � r r0 �� , (3b) where M, Q and Ω ∈ [0,1) are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' When Ω > 0, the electromagnetic potential Aµ acquires a nonzero magnetic part and is given by: Aµdxµ = − Q √ 1−Ω 2 [dt −Ωℓdϕ]ln � r r0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (3c) The mass � M, angular momentum �J and charge �Q of the black hole are given in terms of the parameters M, Q, and Ω as follows [48]: � M = 1 1−Ω 2 � M � 1+Ω 2� − 1 2Q2Ω 2 � , �J = 2Ω 1−Ω 2 � M − 1 4Q2 � , �Q = Q √ 1−Ω 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (4) Unlike the neutral BTZ black hole, the metric (3a) cannot be obtained by identifying points in three-dimensional adS space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, the scalar curvature R is not con- stant: R = Q2 2r2 − 6 ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (5) In the limit Ω → 0, the metric and gauge field potential (3) reduce to those in the static case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' If we set Q = 0, the metric (3a) does not reduce to the original rotating BTZ metric (1a) in (t,r,ϕ) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, using R as the radial coordinate, when Q = 0 the metric (3a) becomes ds2 = −�N(R)dt2 + �N(R)−1 dR2 +R2 [dϕ +Nϕ(r)dt]2 (6a) where we have defined the function �N(R) = N(r) r2 R2 = R2 ℓ2 − M � 1+Ω 2� 1−Ω 2 + M2Ω 2ℓ2 (1−Ω 2)2 R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (6b) We therefore have a metric of the form (1a) with mass � M = M � 1+Ω 2� / � 1−Ω 2�2 and angular momentum �J = 2MΩℓ/ � 1−Ω 2� , in accordance with (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The horizons of the black hole are located at those val- ues of the radial coordinate r for which N(r) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' If M < Q2 [1−2ln(Qℓ/2r0) ]/2 there is a naked singularity at r = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' we do not consider this possibility further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For M > Q2 [1−2ln(Qℓ/2r0)]/2 there is an event horizon at r = rh, the largest zero of N(r) and an inner horizon at the smaller positive zero of N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' These two horizons coincide Charge superradiance on charged BTZ black holes 3 when M = Q2 [1−2ln(Qℓ/2r0)]/2 and in this case we have an extremal black hole [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this paper we focus on the case where the black hole is nonextremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' By making a gauge transformation of the form Aµ → Aµ +∂µχ, χ = Q √ 1−Ω 2 (t −Ωℓϕ)ln �rh r0 � , (7) we may set r0 = rh without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We then find, by considering the zeros of (2c), that M = r2 h ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (8) At the horizon, we have R(rh)2 = r2 h/ � 1−Ω 2� and Nϕ(rh) = −Ω/ℓ, so that Ω/ℓ is the angular speed with which the event horizon rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In our analysis of superradiance, we will be interested in the flux of energy down the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For this analysis, we will require suitable coordinates which are regular across the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We will employ ingo- ing Eddington-Finkelstein (EF) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' First we define an ingoing null coordinate v by dv = dt + 1 r R(r) N(r) dr, (9a) and a new angular coordinate �ϕv by d �ϕv = dϕ − R(r) r Nϕ(r) N(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (9b) Then the coordinates (v,r, �ϕv) are ingoing EF coordinates, in terms of which the metric (3a) becomes ds2 = −N(r) r2 R(r)2 dv2 + 2r R(r) dvdr +R(r)2 [d �ϕv +Nϕ(r)dv]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (10) The metric (10) is regular when r = rh and N(r) = 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Near the horizon, the ingoing EF coordinates take the form v = t + r∗ √ 1−Ω 2 , �ϕv = ϕ − Ωr∗ ℓ √ 1−Ω 2 (11) where r∗ is the usual tortoise coordinate, defined by dr∗ dr = 1 N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 Charged scalar field We consider a scalar field Φ with charge q and mass m prop- agating on the rotating charged black hole (3a), and satisfy- ing the charged scalar field equation � DµDµ −m2� Φ = 0, (13) where Dµ = ∇µ − iqAµ is the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We as- sume that the scalar field is minimally coupled to the ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The stress-energy tensor for the charged scalar field is Tµν = ℜ �� DµΦ �∗ DνΦ − 1 2gµνgρσ � DρΦ �∗ DσΦ −1 2m2gµνΦ∗Φ � , (14) where ℜ is the real part and a star is used to denote complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Mode solutions of the scalar field equation (13) take the form Φωk(t,r,ϕ) = 1 √r e−iωt eikϕ Xωk(r), (15) where ω is the frequency of the wave (which may be com- plex) and k ∈ Z is the azimuthal quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In terms of the tortoise coordinate r∗ (12), the radial function Xωk(r) satisfies the equation � d2 dr2∗ +Vωk(r) � Xωk(r) = 0 (16a) where the potential Vωk(r) takes the form Vωk(r) = �ω −kΩℓ−1 √ 1−Ω 2 −qQln � r rh ��2 −m2N(r) + N(r)2 4r2 − N′(r)N(r) 2r − � ωΩ −kℓ−1�2 ℓ2N(r) (1−Ω 2)r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (16b) As r → rh and the event horizon is approached, we have r∗ → −∞ and Vωk(r) → �ω2 (17) where we have defined �ω = ω −kΩℓ−1 √ 1−Ω 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (18) Therefore, near the horizon, the radial function Xωk(r) takes the form Xωk(r) ∼ Aωkei �ωr∗ +Bωke−i �ωr∗ (19) where Aωk and Bωk are complex constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The frequency of the wave has effectively been shifted due to the rotation of 4 Sebastian Konewko, Elizabeth Winstanley the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The fact that �ω does not depend on the charge stems from our choice of gauge, in that the electromagnetic gauge potential (3c) vanishes at the horizon since we have taken r0 = rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Far from the black hole, as r → ∞, the leading-order be- haviour of the potential (16b) is, in general, Vωk(r) ∼ − � m2 + 3 4ℓ2 � r2 ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (20) This leading-order behaviour is the same as for the neutral scalar field, and does not depend on the frequency ω or the azimuthal quantum number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this regime the tortoise co- ordinate has the following form: r∗ ∼ −ℓ2 r , (21) yielding the equidimensional differential equation � d2 dr2∗ − µ2 r2∗ � Xωk(r) = 0 (22) where µ2 is a constant given by µ2 = m2ℓ2 + 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (23) Let us assume for the moment that µ2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The solutions of (22) are Xωk ∼ rp ∗ ∼ r−p, where p = 1 2 � 1± � 1+4µ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (24) For 4µ2 > −1, the values of p are real and Xωk(r) ∼ Cωkr− 1 2 (1+√ 1+4µ2) +Dωkr− 1 2 (1−√ 1+4µ2) (25) for complex constants Cωk, Dωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The second term gives a ra- dial function which is not square integrable at infinity when 4µ2 > 0 and therefore we set Dωk = 0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this situation there is no choice of boundary conditions which can be imposed on the scalar field at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For −1 < 4µ2 < 0, both solutions in (25) are square- integrable, resulting in some freedom in the choice of bound- ary conditions at infinity [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The solution with Dωk = 0 satisfies Dirichlet boundary conditions, while, following [50], we define Neumann boundary conditions to be such that Cωk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' If both Cωℓ and Dωℓ are nonzero, then we have Robin (mixed) boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this situation we write the solution (25) in the form [41, 50] Xωk(r) ∼ Eωk � r− 1 2 (1+√ 1+4µ2) cosζ +r− 1 2 (1−√ 1+4µ2) sinζ � (26) where Eωk is a complex constant and the real angle ζ (which we term the “Robin parameter”) can be taken to lie in the interval 0 ≤ ζ < π (we could equally well take ζ ∈ (− π 2 , π 2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Setting ζ = 0 yields Dirichlet boundary conditions, while ζ = π 2 corresponds to Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' When 4µ2 = −1, we have Xωk(r) ∼ Cωkr− 1 2 +Dωkr− 1 2 ln � r rh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (27) Both solutions are square-integrable in this case, so again we have a choice of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For 4µ2 < −1, the exponent p (24) is complex and Xωk(r) is oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Once again both linearly independent solutions of the radial equa- tion are square-integrable at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, these values of µ2 violate the Breitenlohner-Freedman bound [52, 53] and therefore we do not consider them further in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The above discussion of the boundary conditions at in- finity is valid only when µ2 ̸= 0, in which case the behaviour of the charged scalar field at infinity is identical to that for a neutral scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In the special case µ2 = 0 the leading order behaviour of the potential (16b) is no longer (20), but instead we have, as r → ∞, Vωk(r) ∼ q2Q2 � ln � r rh ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (28) In this case it is not possible to solve the asymptotic form of the radial equation exactly in terms of elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, it is possible to perform an asymptotic expansion for the radial function Xωk(r) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The first couple of terms in this asymptotic expansion are: Xωℓ(r) ∼ Cωk � 1 r − q2Q2ℓ4 6r3 � ln � r rh ��2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' � +Dωk � 1− q2Q2ℓ4 2r2 � ln � r rh ��2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (29) The second solution gives a mode which is not square in- tegrable at infinity, so we set Dωk = 0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The be- haviour at infinity of a massless and conformally coupled charged scalar field is thus rather different from that seen in the neutral case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 3 Criterion for superradiance We now explore whether superradiance occurs for a charged scalar field, examining separately the cases where the fre- quency ω is real or complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 Wronskian condition We first consider the situation in which the frequency ω is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this case, the potential Vωk(r) (16b) is also real and therefore the Wronskian Wωk = X∗ ωk dXωk dr∗ −Xωk dX∗ ωk dr∗ (30) Charge superradiance on charged BTZ black holes 5 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Near the horizon, using (19) we find Wωk = 2i �ω � |Aωk|2 −|Bωk|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (31) The value of Wωk as r → ∞ depends on the form of the radial function Xωk(r) in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Consider first the solution (25) valid when 4µ2 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this case we have Wωk = 2i ℓ2 ℑ(C∗ ωkDωk) � 1+4µ2, (32) where ℑ denotes the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Therefore, if we spec- ify Dirichlet boundary conditions (for which Dωk = 0) or Neumann boundary conditions (for which Cωk = 0), equat- ing (31, 32) gives that |Aωk|2 = |Bωk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This means that the amplitudes of the ingoing and outgoing waves at the hori- zon are equal and there is no superradiance, generalizing the result of [38] to the charged case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' When −1 < 4µ2 < 0, Dirichlet and Neumann bound- ary conditions are not the only possibility, we can also im- pose Robin boundary conditions for which both Cωk and Dωk are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Using the parameterization (26), we have Cωk = Eωk cosζ and Dωk = Eωk sinζ and the Wronskian (32) becomes Wωk = 2i ℓ2 ℑ � |Eωk|2 cosζ sinζ �� 1+4µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (33) We therefore deduce that there are no superradiant modes having real frequency, even when Robin boundary condi- tions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' There are two special cases which need to be considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' First, when 4µ2 = −1, the radial function Xωk(r) has the form (27) as r → ∞, whence Wωk = 2i ℓ2 ℑ(C∗ ωkDωk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (34) Our conclusions are however unchanged: (34) vanishes for Dirichlet, Neumann and Robin boundary conditions and there is no superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Finally we have the case 4µ2 = 0, for which the radial function takes the form (29) as r → ∞, with Dωk = 0 to ensure square integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this situation the Wronskian tends to zero as r → ∞, so that amplitudes of the ingoing and outgoing waves at the horizon are again equal and there is no superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The inclusion of a scalar field charge has made no dif- ference to the analysis of superradiance using the Wron- skian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, we find that there is no superradiance for modes having real frequency, irrespective of the bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This is in accordance with the results of [41] for the neutral scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In that case there are superradi- ant modes when Robin boundary conditions are applied, but these modes have complex frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 Energy flux down the horizon In this subsection, we take an alternative approach to inves- tigate whether there are superradiant modes having complex frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Following [41], we consider the energy flux down the event horizon due to an ingoing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For the re- mainder of this paper the frequency ω will be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Consider an ingoing mode for which Xωk(r) ∼ e−i �ωr∗as r∗ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (35) In terms of the ingoing EF coordinates (9), the scalar field mode (15) takes the form φωk ∼ Bωk √rh exp[−i(ωt + �ωr∗ −kϕ)] = Bωk √rh exp[−i(ωv−k�ϕv)] (36) where Bωk is a complex constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In terms of the Killing vectors ξ = ∂v and χ = ∂v +Ω/ℓ∂�ϕv, where Ω/ℓ is the an- gular speed of the event horizon, the flux of energy down the black hole is [41] FE = � 2π 0 d �ϕv rhχµT µ ν ξ ν = 2πrh T r t |r=rh � 1−Ω 2, (37) where the stress-energy tensor for the charged scalar field is given by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Evaluating the required components of the stress-energy tensor (14) gives FE F = ℜ(ω) � ℜ(ω)− kΩ ℓ � +ℑ(ω)2, (38) where F = 2πrh |Bωk|2 e2vℑ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (39) Thus the flux of energy down the horizon due to an ingoing mode will be positive unless ℜ(ω) � ℜ(ω)− kΩ ℓ � +ℑ(ω)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' (40) This is exactly the same condition for superradiance as in the neutral case [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, for the nonrotating charged black hole we have FE,Ω=0 F = ℜ(ω)2 +ℑ(ω)2 ≥ 0 (41) and there is no superradiance for a charged scalar field on a static, charged BTZ black hole, irrespective of the boundary conditions applied to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For the rotating black hole, the presence of charge will affect the frequencies ω of the modes, so in the next section we investigate whether there are charged scalar field modes for which (40) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 6 Sebastian Konewko, Elizabeth Winstanley 4 Superradiance on charged rotating BTZ black holes We now demonstrate the existence of superradiant modes satisfying the condition (40), using a numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We restrict attention to the regime −1 < 4µ2 < 0, for which there is a choice of boundary conditions that can be applied to the scalar field at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We briefly outline our numer- ical method before discussing a selection of results for the flux of energy (38) from superradiant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 Numerical method We seek to solve the radial equation (16) to find complex frequencies ω for which the radial function Xωℓ(r) satisfies ingoing boundary conditions (35) at the horizon and Robin boundary conditions (26) at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In other words, we are seeking quasi-normal modes (QNMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' There are many meth- ods in the literature for the accurate computation of QNM frequencies (see, for example [54–57] for reviews and [58– 65] for a selection of references concerning QNMs on BTZ black holes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, the form of the potential (16b) (in particular, the presence of the nonanalytic ln � r rh � term) hin- ders implementing these in our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Our aims in this section are rather less ambitious than the high-precision com- putation of QNM frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Instead, we are looking for numerical evidence for the existence of superradiant modes, and some qualitative information about the energy flux (38) for these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' With this in mind, we employ a rather naive direct integration method, which is sufficiently accurate for our purposes for modes lying in the region (40) for which su- perradiance is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Our computations are implemented in MATHEMATICA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Given a complex frequency ω, we impose ingoing bound- ary conditions (35) on the radial function Xωk(r) at r = rh + ε, where ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For r ≫ rh, the function Xωk(r) takes the form (25) for complex constants Cωk, Dωℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We rewrite the radial equation (16) in terms of the radial coordinate r and a new dependent variable Yωk(r) = r 1 2 (1−√ 1+4µ2)Xωk(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We numerically integrate this new radial equation from r = rh + ε to r = rmax, where rmax ≫ rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The value of Dωk can be found as the limit of Yωk(r) as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The value of Cωk is found from the limit of r1+√ 1+4µ2Y ′ ωk(r) as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For a general frequency ω, the constants Cωk and Dωk thus found will be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' To apply Robin boundary con- ditions (26), we require the ratio Dωk/Cωk to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Ac- cording to (40), superradiant modes can exist only for fre- quencies have a real part ℜ(ω) satisfying 0 < ℜ(ω) < kΩ/ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We therefore consider only frequencies whose real parts lie in this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For fixed ℜ(ω), we use MATHEMATICA’s in- built root-finding command FindRoot to find the value of ℑ(ω) for which the imaginary part of Dωk/Cωk vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We then determine the parameter ζ (26) governing the Robin boundary conditions from ζ = arctan �Dωk Cωk � , (42) and the energy flux FE/F using (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We mostly take a branch of the arctan function such that ζ ∈ [0,π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' however in part of our analysis it will be helpful to consider instead ζ ∈ (− π 2 , π 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This direct integration method has a number of draw- backs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' First, we require the numerical integration of the ra- dial equation to very high precision in order to extract the constants Cωk and Dωk to a reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Second, we find that the method yields satisfactory results only when either the scalar field charge vanishes (q = 0) or for reason- ably large values of at least one of the charges Q, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In or- der to obtain good results for a wider range of values of the charges, and for nonsuperradiant modes, a more sophisti- cated method would be needed to find the QNM frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, our method is sufficiently accurate to give a selection of superradiant modes which enables us to quali- tatively explore the effect of black hole and/or scalar field charge on superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 Numerical results In Figures 1 and 2 we present our numerical results demon- strating the existence of superradiant modes for a charged scalar field on a charged, rotating BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' To aid comparison with the results for a neutral scalar field in [41], we set µ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='65, M = 16 (which corresponds to Figure 2 in [41]) and consider only modes with azimuthal quan- tum number k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In all our plots we show the energy flux FE/F (38) as a function of the Robin parameter ζ (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' A negative energy flux corresponds to a superradiant mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We begin by setting the scalar field charge q = 0, see Figure 1(a), where we have also fixed the rotation parame- ter Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6 and varied the black hole charge parameter Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' When Q = 0, we reproduce the results in [41], which pro- vides verification of our numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' QNM for Q ̸= 0 and q = 0 were studied in [49], although the focus in that work was the mode frequencies rather than the energy flux, as is the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' As the black hole charge parameter Q in- creases, we find that the energy flux FE/F in superradiant modes decreases in magnitude, and that superradiant modes exist for smaller values of the Robin parameter ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Superra- diant effects are small in this situation: there is only a nar- row interval of values of ζ for which there are superradiant modes, and the resulting fluxes of energy have small magni- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We also note that all values of the Robin parameter ζ for the superradiant modes when q = 0 are greater than π/2, the value corresponding to Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We now examine the effect of the scalar field charge on the superradiant energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In Figure 1(b) we set the scalar Charge superradiance on charged BTZ black holes 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='60 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='025 FE/F Energy Flux (q=0) Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='00 FE/F Energy Flux (q=1) Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 Q = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 Q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='50 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='00 FE/F Energy Flux (Q=2) q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='7 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='7 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='20 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='00 FE/F Energy Flux (Q=3, q=1) Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='4 Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6 Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='8 (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 1: Energy flux FE/F (38) for superradiant charged scalar field modes as a function of the Robin parameter ζ (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' A negative energy flux corresponds to superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have fixed µ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='65, M = 16 and k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In plots (a), (b) and (c) the rotation parameter Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6, and in (d) a selection of values of Ω are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The values of the black hole charge parameter Q and scalar field charge q are as given in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' field charge q = 1 and consider a selection of values of the black hole charge parameter Q, again for fixed rotation pa- rameter Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For fixed Q, superradiant modes exist only in a narrow interval of values of the Robin parameter ζ, with the width of this interval decreasing as Q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' As Q varies, the possible values of ζ for which there are super- radiant modes is much broader than in the case q = 0, and we find superradiant modes with boundary conditions close to Dirichlet (ζ = 0) when Q is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The magnitude of the energy flux FE/F for the superradiant modes with q = 1 in Figure 1(b) is roughly an order of magnitude greater than those in Figure 1(a) for q = 0, indicating a significant en- hancement in superradiance due to the scalar field charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The values of the Robin parameter ζ for the superradiant modes in Figure 1(b) mostly lie between the Dirichlet value ζ = 0 and that for Neumann boundary conditions ζ = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We see that there is a superradiant mode with ζ = π 2 (Neumann boundary conditions) for Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This does not contradict our analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1, as this mode will have a complex frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In Figure 1(c), with the rotation parameter again set to be Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6, we fix the black hole charge parameter Q = 2 and vary the scalar field charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The interval of values of the Robin parameter ζ for which superradiant modes exists shrinks as the scalar field charge q increases, and moves to smaller values of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' At the same time, the maximum magni- 8 Sebastian Konewko, Elizabeth Winstanley 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 FE/F Energy Flux (Q=5) q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='8 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='9 q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0 q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='020 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='010 ζ/π −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='01 FE/F Energy Flux (Q=5) q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='3 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='7 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 2: Energy flux FE/F (38) for superradiant charged scalar field modes as a function of the Robin parameter ζ (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' A negative energy flux corresponds to superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have set µ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='65, M = 16, Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6 and k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The black hole charge parameter is fixed to be Q = 5, and a selection of values of the scalar field charge q are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' tude of the superradiant energy flux FE/F increases as q in- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Combining the results in Figures 1(b,c), we deduce that increasing either the black hole or scalar field charge gives a narrower interval of values of ζ yielding superradi- ant modes, with that interval being closer to Dirichlet bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The maximum magnitude of the superradiant energy flux generally increases as either q or Q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' So far we have studied superradiant modes with the ro- tation parameter Ω fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In Figure 1(d) we fix the black hole charge parameter Q = 3 and scalar field charge q = 1, and consider a selection of values of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Increasing the ro- tation parameter results in large increases in both the width of the interval of values of ζ for which there are superradi- ant modes, and the maximum magnitude of the superradiant energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' These effects are significantly larger than those resulting from changing either the scalar field or black hole charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We deduce from this that the most important factor influencing superradiance is the rotation of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We close our discussion of superradiant modes by ex- ploring, in Figure 2, some results for a large value of the black hole charge parameter, namely Q = 5, again with the rotation parameter Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Here we find behaviour which is qualitatively different from that shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For larger values of the scalar field charge q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 (see the left- hand plot), we find a narrow interval of values of the Robin parameter ζ which yield superradiant modes, and further- more these values of ζ lie close to the Dirichlet value ζ = 0, similarly to the results in Figure 1(c) for Q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, as q decreases (again in the left-hand-plot), the interval of val- ues of ζ for which there are superradiant modes widens sig- nificantly, and comprises the majority of the interval 0 < ζ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, for q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2 we find superradiant modes for which ζ = π 2 , corresponding to Neumann bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='8 < q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='2, the left-hand plot shows that the value of the Robin parameter ζ at which the en- ergy flux has its maximum magnitude shifts from a location close to ζ = 0 to a location close to ζ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' On decreasing q further, for fixed q we find two “branches” of superradiant modes, one in a neighbourhood of ζ = π and one in a neigh- bourhood of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' These superradiant modes are depicted in the right-hand plot in Figure 2, where we have chosen a branch of the arctan function in (42) for which − π 2 < ζ < π 2 instead of 0 < ζ < π as used elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In the right-hand plot, we can see that for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='7, there are super- radiant modes for which ζ = 0, corresponding to Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We therefore find that, unlike the situation for a neutral scalar field, for a charged scalar field on a charged BTZ black hole background, at least for a small subset of the (Q,q)-parameter space, there are superradiant modes with complex frequencies satisfying either Dirichlet or Neumann boundary conditions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 5 Conclusions In this paper we have explored the effect of black hole and scalar field charge on superradiance on three-dimensional BTZ black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We considered separable mode solutions of the charged scalar field equation on the charged general- ization of the rotating BTZ black hole metric [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Working in the frequency domain, we find, as in the neutral scalar Charge superradiance on charged BTZ black holes 9 field case, that modes with real frequency do not exhibit su- perradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For modes with complex frequencies, follow- ing [41], we define superradiance as occurring if the ingoing flux of energy due to an ingoing scalar field mode is nega- tive (in other words, if an ingoing mode results in an out- going flux of energy it is said to be superradiant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We find that it is necessary for the black hole to be rotating in order for superradiance to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Therefore, there is no charge su- perradiance for nonrotating BTZ black holes, unlike the sit- uation for four-dimensional, Reissner-Nordstr¨om-adS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Superradiant modes lie in a region of the complex fre- quency plane satisfying the inequality (40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' however only a small proportion of modes in this region are superradiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We use a simple numerical method, applicable to modes in the superradiant regime, to demonstrate the existence of superradiant charged scalar field modes when the black hole charge is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have not performed an exhaustive search of the parameter space, but instead considered a sam- ple of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The presence of black hole and scalar field charges results in a flux of outgoing energy which is about an order of magnitude larger than in the uncharged case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' However, the dominant parameter affecting the mag- nitude of the outgoing energy flux is the black hole rotation rather than the charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have also examined the range of boundary condi- tions satisfied by the superradiant modes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' These boundary conditions are labelled by the Robin parameter ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For most fixed values of the black hole and scalar field charge, we find that superradiant modes correspond to val- ues of ζ lying in a narrow interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For a large black hole charge parameter Q = 5, we have found some values of the scalar field charge q ∼ 2 where the interval of values of ζ is considerably wider than in the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have also found some values of Q and q for which there are superra- diant modes satisfying either Dirichlet or Neumann bound- ary conditions, which are absent in the neutral scalar field case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Superradiant modes satisfying Dirichlet boundary con- ditions have also been found for charged scalar perturbations of a Coulomb-like adS black hole in nonlinear electrody- namics in three dimensions [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Our numerical method has limited us to exploring a com- paratively small region of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, we find reliable numerical results only when at least one of the scalar field charge q or black hole charge parameter Q is comparatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We have also fixed the black hole mass parameter M and azimuthal quantum number k, as well as the scalar field mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Furthermore, we have restricted our attention to a charged scalar field minimally coupled to the spacetime curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' With a more sophisticated numerical method, it would be interesting to probe the parameter space more widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In this paper we have studied a classical charged scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' A natural extension of our work would be to con- sider a quantum charged scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The study of a mass- less, conformally coupled quantum scalar field on a neu- tral BTZ black hole is comparatively straightforward due to the construction of the BTZ metric by identifying points in adS space-time [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' In particular, when either Dirichlet or Neumann boundary conditions are applied, the maximal symmetry of adS can be exploited to enable the computation of the renormalized expectation value of the stress-energy tensor using the method of images [67, 68], see also [69– 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This method is not applicable when Robin boundary conditions are applied as these break the maximal symmetry of the underlying adS geometry [72–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The ground state Green’s function for a neutral scalar field with Robin bound- ary conditions applied is constructed in [50] using a mode sum decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' It would be interesting to explore what effect the su- perradiant modes we have found in this paper have on the definition of quantum states for a charged scalar field on a charged BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' On four-dimensional asymptoti- cally flat black holes, the presence of superradiant modes introduces subtleties in the construction of quantum states, both in the rotating [75–79] and charged scenarios [80], and one might anticipate similar challenges on a BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Neutral scalar field modes on a neutral BTZ black hole are given by hypergeometric functions which simplifies the anal- ysis [41, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' For a charged scalar field on a charged BTZ background there appears to be no simple closed-form ex- pression for the modes, which will complicate the construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' We therefore postpone further consideration of the quan- tum charged scalar field to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Acknowledgements We thank Sam Dolan for helpful discussions re- garding the numerical computation of QNM frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' The work of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/T001038/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' This re- search has also received funding from the European Union’s Horizon 2020 research and innovation program under the H2020-MSCA-RISE- 2017 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' FunFiCO-777740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Conflict of interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' References 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Aliev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Delice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' D 79, 024013 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='084003 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Garcia, (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Tang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Kord Zangeneh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Saavedra, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' La Madrid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Leston, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' C 77(11), 807 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1088/0264- 9381/26/16/163001 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Konoplya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Zhidenko, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} 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Nay- lor, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' 2012, 281705 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1155/2012/281705 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1007/s10714-018-2381-5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Singha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Chakraborty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Dadhich, JHEP 06, 028 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Bernar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Winstanley, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' D 106(12), 125013 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='1103/Phys- RevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} +page_content='125013' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdAzT4oBgHgl3EQfOvsC/content/2301.01169v1.pdf'} diff --git a/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/2301.13024v1.pdf.txt b/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/2301.13024v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..21dece5ccb222f98347a643b72620cae72bc9bfb --- /dev/null +++ b/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/2301.13024v1.pdf.txt @@ -0,0 +1,1428 @@ +Heavy quasiparticles and cascades without symmetry breaking in twisted bilayer +graphene +Anushree Datta1,2,3, M.J. Calder´on1, A. Camjayi4, and E. Bascones1∗ +1Instituto de Ciencia de Materiales de Madrid (ICMM). Consejo Superior de Investigaciones Cient´ıficas (CSIC), +Sor Juana In´es de la Cruz 3, 28049 Madrid (Spain), +2 Universit´e de Paris, Laboratoire Mat´eriaux et Ph´enomenes Quantiques, CNRS, F-75013, Paris, France, +3 Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France, +4Ciclo B´asico Com´un, Universidad de Buenos Aires and IFIBA, +Conicet, Pabell´on 1, Ciudad Universitaria, 1428 CABA, Argentina +Twisted bilayer graphene (TBG) exhibits a +plethora of electronic phases. Among the variety +of correlated states, the cascades in the spectro- +scopic properties and in the compressibility hap- +pen in a much larger energy, twist angle and tem- +perature range than other effects, pointing to a +hierarchy of phenomena. In this work, by apply- +ing a self-consistent Dynamical Mean Field The- +ory + Hartree approximation to a multi-orbital +model for TBG, we show that the spectral weight +reorganization associated to the formation of lo- +cal moments and heavy quasiparticles, and not +a symmetry breaking process, is responsible for +the cascade phenonema. Due to the fragile topol- +ogy of TBG, a strong momentum differentiation +is found in the incoherent spectral weight. The +phenomena reproduced here include the cascade +flow of spectral weight, the oscillations of the re- +mote band energies and the asymmetric jumps of +the inverse compressibility. +A large variety of correlated states, including cor- +related insulators, +superconductivity and topological +phases[1–14], have been found in TBG. Among these phe- +nomena, early scanning tunneling microscopy (STM) ex- +periments found a strong doping dependent broadening +of the density of states (DOS) with a spectral weight re- +organization up to several tens meV and minima of the +DOS at the Fermi level at integer fillings[6–8, 15]. This +reorganization was later found to happen in the form of +cascades of spectral weight with resets[10, 11] and band +flattening at the integers[16] and is accompanied by oscil- +lations in the energies of the remote bands[10]. The phe- +nomelogy at the charge neutrality point (CNP) was found +to be different to the one at other integer fillings with the +spectral weight pushed away from low energies[10]. The +cascades are also revealed in scanning electron transistor +(SET) experiments as strong asymmetric jumps of the +local electronic compressibility [17]. +The cascade phenomena have been detected up to +much higher temperatures than the superconductivity, +the correlated insulators and the anomalous Hall effects +[17, 18], and they are also found in a wider range of an- +gles, suggesting that the cascades constitute the parent +state in which these low temperature phenomena emerge. +The cascades have been primarily interpreted in terms of +symmetry breaking states[8, 11, 16, 17] but also as a con- +sequence of strong correlations in the normal state[7, 10]. +The discussion on whether the phenomenology of TBG +can be explained just in terms of ordered states or it is +necessary to take into account the strong modification +of the non-ordered normal state was put forward early +on[1, 7, 8, 19, 20] and it is still unsettled. Advancing in +this field has been hindered by the complexity of min- +imal models in TBG and of the techniques required to +address the role of strong correlations beyond the sym- +metry breaking transitions. +The heavy fermion description for TBG, early intro- +duced in [20], and supported by an analysis of the in- +teractions in TBG [21], is now receiving more attention, +particularly after the publication of Ref.[22]. In the dif- +ferent heavy fermion models for TBG the flat bands of +each valley are spanned by two p+ and p− orbitals cen- +tered at the AA region of the moir´e unit cell, except +close to Γ [20–24], extended Fig. S1b. These AAp or- +bitals, strongly sensitive to electronic interactions, are +coupled to less correlated electrons. In such a system, +a strong spectral weight reorganization is expected even +in the non-ordered normal state[25]. +Studying the ef- +fect of strong local correlations, such as Mott physics +or the formation of heavy quasiparticles, requires tech- +niques like dynamical mean field theory (DMFT)[26, 27]. +However, even if the intra unit cell interaction U among +these AAp orbitals is expected to be responsible for the +unconventional behavior, other interactions in the model +are sizable and cannot be neglected[21], making a com- +plete DMFT treatment out of reach. +Here we use a +self-consistent DMFT+Hartree approximation to study +a multi-orbital model for TBG[21, 24]. +The intra and +inter-orbital interactions U involving the correlated AAp +orbitals within the same unit cell are treated with DMFT +and the other interactions are included at the Hartree +level. Self-consistency is enforced both within the DMFT +loop and between DMFT and Hartree. +We start from a θ = 1.08◦ TBG, Fig. 1a, with the +ratio between the interlayer tunneling at AA and AB +w0/w1 = 0.78 [28, 29]. The low energy band structure +is fitted with an eight orbital model per valley and spin, +adapted from[24]. +Having into account the valley de- +arXiv:2301.13024v1 [cond-mat.str-el] 30 Jan 2023 + +2 +FIG. 1. +Cascades in the spectral weight. (a) Bands of TBG obtained from the continuum model (black) and 8 orbital +model fitting used in the calculations (blue). (b) Color plot of the density of states obtained in the DMFT+Hartree calculations +as a function of the doping and the energy. Doping and energy are respectively measured with respect to the CNP and to the +chemical potential. Cascades of incoherent spectral weight flow from an energy ≈ U=44.5 meV towards the chemical potential. +The cascades are shifted in doping at positive and negative energies with resets in both cases at integer fillings. Oscillations in +the remote bands are clearly distinguished between −2 < ν < 2. The cascades show strong resemblance with STM experiments. +(c) Line cuts of the density of states for selected dopings within the range −0.5 < ν < 2 with primarily a three peak structure: +quasiparticle peak around zero energy and broader Hubbard bands. The Hubbard bands shift with electronic filling and merge +with the quasiparticle peak at certain dopings. Curves are shifted vertically for clarity. Notice that this is the total density of +states, including the whole moir´e unit cell. For a DOS with a partial spatial differentiation see Fig. S2. +generacy, for each spin there are 4 strongly correlated +AAp orbitals, degenerate in the absence of inversion or +time-reversal symmetry breaking, and 12 less correlated +orbitals, named lc orbitals in the following. The inter- +actions between all the orbitals are calculated assuming +a 1/r interaction between the electrons in the carbon +atoms[21]. For the dielectric constant ϵ = 12, used here, +we obtain U=44.5 meV, larger than the gap between the +flat and the remote bands ∆ =22 meV. Calculations are +performed at T=6K, except otherwise indicated, and we +do not allow states with spontaneous symmetry break- +ing. See methods and Extended Fig. S1 for more details +on the model and on the theoretical technique. +Fig. 1b shows the DOS for dopings −4 < ν < 4. The +DOS displays a strong energy, ω, and doping dependence +with resets of spectral weight and minima at the Fermi +level ω = 0 at integer dopings, Fig. 2a. In spite of the nar- +row bandwidth of the flat bands in the non-interacting +model, 1 meV at M and 8 meV at Γ (Fig. 1a and Ex- +tended Fig. S1c), a large amount of spectral weight is +found in Fig. 1b, within a range of 50 meV around the +Fermi level. This spectral weight shows up in the form of + +a +250 +0.28 +200 +150 +100 +(meV) +50 +3 +0 +-50 +3 +-100 +-150 +2 +0.21 +-200 +-250 +K +M +K +1 +C +2.01 +.86 +1.76 +0 +0.14 +1.61 +1.45 +DOS (arb. units) +1.27 +-1 +1.1 +1.0 +0.88 +0.8 +0.68 +0.07 +0.58 +0.46 +0.36 +0.23 +-3 +0.09 +0.01 +-0.11 +-0.27 +0.00 +-0.37 +-135-90 -45 +45 +0 +90 +135 +-45 +0 +45 +w (meV) +w (meV)3 +cascades at positive and negative energies flowing from +an energy of order ±U towards the chemical potential. +For hole doping the spectral weight of the cascade at +positive energy is larger than the one at negative ener- +gies and it increases with doping. This positive energy +cascade starts forming at energy U around a given integer +doping and becomes very close to the chemical potential +at the next smaller integer dopings. The negative energy +cascade is shifted in doping with respect to the one at +positive energies. It reaches the chemical potential at in- +termediate fillings, showing resets at finite energy at the +integer ones. The doping and energy dependence of the +cascades is reversed for electron-hole doping. No cascade +extends up to zero energy at the CNP, ν = 0. At the +CNP the cascades do not converge towards the chemical +potential but remain separated. The energy separation +of the Hubbard peaks at CNP is around U, probably a +bit larger, but it is difficult to identify the exact maxima +of these peaks due to their broadening and a double peak +shape in some cases. The cascades of spectral weight in +Fig. 1b and c look remarkably similar to the ones ob- +served in STM experiments[10, 18]. This strong similar- +ity imply that the cascades are not the consequence of a +symmetry breaking order, prohibited in our calculations. +As shown in extended Fig. S2, the cascades originate in a +reorganization of the spectral weight mostly coming from +the correlated AAp orbitals with a finite but smaller con- +tribution of the lc orbitals to the DOS at low energies. +The cascades within the range −0.5 < ν < 2 can be +also visualized in the line plots of the density of states +in Fig. 1c. In a wide range of dopings, the DOS show a +three peak structure, similar to the ones observed in other +strongly correlated systems with a quasiparticle peak at +zero energy and bumps at positive and negative energies. +A broad peak, the lower Hubbard band, starts forming at +energies slightly larger than -U at ν = 2 (top curve). For +smaller dopings, the Hubbard band starts approaching +the chemical potential, first slowly and a bit faster from +approximately ν = 1.5. When approaching the chemical +potential the peak height increases and acquires a more +asymmetric shape. +This cascade peak coexists with a +quasiparticle peak at zero energy, and with a weak and +broad upper Hubbard band. Close to ν = 1 the quasipar- +ticle peak at zero energy becomes very small and vanishes +at the integer doping. The flow of the cascade starting +at ν = 1 is similar to the previous one, except close to +the CNP, where the flow stops at a finite energy. +In agreement with STM experiments[10], in Fig. 1b +the cascades are accompanied by oscillations of the re- +mote band energies with respect to the chemical poten- +tial. These oscillations are best visualized at the energy +of the van Hove peaks, in darker grey between -50 meV +and -150 meV and between 50 and 150 meV, coming from +the flat band sections along the M-K direction at positive +and negative energies, approximately 100 meV far from +the CNP in Fig. 1a. These flat bands are contributed +by the lc orbitals, see extended Fig. S1b and Fig. S2b. +The shift of the van Hove peaks reflects an average dif- +ference in charging energy cost suffered by the AAp and +the lc electrons when an electron is added or removed +producing a relative shift of the onsite energies of both +type of electrons. It depends on whether the electron or +hole added dopes the AAp and the lc orbitals and on the +value of the intra and inter-orbital interactions between +both the AAp and the lc orbitals at the same or different +moir´e unit cells[21]. Its magnitude is not simply related +to a single interaction. For −2 < ν < 2 the van Hove +peaks oscillate as a function of doping on top of an ap- +proximately linear background. The linear contribution +with slope ≈ 11 meV per electron doped, is accounted +for by the Hartree approximation, extended data Fig. S3 +and Fig. S4. As discussed below, the oscillations on top +of this linear background originate in a non-monotonous +filling of the lc orbitals with doping. +The oscillations are well defined only for dopings ν < +|2|. +Beyond this doping the remote bands cross the +Fermi level modifying the doping dependence of the spec- +tral weight (extended Fig. S5), particularly the remote +band oscillations and the DOS at the Fermi level. The +DOS(w = 0) does not show either a minimum at |ν| = 3 +or a gap at |ν| = 4. The crossing of the bands would +happen at larger dopings in calculations starting from +a tight binding model for TBG with a larger gap ∆ be- +tween flat and remote bands or for larger ϵ. We note that +the contribution of the remote band to the Fermi level +does not prevent the AAp orbitals from being strongly +correlated beyond ν > |2|. This is confirmed by the cas- +cade at n = |3| in Fig. 1b and the strong incoherence in +extended Fig. S5. +The energy dependence and the strong doping evolu- +tion of the DOS in Fig. 1b with resets at integer fillings +contrast with the much less doping dependent DOS ob- +tained if all the interactions, including the interaction U +between the AAp orbitals, are treated in the Hartree ap- +proximation, extended data Fig. S3a. As it was widely +discussed in the past [21, 30, 31], when the interactions +in TBG are treated at the Hartree level, the shape of +the bands changes, extended data Fig. S3b and Fig. S3c. +Nevertheless, the spectral weight shifted to energies of or- +der U is very small, the changes in the density of states +are monotonous in doping, and there are no oscillations +of the remote bands. +We note that the tight-binding +model and the interactions used in Fig.1b and in ex- +tended Fig. S3a are the same. The only difference be- +tween both figures relies in the use of DMFT, instead +of Hartree, to treat the intra-moir´e unit cell interactions +between the strongly correlated AAp orbitals. The dif- +ferences in the spectral weight calculated with both tech- +niques make clear that the local correlations are behind +the formation of the cascades and the oscillations. These +correlations are properly described by DMFT and not by +other techniques, including Hartree or Hartree-Fock. + +4 +FIG. 2. Asymmetric peaks in the inverse compressibility and in the density of states. (a) Density of states at the +Fermi level as a function of doping, showing asymmetric minima at integer fillings within the range −2 < ν < 2, except at +CNP that is symmetric. (b) Local inverse compressibility as a function of doping with strongly asymmetric peaks. Its shape +strongly resembles the ones obtained in SET measurements[17]. (c) Inverse of the density of states at the Fermi level shown +in (a) with asymmetric peaks similar to the ones in the local inverse compressibility. (d) Density of states for several dopings +close to ν = −2. All calculations are at T=6K. +Fig. 2b shows the local inverse compressibility as a +function of ν. In spite of some noise, strong asymmetric +peaks with maxima at integer fillings are clearly observed +for |ν| < 3 except at CNP where the peak is approxi- +mately symmetric. For simplicity let’s focus on ν = −2. +For dopings between ν = −3 and ν = −2 the local +inverse compressibility increases steadily as the integer +is approached. Decreasing the doping beyond ν = −2, +the inverse compressibility shows a fast decrease with a +minimum around ν = −1.8 and then it starts increasing +more slowly towards ν = −1. This sawtooth-like behav- +ior of the inverse compressibility strongly resembles the +one observed experimentally in SET measurements[17]. +The SET experiments were originally interpreted as a +signature of ferromagnetic polarization involving Dirac +revivals. We emphasize that in our calculations we do not +have any ferromagnetic polarization or any other spon- +taneous symmetry breaking. +An asymmetry with the same sign, and peaking around +the same values is found in the inverse DOS at the Fermi +level, Fig. 2c. +Therefore, looking at the evolution of +the DOS(ω = 0) with doping can help understanding +the behavior of the compressibility and its connection +with the spectral weight reorganization observed in STM. +Consistent with the previous results, the minima in the +DOS(ω = 0) at integers dopings are strongly asymmet- +ric, except around CNP where it has a V-shape depen- +dence, Fig. 2a. For dopings between ν = −3 and ν = −2, +the DOS(ω = 0) decreases slowly as the integer is ap- +proached. Decreasing the doping beyond ν = −2, the +DOS(ω = 0) shows a fast rise with a maxima around +ν = −1.7 and then a slow decrease towards ν = −1. +Similar dependence is found around ν = −1, 1, 2. +Fig. 2d shows the evolution of the DOS close to ν = +−2. For dopings slightly above the integer, ν = −2.02, +the DOS shows a broad and slightly asymmetric peak at +energies ω = 7 meV and a small quasiparticle peak at +zero energy, whose height controls the DOS(ω = 0) and +its inverse. The broad peak has evolved from the Hub- +bard bands, as described above. At ν = −2 this broad +peak has shifted towards smaller energies ω = 3 meV +becoming narrower, higher and more asymmetric. Close +to integer doping even tiny changes in doping produce +a strong spectral weight reorganization. At ν = −2 the +peak has not reached zero energy yet and there is not +a quasiparticle peak. Correspondingly, the DOS(ω = 0) + +100 +b +a +0.25 +) (mevl) +75 +. +.. +0.2 +local' +0.15 +50 +0.1 +1 +25 +0.05 +0 +-4 +-3 +-2 +0 +2 +3 +-3 +-2 +-1 +0 +2 +1 +1 +4 +4 +1 +3 +2 +0.3 +50 +d +C +v=-1.45 +0.25 +(aw) ( +40 +V=-1.79 +v=-1.98 +0.2 +v=-2 +1/DOS(α=0) ( +30 +v=-2.02 +: +DOS +20 +0.1 +10 +: +-0.. +0.05 +0 +0 +.4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +-5 +0 +5 +10 +15 +w(meV)5 +FIG. 3. Temperature dependence of the density of states, orbital filling and doping dependent correlated bands. +(a) Density of states around ν − 1.96 for different temperatures. Densities are ν = −1.972 for 1.2 and 6 K, ν = −1.965 for 12 +K and ν = −1.955 for 24 K showing that the incoherence survives as the temperature rises but the tendency to form a heavy +quasiparticle does not. The shift and change in peak height is a clear effect of temperature and it cannot be explained by the +slight decrease in density in the curves with increasing temperature, as such decrease in density would shift the peak in opposite +direction. (b) Filling of the correlated AAp orbitals as a function of doping, showing a monotonous increase of the AAp filling +with weak steps at integer dopings. Inset: Doping of the less correlated orbitals measured with respect to CNP as a function +of total doping. The resets in the cascades and fillings happen at integer values of the total doping and not at integer fillings +of the correlated AAp orbitals emphasizing the important role of the hybridization between the AAp and the lc orbitals. (c) to +(h) Momentum resolved spectral weight respectively for ν =0.01, 0.22, 0.58, 1.0, 1.27 and 1.61, showing the modifications in +the band-structure induced by the correlations. Blurred spectra indicates incoherent spectral weight. The strong momentum +differentiation of incoherence in these figures emphasizes the role played by the fragile topology and the hybridization between +the correlated AAp and the less correlated electrons. +has a minimum and its inverse a maximum. When de- +creasing the doping further below the integer, the peak +continues shifting towards smaller energies and narrowing +without crossing to negative energies, but contributing to +the DOS(ω = 0), initially via its tail. As the peak ap- +proaches the chemical potential its shape becomes even +more asymmetric. The DOS(ω = 0) is maximum (its in- +verse minimum) at an electronic filling around ν = −1.7, +close to the fillling at which the peak becomes pinned at +zero energy. If doping decreases beyond this value, the + +8 +1.2F +b +a +T=1.2 K +0.25 +T=6 K +T=12 K +6 +T=24 K +0.2 +2V +3 +DOS +0.1 +2 +0.05 +10 +-4 +-3 +-2 +-20 +-15 +-10 +-5 +0 +10 +15 +20 +-1 +0 +1 +2 +3 +4 +5 +3 +(meV) +75 +75 +75 +0.200 +v=0.01 +v=0.22 +y=0.58 +50 +0.175 +50 +50 +0.150 +25 +25 +25 +w (meV) +w (meV) +0.125 +(meV) +0 +0 +0 +0.100 +3 +0.075 +-25 +-25 +-25 +0.050 +-50 +-50 +c +-50 +d +e +0.025 +-75, +-75 +-75 +0.000 +K +M +K +K +M +K +K +M +K +75 +75 +75 +0.200 +f +09 +0.175 +50 +50 +50 +0.150 +25 +25 +25 +w (meV) +w (meV) +w (meV) +0.125 +0 +0 +0 +0.100 +0.075 +-25 +-25 +-25 +0.050 +-50 +-50 +-50 +v=1.0 +v=1.27 +0.025 +v=1.61 +-75. +-75. +-75 +0.000 +K +M +K +K +M +k +K +M +K6 +DOS(ω = 0) decreases. The spectral weight is shifted +from the quasiparticle peak to a newly formed Hubbard +band. +With decreasing temperature from 6 K to 1.2 K the +sharp asymmetric peak at finite energy slightly ap- +proaches the chemical potential, Fig. 3a and extended +Fig. S7. Therefore the maximum in the DOS(ω = 0), +and the minimum in the inverse compressibility, are ex- +pected to appear slightly closer to the integer filling. We +find pseudogaps in the DOS at zero energy ν = 1, −1 and +−2 which shift with slight doping and are smaller when +the temperature is reduced, suggesting that they could +close at zero temperature. However, we emphasize that +the physics of the cascades discussed here is not linked to +whether there is a a pseudogap or small gap at exactly in- +teger fillings but to a reorganization of the spectral weight +at all dopings of the flat bands in a large energy range, +with incoherent bands starting from interactions around +U and progressively approaching the chemical potential. +With an increase of the temperature to 12K or 24 K the +spectrum is more sensitive. The peak shifts to higher en- +ergies, becomes broader, more symmetric and its height +decreases. Close to ν = −2, the peak height is reduced +approximately to half of its value when increasing the +temperature from 6K to 24 K. Notably, the spectrum +does not return to its non-interacting or Hartree shape +at high temperatures, Fig. S7. The incoherent spectral +weight is resilient with temperature, becoming broadened +and less defined around integer values. This is consistent +with experimental findings which show that broadened +cascade signatures persist to at least 50 K[17, 18]. +The cascade energy and doping dependence can be un- +derstood in terms of the tendency of the AAp orbitals to +form local moments and become incoherent, and their +hybridization to the lc orbitals which accomodate part +of the doping and tend to screen the local moments. In +the absence of coupling between the AAp and the less +correlated electrons at integer fillings, the AAp orbitals +would become Mott insulators with large gaps and their +spectral weight would become incoherent and shifted to +the Hubbard bands, Fig. S8. Doping these Mott insula- +tors drives them metallic creating a heavy quasiparticle +at the Fermi level. Resets in the shape and position of the +Hubbard bands would appear around integer fillings, as +with hole doping the quasiparticle is created in the lower +Hubbard band, while with electron doping the quasipar- +ticle appears in the upper Hubbard band[26]. The reor- +ganization of the spectral weight as a function of energy +and doping shows a qualitatively different shape, Fig. S8, +compared to the cascades that arise when the coupling +between the AAp and the lc orbitals is included. This +can be better seen comparing the evolution of the DOS +in Fig.1 with the momentum resolved bands in Fig. 3. +Around the CNP, the screening of the local moments +is not operative, at least at the considered temperatures. +The AAp orbitals become incoherent and shift their spec- +tral weight towards the broad Hubbard bands with barely +no spectral weight around zero energy, extended Fig.S2a. +However, the DOS does not present a Mott gap. Due to +the fragile topology of TBG, the lc electrons contribute +around Γ, Fig. 3c. When doping the system with elec- +trons around the CNP it is energetically more favorable +for the system to add these electrons to the lc orbitals, +producing a band deformation around Γ, Fig.3d. +The +spectral weight of the AAp is still highly incoherent at +this doping. +The lc orbitals contribute in a small k- +region and the density of states is small at the Fermi +level. As the lc orbitals are doped, the onsite energies of +the two type of electrons are shifted. The upper Hubbard +band of the AAp orbitals changes its shape with a main +peak approaching the chemical potential, while the lower +Hubbard band shifts downwards in energy. This explains +the evolution of the cascade peaks close to CNP. With +further doping, the AAp upper Hubbard band reaches +the chemical potential and a quasiparticle peak pinned +at the Fermi level with primary contribution of the AAp +orbitals is formed, Fig. 3e. The AAp quasiparticle band +is very flat and its contribution to the density of states +high. Once the heavy AAp quasiparticle is formed, it be- +comes advantageous to dope the AAp orbitals and the lc +orbitals are slightly emptied, modifying again the band +structure at Γ. +At ν = 1 there is no quasiparticle at +the Fermi level, at least at the temperature of the cal- +culations, but the hybridization between the lc and the +AAp orbitals prevents the latter from being incoherent, +Fig. 3f. There is a reset in the spectral weight accompa- +nied by a small step in the filling of the correlated AAp +orbitals and a small pseudogap. Doping beyond the in- +teger, the AAp electrons become incoherent and the lc +orbitals are doped again giving small spectral weight to +the DOS(ω = 0), Fig.3g, until a weak peak emerging +from the upper Hubbard band reaches the chemical po- +tential and a heavy quasiparticle with main contribution +from AAp forms, Fig. 3h. The process repeats for higher +dopings. The strong momentum differentiation of inco- +herence in Fig. 3c to h emphasizes the role played by +the fragile topology and the hybridization between the +correlated AAp and the less correlated electrons. +The changes in the filling of the AAp and lc orbitals +can be seen in Fig. 3b. In spite of the strong correla- +tions, the filling of the AAp does not remain constant +between integer dopings. On the contrary, as described +above, the doping goes primarily to these AAp orbitals +with just weak steps at integer dopings. +The gap be- +tween the flat and remote bands prevents more abrupt +steps as charging the lc orbitals costs kinetic energy. The +role of the hybridization is clear in that the steps and the +resets happen at integer doping, not at integer filling of +the AAp orbitals. The oscillations of the remote bands +with respect to the chemical potential follow the non- +monotonic occupation of the lc orbitals. The tempera- +ture dependence is also consistent with this picture. In + +7 +the calculations the screening of the local moment is not +complete. As the temperature is reduced the high peak +at integer values approaches the Fermi level towards the +formation of a heavy quasiparticle. With increasing tem- +perature, the screening of the local moment is reduced as +the entropy of the local moment is higher than the Fermi +liquid one. The formation of the heavy quasiparticles is +expected to happen only at low temperatures while Mott +physics, responsible for the incoherence, remains at high +temperatures. +In conclusion, our work shows that the experimen- +tally cascade phenomenon is not associated to a sym- +metry breaking transition but to the reorganization of +spectral weight due to strong correlations. +As it hap- +pens in Mott-like systems, the spectral weight becomes +incoherent as correlated AAp electrons form local mo- +ments. However, due to the fragile topology of TBG, the +coupling between the AAp orbitals to the less correlated +electrons introduces a strong momentum differentiation +in the incoherence, suppressing the insulating tendencies +and promoting the formation of a heavy quasiparticle +around integer fillings away from CNP. The cascades con- +nect TBG with other strongly correlated systems such as +high temperature superconductors, heavy fermion sys- +tems and oxides through their hallmark: the incoherent +spectral weight reorganization. The cascades constitute +the normal state in which low temperature phenomena +such as ferromagnetism, Chern insulators, superconduc- +tivity or nematicity emerge. As shown recently, a strong +reorganization of the spectral weight with heavy quasi- +particles but without the effects of the fragile topology +is expected in other moir´e heterostructures, such as the +ABC/hBN trilayer[32]. +The cascade phenomena origi- +nates in the intra moir´e unit cell interaction U between +the AAp orbitals within the unit cell, specifically to the +local correlations. This interaction is not much screened +by gates farther than 5 nm[21] therefore the cascades are +expected to survive to the proximity of gates above this +distance, contrary to other effects controlled by longer +range interactions. +When we were completing this manuscript some re- +lated works appeared on the arXiv[33–35] +We thank conversations with M. Rozenberg, +M. +Civelli, E. Miranda, M.C.O. Aguiar, A. Yazdani, A. +V. Chubukov, F. de Juan and B.A. Bernevig. +M.J.C, +E.B. and A.D. acknowledge funding from PGC2018- +097018-B-I00 +(MCIN/AEI/FEDER, +EU), +M.J. +and +E.B. +PID2021-125343NB-100 +(MCIN/AEI/FEDER, +EU). A.C. acknowledges support from UBACyT (Grant +No. +20020170100284BA) and Agencia Nacional de +Promoci´on de la Investigaci´on, el Desarrollo Tecnol´ogico +y la Innovaci´on (Grant No. +PICT-2018-04536). +A.D. +acknowledges the French National Research Agency +(project TWIST- GRAPH, ANR-21-CE47-0018). +Methods +We start from a θ = 1.08◦ TBG with interlayer tun- +neling ratio between AA and AB w0/w1 = 0.78 keep- +ing the sin(θ/2) term which slightly breaks particle-hole +symmetry[28, 29]. +We have adapted the eight orbital +model per valley and spin from [24] to fit the low en- +ergy band structure. +We keep intra and inter-orbital +density-density interactions within a distance equal to +7 moir´e lattice constant, including up to 54 neighboring +cells. +Hund’s coupling, pair-hopping and exchange in- +teractions, considerably smaller than the density-density +included here[21], are neglected and the intra moir´e unit +cell intra and inter-orbital interactions between the AAp +orbitals are taken to be equal. Based on the difference +between the orbital dependent onsite interactions and +bandwidths[21] we distinguish between strongly corre- +lated AA p+ and p−, named AAp, and less correlated +AA s, AB pz and DW s lc orbitals. The later orbitals +are treated at the Hartree level while the DMFT ap- +proximation [26, 27] is used on the strongy correlated +AAp orbitals, which acquire a frequency dependent self +energy. In the DMFT calculation, the coupling to the +lc orbitals enters via an effective hybridization function +which depends on the Hartree onsite potentials. Due to +the coupling with the correlated AAp orbitals, the lc or- +bitals also get a frequency dependent self-energy. A dou- +ble step self-consistency loop is implemented. A number +of DMFT iterations are run starting from a given set of +Hartree onsite potentials and taking care to avoid double +counting the effect of the interaction U. As an outcome +of the DMFT, the self energy, the Green’s function and +the charge of the AAp orbitals are calculated. Due to +the hybridization between the two types of orbitals, the +orbital filling of all the orbitals is modified. The Hartree +onsite shifts are re-calculated with the new fillings and +new DMFT iterations are run with the new calculated on- +site potentials. Convergence is reached when the Green’s +functions, self-energy and orbital fillings do not change +between either DMFT or Hartree iterations. To avoid +double counting the interaction effectively included in the +tight binding model parameters, we substract the onsite +potentials obtained at the CNP when the Hartree ap- +proximation is used for all the orbitals[31]. The single- +site DMFT calculations are performed at a given chem- +ical potential and finite temperature using a continuous +time quantum Monte Carlo impurity solver [36] as imple- +mented in Ref. [37]. 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Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, +M. Troyer, and P. Werner, Continuous-time monte carlo +methods for quantum impurity models, Rev. Mod. Phys. +83, 349 (2011). +[37] K. Haule, Quantum monte carlo impurity solver for clus- +ter dynamical mean-field theory and electronic structure +calculations with adjustable cluster base, Phys. Rev. B +75, 155113 (2007). +[38] M. Jarrell and J. E. Gubernatis, Bayesian inference and +the analytic continuation of imaginary-time quantum +monte carlo data, Phys. Rep. 269, 133 (1996). + +10 +FIG. S1. Model details (a) Schematic plot of the triangular AA, honeycomb AB/BA, and kagome DW lattices where the +eight orbitals per valley and spin are centered. Besides the two correlated AAp orbitals (orange), at the AA regions of the +moir´e unit cell another orbital with s character AAs is centered (red). The model also includes two pz orbitals centered at the +honeycomb lattice formed by the AB/BA regions (green) and three s orbitals at the kagome lattice (purple) formed by domain +wall DW regions separating two AB/BA points. (b) Orbital weight of the two correlated AAp (left) and the six less correlated +orbitals (right) to the band structure of the eight orbital model used in the calculation. (c) Zoom of Fig.1a corresponding to +the non-interacting flat band obtained from the continuum model (black) for an angle θ = 1.08◦ and its fitting with the eight +orbital model. (d) Intra and inter-orbital density-density interactions as a function of the distance between the orbitals. Here +λ is the moir´e unit cell. + +a +200 +100 +(meV) +3 +-100 +-200 +K +r +M +K +r +M +K +8 +d +AAp-AAp +8=12 +6 +AAp-AAs +AAp-AB/BA +AAp-DW +AAs-AAs +Interactions ( +30 +2 +AAs-AB/BA +0 +AAs-DW +20 +AB/BA-AB/BA +-2 +AB/BA-DW +DW-DW +-4 +10 +0=1.08 +-6 +0 +0 +0.5 +1 +1.5 +2 +K +M +K +r/△11 +FIG. S2. Contribution to the spectral weight of the two types of orbital. Contribution to the density of states in +Fig.1b of the (a) two strongly correlated AAp orbitals. (b) Same as (a) but for the six less correlated AAs, AB / BApz and +DWs orbitals. The lc contribution is reduced at small energies. Nevertheless some effects of the resets can be also appreciated +in their spectral weight at low energies. The contribution of the AAp orbitals will be more visible in STM measurements at +the AA site, while the spectral weight of the less correlated orbitals will have larger weight at the AB position (the AAs has +an annular shape). Due to the finite extension of the orbitals, this separation of the orbitals in spatial regions should not be +taken too literal. + +4 +4 +0.28 +a +b +3 +3. +0.06 +2 +0.21 +2 +1 +0.04 +0 +0.14 +0 +-1 +-1 +0.02 +-2 +0.07 +-2 +-3 +-3 +-4 1 +0.00 +-4 - +0.00 +-135-90 -45 +¥045 +90135 +-135-90 -45 +50459 +90135 +w (meV) +w (meV)12 +FIG. S3. Density of states and band deformation in the Hartree approximation. (a) Same as Fig.1b but treating +all the interactions in the Hartree approximation. The cascades of spectral weight and the oscillations in the remote band van +Hove peaks have disappeared. (b) and (c) Low energy bands calculated for ν = −2 and ν = 2 in the Hartree approximation. +The band shape has changed with respect to the non-interacting one in (d) as discussed extensively[21, 30, 31]. Nevertheless, +the effect of such changes on the spectral weight reorganization in (a) is small. +FIG. S4. Oscillations of the van Hove of the remote bands . Dependence of the remote band van Hove energies as a +function of doping. The red lines are an approximation to the doping dependent shift of the van Hove positions in Fig.S3. + +4 +0.75 +b +v=-2 +a +40 +3 +20 +w (meV) +0 +2 +-20 +0.50 +-40 +1 +K +M +K +C +v=2 +-1 +40 +0.25 +20 + (meV) +-2 +0 +3 +-20 +-3 +-40 +-4 +0.00 +90 135 +K +M +K +-135-90 -45 +045 +w (meV)135 +90 +45 +(meV) +0 +3 +-45 +90 +-135 +-4 +3 +-2 +-1 +0 +i +2 +3 +4 +V13 +FIG. S5. Momentum resolved spectral weight for |ν| > 2. Interacting bandstructure obtained from the DMFT+Hartree +calculations for dopings ν < −2. (a) ν = −2.27, (b) ν = −3.28 and (c) ν = −3.97. It can be observed how the remote +bands cross the chemical potential contributing significantly to the DOS at the Fermi level and to the doping dependence of +the spectral weight. The contribution of the remote bands here resembles the one in charge transfer insulators. Due to this +crossing at large dopings the filling of the lc orbitals is less sensitive to the integer dopings in Fig.3 and the oscillations of the +van Hove energies in Fig.1 are not so well defined. +FIG. S6. Chemical potential as a function of doping. In our calculation we can define two different chemical potentials. +The local chemical potential µlocal in (a) enters into the DMFT calculation. Its value is primarily controlled by the interaction +U and the filling of the AAp orbitals. In (b) The global chemical potential µtotal strongly depends on the long range interactions +and account primarily for a global shift of all the bands. µtotal has a very large magnitude and an almost linear dependence on +the density which closely follows the value obtained when all the interactions, including U, are treated at the Hartree level. In +the DMFT+Hartree calculation µtotal has small steps at the integer dopings. The derivative of µtotal (not shown) is dominated +by the linear dependence and the features around the integers are quite noisy. +FIG. S7. Temperature dependence of the spectral weight. Dependence of the density of states on temperatures for +dopings close to ν = −2. +(a) ν=2 and (b) ν ≈1.77-1.78 showing a similar behavior as discussed in Fig. 3 for ν +− 1.96. +With increasing temperature the peak shifts towards positive energies, its height decreases and the shape is slightly changed. +The pseudogap around zero or small energies slightly decreases with decreasing temperature. +(c) and (d) Band structure +corresponding to the DOS in at T=1.2K and T=24K discussed in Fig. 3. With increasing temperature, the bands become more +incoherent and small band shifts are observed. + +75 +75 +0.200 +V~ -1.78 +V~-2.0 +v=-1.97 +T=1.2 K +C +d +v=-1.95 + 0.175 +0.25 +T=6 K +0.25 +T=6 K + 50 +T=1.2K + 50 +T=24 K +. T=12 K + T=12 K +0.150 +0.2 + T=24 K +0.2 +T=24 K +25 +25 +w (meV) +w (meV) +0.125 +三 0.15 +0.15 +0 +0.100 +DOS +0.1 +0.075 +0.1 +25 +-25 + 0.050 +0.05 +0.05 +50 +So +0.025 +-75 +20 +0.000 +-10 +-5 +0 +10 +15 +20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +5 +K +M +K +M +K +K +w (meV) +w (meV)75 +75 +75 +0.200 +v=-2.27 +v=-3.28 +a +v=-3.97 +C +0.175 +50 +50 +50 +0.150 +25 +25 +25 +w (meV) +w (meV) +w (meV) +0.125 +0 +0 +0 +0.100 +0.075 +-25 +-25 +-25 +0.050 +-50 +-50 +-50 +0.025 +-75. +-75 - +-75- +0.000 +K +M +K +K +M +K +K +M +K300 +1000 +a +b +750 +250 +500 +(meV) +200 +(meV) +250 +150 +0 +local +total +-250 +100 +U +-500 + DMFT+Hartree +50 +-750 +All Hartree +-1000 +0 +-4 +-3 +-2 +-1 +0 +1 +2 +2 +3 +-4 +-3 +-2 +-1 +1 +3 +4 +4 +014 +FIG. S8. Density of states of an only AAp model. (a) Colour plot corresponding to the density of states calculated with +DMFT of a Hubbard model containing only the AAp orbitals (four orbital model degenerate in spin) interacting with both +intra and inter-orbital interaction U=44.5 meV, as in the eight orbital model, for fillings n within the range −3.5 < n < 6. Here +half-filling (the equivalent for this model to the CNP) is 4. A clear reorganization of the spectral weight with doping extending +up to energies of order U is observed. At each integer the lower (upper) Hubbard band approaches the chemical potential at +integer fillings if the system is doped with holes (electrons). Strong resets in the spectral weight are found associated to the +gaps at integer fillings. At non-integer fillings a large density of states associated to the formation of a heavy quasiparticle, +typical of doped Mott insulators, is found. (b) Line cuts of the density of states for selected dopings. Curves are shifted for +clarity. The typical three peak structure is observed at partial non- integer fillings, being the thin peak at zero energy the +quasiparticle peak and the two broad peaks at a few tens meV the Hubbard bands. The Hubbard bands shift with doping. At +integer values there are large Mott gaps and resets in the Hubbard bands. + +6.0 +a +n=4 +0.6 +4.31 +4.53 +6 +4.71 +5.5 +4.79 +5 +5 +5.22 +5.32 +0.4 +5.0 +(arb. +n +4.5 +0.2 +2 +4.0 +1 +3.5 +0.0 +0 +-60 +-40 +-20 +0 +20 +40 +60 +-40 +-20 +20 +40 +0 +w (meV) +w (meV) \ No newline at end of file diff --git a/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/load_file.txt b/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4229a13fb6357641c9fda17c3fd04248e87993aa --- /dev/null +++ b/pdFPT4oBgHgl3EQfLzS8/content/tmp_files/load_file.txt @@ -0,0 +1,969 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf,len=968 +page_content='Heavy quasiparticles and cascades without symmetry breaking in twisted bilayer graphene Anushree Datta1,2,3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Calder´on1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Camjayi4, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Bascones1∗ 1Instituto de Ciencia de Materiales de Madrid (ICMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Consejo Superior de Investigaciones Cient´ıficas (CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Sor Juana In´es de la Cruz 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 28049 Madrid (Spain),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2 Universit´e de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Laboratoire Mat´eriaux et Ph´enomenes Quantiques,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' F-75013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3 Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Laboratoire de Physique des Solides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 91405,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 4Ciclo B´asico Com´un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Universidad de Buenos Aires and IFIBA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Conicet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Pabell´on 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Ciudad Universitaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1428 CABA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Argentina Twisted bilayer graphene (TBG) exhibits a plethora of electronic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Among the variety of correlated states, the cascades in the spectro- scopic properties and in the compressibility hap- pen in a much larger energy, twist angle and tem- perature range than other effects, pointing to a hierarchy of phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In this work, by apply- ing a self-consistent Dynamical Mean Field The- ory + Hartree approximation to a multi-orbital model for TBG, we show that the spectral weight reorganization associated to the formation of lo- cal moments and heavy quasiparticles, and not a symmetry breaking process, is responsible for the cascade phenonema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to the fragile topol- ogy of TBG, a strong momentum differentiation is found in the incoherent spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The phenomena reproduced here include the cascade flow of spectral weight, the oscillations of the re- mote band energies and the asymmetric jumps of the inverse compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A large variety of correlated states, including cor- related insulators, superconductivity and topological phases[1–14], have been found in TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Among these phe- nomena, early scanning tunneling microscopy (STM) ex- periments found a strong doping dependent broadening of the density of states (DOS) with a spectral weight re- organization up to several tens meV and minima of the DOS at the Fermi level at integer fillings[6–8, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This reorganization was later found to happen in the form of cascades of spectral weight with resets[10, 11] and band flattening at the integers[16] and is accompanied by oscil- lations in the energies of the remote bands[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The phe- nomelogy at the charge neutrality point (CNP) was found to be different to the one at other integer fillings with the spectral weight pushed away from low energies[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades are also revealed in scanning electron transistor (SET) experiments as strong asymmetric jumps of the local electronic compressibility [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascade phenomena have been detected up to much higher temperatures than the superconductivity, the correlated insulators and the anomalous Hall effects [17, 18], and they are also found in a wider range of an- gles, suggesting that the cascades constitute the parent state in which these low temperature phenomena emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades have been primarily interpreted in terms of symmetry breaking states[8, 11, 16, 17] but also as a con- sequence of strong correlations in the normal state[7, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The discussion on whether the phenomenology of TBG can be explained just in terms of ordered states or it is necessary to take into account the strong modification of the non-ordered normal state was put forward early on[1, 7, 8, 19, 20] and it is still unsettled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Advancing in this field has been hindered by the complexity of min- imal models in TBG and of the techniques required to address the role of strong correlations beyond the sym- metry breaking transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The heavy fermion description for TBG, early intro- duced in [20], and supported by an analysis of the in- teractions in TBG [21], is now receiving more attention, particularly after the publication of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In the dif- ferent heavy fermion models for TBG the flat bands of each valley are spanned by two p+ and p− orbitals cen- tered at the AA region of the moir´e unit cell, except close to Γ [20–24], extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' These AAp or- bitals, strongly sensitive to electronic interactions, are coupled to less correlated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In such a system, a strong spectral weight reorganization is expected even in the non-ordered normal state[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Studying the ef- fect of strong local correlations, such as Mott physics or the formation of heavy quasiparticles, requires tech- niques like dynamical mean field theory (DMFT)[26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' However, even if the intra unit cell interaction U among these AAp orbitals is expected to be responsible for the unconventional behavior, other interactions in the model are sizable and cannot be neglected[21], making a com- plete DMFT treatment out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Here we use a self-consistent DMFT+Hartree approximation to study a multi-orbital model for TBG[21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The intra and inter-orbital interactions U involving the correlated AAp orbitals within the same unit cell are treated with DMFT and the other interactions are included at the Hartree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Self-consistency is enforced both within the DMFT loop and between DMFT and Hartree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We start from a θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='08◦ TBG, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1a, with the ratio between the interlayer tunneling at AA and AB w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='78 [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The low energy band structure is fitted with an eight orbital model per valley and spin, adapted from[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Having into account the valley de- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='13024v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='str-el] 30 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Cascades in the spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) Bands of TBG obtained from the continuum model (black) and 8 orbital model fitting used in the calculations (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Color plot of the density of states obtained in the DMFT+Hartree calculations as a function of the doping and the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Doping and energy are respectively measured with respect to the CNP and to the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Cascades of incoherent spectral weight flow from an energy ≈ U=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 meV towards the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades are shifted in doping at positive and negative energies with resets in both cases at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Oscillations in the remote bands are clearly distinguished between −2 < ν < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades show strong resemblance with STM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (c) Line cuts of the density of states for selected dopings within the range −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 < ν < 2 with primarily a three peak structure: quasiparticle peak around zero energy and broader Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The Hubbard bands shift with electronic filling and merge with the quasiparticle peak at certain dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Curves are shifted vertically for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Notice that this is the total density of states, including the whole moir´e unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For a DOS with a partial spatial differentiation see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' generacy, for each spin there are 4 strongly correlated AAp orbitals, degenerate in the absence of inversion or time-reversal symmetry breaking, and 12 less correlated orbitals, named lc orbitals in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The inter- actions between all the orbitals are calculated assuming a 1/r interaction between the electrons in the carbon atoms[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For the dielectric constant ϵ = 12, used here, we obtain U=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 meV, larger than the gap between the flat and the remote bands ∆ =22 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Calculations are performed at T=6K, except otherwise indicated, and we do not allow states with spontaneous symmetry break- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' See methods and Extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S1 for more details on the model and on the theoretical technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b shows the DOS for dopings −4 < ν < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The DOS displays a strong energy, ω, and doping dependence with resets of spectral weight and minima at the Fermi level ω = 0 at integer dopings, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In spite of the nar- row bandwidth of the flat bands in the non-interacting model, 1 meV at M and 8 meV at Γ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1a and Ex- tended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S1c), a large amount of spectral weight is found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b, within a range of 50 meV around the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This spectral weight shows up in the form of a 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='28 200 150 100 (meV) 50 3 0 50 3 100 150 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='21 200 250 K M K 1 C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='76 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='45 DOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' units) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='27 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='23 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='37 135-90 -45 45 0 90 135 45 0 45 w (meV) w (meV)3 cascades at positive and negative energies flowing from an energy of order ±U towards the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For hole doping the spectral weight of the cascade at positive energy is larger than the one at negative ener- gies and it increases with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This positive energy cascade starts forming at energy U around a given integer doping and becomes very close to the chemical potential at the next smaller integer dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The negative energy cascade is shifted in doping with respect to the one at positive energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' It reaches the chemical potential at in- termediate fillings, showing resets at finite energy at the integer ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The doping and energy dependence of the cascades is reversed for electron-hole doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' No cascade extends up to zero energy at the CNP, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At the CNP the cascades do not converge towards the chemical potential but remain separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The energy separation of the Hubbard peaks at CNP is around U, probably a bit larger, but it is difficult to identify the exact maxima of these peaks due to their broadening and a double peak shape in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades of spectral weight in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b and c look remarkably similar to the ones ob- served in STM experiments[10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This strong similar- ity imply that the cascades are not the consequence of a symmetry breaking order, prohibited in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As shown in extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S2, the cascades originate in a reorganization of the spectral weight mostly coming from the correlated AAp orbitals with a finite but smaller con- tribution of the lc orbitals to the DOS at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades within the range −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 < ν < 2 can be also visualized in the line plots of the density of states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In a wide range of dopings, the DOS show a three peak structure, similar to the ones observed in other strongly correlated systems with a quasiparticle peak at zero energy and bumps at positive and negative energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A broad peak, the lower Hubbard band, starts forming at energies slightly larger than -U at ν = 2 (top curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For smaller dopings, the Hubbard band starts approaching the chemical potential, first slowly and a bit faster from approximately ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' When approaching the chemical potential the peak height increases and acquires a more asymmetric shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This cascade peak coexists with a quasiparticle peak at zero energy, and with a weak and broad upper Hubbard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Close to ν = 1 the quasipar- ticle peak at zero energy becomes very small and vanishes at the integer doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The flow of the cascade starting at ν = 1 is similar to the previous one, except close to the CNP, where the flow stops at a finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In agreement with STM experiments[10], in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b the cascades are accompanied by oscillations of the re- mote band energies with respect to the chemical poten- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' These oscillations are best visualized at the energy of the van Hove peaks, in darker grey between -50 meV and -150 meV and between 50 and 150 meV, coming from the flat band sections along the M-K direction at positive and negative energies, approximately 100 meV far from the CNP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' These flat bands are contributed by the lc orbitals, see extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S1b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The shift of the van Hove peaks reflects an average dif- ference in charging energy cost suffered by the AAp and the lc electrons when an electron is added or removed producing a relative shift of the onsite energies of both type of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' It depends on whether the electron or hole added dopes the AAp and the lc orbitals and on the value of the intra and inter-orbital interactions between both the AAp and the lc orbitals at the same or different moir´e unit cells[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Its magnitude is not simply related to a single interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For −2 < ν < 2 the van Hove peaks oscillate as a function of doping on top of an ap- proximately linear background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The linear contribution with slope ≈ 11 meV per electron doped, is accounted for by the Hartree approximation, extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As discussed below, the oscillations on top of this linear background originate in a non-monotonous filling of the lc orbitals with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The oscillations are well defined only for dopings ν < |2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Beyond this doping the remote bands cross the Fermi level modifying the doping dependence of the spec- tral weight (extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S5), particularly the remote band oscillations and the DOS at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The DOS(w = 0) does not show either a minimum at |ν| = 3 or a gap at |ν| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The crossing of the bands would happen at larger dopings in calculations starting from a tight binding model for TBG with a larger gap ∆ be- tween flat and remote bands or for larger ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We note that the contribution of the remote band to the Fermi level does not prevent the AAp orbitals from being strongly correlated beyond ν > |2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This is confirmed by the cas- cade at n = |3| in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b and the strong incoherence in extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The energy dependence and the strong doping evolu- tion of the DOS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 1b with resets at integer fillings contrast with the much less doping dependent DOS ob- tained if all the interactions, including the interaction U between the AAp orbitals, are treated in the Hartree ap- proximation, extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As it was widely discussed in the past [21, 30, 31], when the interactions in TBG are treated at the Hartree level, the shape of the bands changes, extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Nevertheless, the spectral weight shifted to energies of or- der U is very small, the changes in the density of states are monotonous in doping, and there are no oscillations of the remote bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We note that the tight-binding model and the interactions used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1b and in ex- tended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3a are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The only difference be- tween both figures relies in the use of DMFT, instead of Hartree, to treat the intra-moir´e unit cell interactions between the strongly correlated AAp orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The dif- ferences in the spectral weight calculated with both tech- niques make clear that the local correlations are behind the formation of the cascades and the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' These correlations are properly described by DMFT and not by other techniques, including Hartree or Hartree-Fock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Asymmetric peaks in the inverse compressibility and in the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) Density of states at the Fermi level as a function of doping, showing asymmetric minima at integer fillings within the range −2 < ν < 2, except at CNP that is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Local inverse compressibility as a function of doping with strongly asymmetric peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Its shape strongly resembles the ones obtained in SET measurements[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (c) Inverse of the density of states at the Fermi level shown in (a) with asymmetric peaks similar to the ones in the local inverse compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (d) Density of states for several dopings close to ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' All calculations are at T=6K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2b shows the local inverse compressibility as a function of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In spite of some noise, strong asymmetric peaks with maxima at integer fillings are clearly observed for |ν| < 3 except at CNP where the peak is approxi- mately symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For simplicity let’s focus on ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For dopings between ν = −3 and ν = −2 the local inverse compressibility increases steadily as the integer is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Decreasing the doping beyond ν = −2, the inverse compressibility shows a fast decrease with a minimum around ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='8 and then it starts increasing more slowly towards ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This sawtooth-like behav- ior of the inverse compressibility strongly resembles the one observed experimentally in SET measurements[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The SET experiments were originally interpreted as a signature of ferromagnetic polarization involving Dirac revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We emphasize that in our calculations we do not have any ferromagnetic polarization or any other spon- taneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' An asymmetry with the same sign, and peaking around the same values is found in the inverse DOS at the Fermi level, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Therefore, looking at the evolution of the DOS(ω = 0) with doping can help understanding the behavior of the compressibility and its connection with the spectral weight reorganization observed in STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Consistent with the previous results, the minima in the DOS(ω = 0) at integers dopings are strongly asymmet- ric, except around CNP where it has a V-shape depen- dence, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For dopings between ν = −3 and ν = −2, the DOS(ω = 0) decreases slowly as the integer is ap- proached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Decreasing the doping beyond ν = −2, the DOS(ω = 0) shows a fast rise with a maxima around ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='7 and then a slow decrease towards ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Similar dependence is found around ν = −1, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 2d shows the evolution of the DOS close to ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' For dopings slightly above the integer, ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='02, the DOS shows a broad and slightly asymmetric peak at energies ω = 7 meV and a small quasiparticle peak at zero energy, whose height controls the DOS(ω = 0) and its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The broad peak has evolved from the Hub- bard bands, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At ν = −2 this broad peak has shifted towards smaller energies ω = 3 meV becoming narrower, higher and more asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Close to integer doping even tiny changes in doping produce a strong spectral weight reorganization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At ν = −2 the peak has not reached zero energy yet and there is not a quasiparticle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Correspondingly, the DOS(ω = 0) 100 b a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='25 ) (mevl) 75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content="2 local' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='15 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 1 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='05 0 4 3 2 0 2 3 3 2 1 0 2 1 1 4 4 1 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='3 50 d C v=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='25 (aw) ( 40 V=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='79 v=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2 v=-2 1/DOS(α=0) ( 30 v=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='02 : DOS 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 10 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='05 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='4 3 2 1 0 1 2 3 4 5 0 5 10 15 w(meV)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Temperature dependence of the density of states, orbital filling and doping dependent correlated bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) Density of states around ν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='96 for different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Densities are ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='972 for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2 and 6 K, ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='965 for 12 K and ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='955 for 24 K showing that the incoherence survives as the temperature rises but the tendency to form a heavy quasiparticle does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The shift and change in peak height is a clear effect of temperature and it cannot be explained by the slight decrease in density in the curves with increasing temperature, as such decrease in density would shift the peak in opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Filling of the correlated AAp orbitals as a function of doping, showing a monotonous increase of the AAp filling with weak steps at integer dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Inset: Doping of the less correlated orbitals measured with respect to CNP as a function of total doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The resets in the cascades and fillings happen at integer values of the total doping and not at integer fillings of the correlated AAp orbitals emphasizing the important role of the hybridization between the AAp and the lc orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (c) to (h) Momentum resolved spectral weight respectively for ν =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='58, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='27 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='61, showing the modifications in the band-structure induced by the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Blurred spectra indicates incoherent spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The strong momentum differentiation of incoherence in these figures emphasizes the role played by the fragile topology and the hybridization between the correlated AAp and the less correlated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' has a minimum and its inverse a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' When de- creasing the doping further below the integer, the peak continues shifting towards smaller energies and narrowing without crossing to negative energies, but contributing to the DOS(ω = 0), initially via its tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As the peak ap- proaches the chemical potential its shape becomes even more asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The DOS(ω = 0) is maximum (its in- verse minimum) at an electronic filling around ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='7, close to the fillling at which the peak becomes pinned at zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' If doping decreases beyond this value, the 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2F b a T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='25 T=6 K T=12 K 6 T=24 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2 2V 3 DOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='05 10 4 3 2 20 15 10 5 0 10 15 20 1 0 1 2 3 4 5 3 (meV) 75 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='200 v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='01 v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='22 y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='58 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='175 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='150 25 25 25 w (meV) w (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='125 (meV) 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='100 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='075 25 25 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='050 50 50 c 50 d e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='025 75, 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='000 K M K K M K K M K 75 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='200 f 09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='175 50 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='150 25 25 25 w (meV) w (meV) w (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='125 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='075 25 25 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='050 50 50 50 v=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='0 v=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='025 v=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='61 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='000 K M K K M k K M K6 DOS(ω = 0) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The spectral weight is shifted from the quasiparticle peak to a newly formed Hubbard band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With decreasing temperature from 6 K to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2 K the sharp asymmetric peak at finite energy slightly ap- proaches the chemical potential, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3a and extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Therefore the maximum in the DOS(ω = 0), and the minimum in the inverse compressibility, are ex- pected to appear slightly closer to the integer filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We find pseudogaps in the DOS at zero energy ν = 1, −1 and −2 which shift with slight doping and are smaller when the temperature is reduced, suggesting that they could close at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' However, we emphasize that the physics of the cascades discussed here is not linked to whether there is a a pseudogap or small gap at exactly in- teger fillings but to a reorganization of the spectral weight at all dopings of the flat bands in a large energy range, with incoherent bands starting from interactions around U and progressively approaching the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With an increase of the temperature to 12K or 24 K the spectrum is more sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The peak shifts to higher en- ergies, becomes broader, more symmetric and its height decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Close to ν = −2, the peak height is reduced approximately to half of its value when increasing the temperature from 6K to 24 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Notably, the spectrum does not return to its non-interacting or Hartree shape at high temperatures, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The incoherent spectral weight is resilient with temperature, becoming broadened and less defined around integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This is consistent with experimental findings which show that broadened cascade signatures persist to at least 50 K[17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascade energy and doping dependence can be un- derstood in terms of the tendency of the AAp orbitals to form local moments and become incoherent, and their hybridization to the lc orbitals which accomodate part of the doping and tend to screen the local moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In the absence of coupling between the AAp and the less correlated electrons at integer fillings, the AAp orbitals would become Mott insulators with large gaps and their spectral weight would become incoherent and shifted to the Hubbard bands, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Doping these Mott insula- tors drives them metallic creating a heavy quasiparticle at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Resets in the shape and position of the Hubbard bands would appear around integer fillings, as with hole doping the quasiparticle is created in the lower Hubbard band, while with electron doping the quasipar- ticle appears in the upper Hubbard band[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The reor- ganization of the spectral weight as a function of energy and doping shows a qualitatively different shape, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S8, compared to the cascades that arise when the coupling between the AAp and the lc orbitals is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This can be better seen comparing the evolution of the DOS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 with the momentum resolved bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Around the CNP, the screening of the local moments is not operative, at least at the considered temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The AAp orbitals become incoherent and shift their spec- tral weight towards the broad Hubbard bands with barely no spectral weight around zero energy, extended Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='S2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' However, the DOS does not present a Mott gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to the fragile topology of TBG, the lc electrons contribute around Γ, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' When doping the system with elec- trons around the CNP it is energetically more favorable for the system to add these electrons to the lc orbitals, producing a band deformation around Γ, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The spectral weight of the AAp is still highly incoherent at this doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The lc orbitals contribute in a small k- region and the density of states is small at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As the lc orbitals are doped, the onsite energies of the two type of electrons are shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The upper Hubbard band of the AAp orbitals changes its shape with a main peak approaching the chemical potential, while the lower Hubbard band shifts downwards in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This explains the evolution of the cascade peaks close to CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With further doping, the AAp upper Hubbard band reaches the chemical potential and a quasiparticle peak pinned at the Fermi level with primary contribution of the AAp orbitals is formed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The AAp quasiparticle band is very flat and its contribution to the density of states high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Once the heavy AAp quasiparticle is formed, it be- comes advantageous to dope the AAp orbitals and the lc orbitals are slightly emptied, modifying again the band structure at Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At ν = 1 there is no quasiparticle at the Fermi level, at least at the temperature of the cal- culations, but the hybridization between the lc and the AAp orbitals prevents the latter from being incoherent, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' There is a reset in the spectral weight accompa- nied by a small step in the filling of the correlated AAp orbitals and a small pseudogap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Doping beyond the in- teger, the AAp electrons become incoherent and the lc orbitals are doped again giving small spectral weight to the DOS(ω = 0), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='3g, until a weak peak emerging from the upper Hubbard band reaches the chemical po- tential and a heavy quasiparticle with main contribution from AAp forms, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The process repeats for higher dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The strong momentum differentiation of inco- herence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3c to h emphasizes the role played by the fragile topology and the hybridization between the correlated AAp and the less correlated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The changes in the filling of the AAp and lc orbitals can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In spite of the strong correla- tions, the filling of the AAp does not remain constant between integer dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' On the contrary, as described above, the doping goes primarily to these AAp orbitals with just weak steps at integer dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The gap be- tween the flat and remote bands prevents more abrupt steps as charging the lc orbitals costs kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The role of the hybridization is clear in that the steps and the resets happen at integer doping, not at integer filling of the AAp orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The oscillations of the remote bands with respect to the chemical potential follow the non- monotonic occupation of the lc orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The tempera- ture dependence is also consistent with this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In 7 the calculations the screening of the local moment is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As the temperature is reduced the high peak at integer values approaches the Fermi level towards the formation of a heavy quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With increasing tem- perature, the screening of the local moment is reduced as the entropy of the local moment is higher than the Fermi liquid one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The formation of the heavy quasiparticles is expected to happen only at low temperatures while Mott physics, responsible for the incoherence, remains at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In conclusion, our work shows that the experimen- tally cascade phenomenon is not associated to a sym- metry breaking transition but to the reorganization of spectral weight due to strong correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As it hap- pens in Mott-like systems, the spectral weight becomes incoherent as correlated AAp electrons form local mo- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' However, due to the fragile topology of TBG, the coupling between the AAp orbitals to the less correlated electrons introduces a strong momentum differentiation in the incoherence, suppressing the insulating tendencies and promoting the formation of a heavy quasiparticle around integer fillings away from CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades con- nect TBG with other strongly correlated systems such as high temperature superconductors, heavy fermion sys- tems and oxides through their hallmark: the incoherent spectral weight reorganization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades constitute the normal state in which low temperature phenomena such as ferromagnetism, Chern insulators, superconduc- tivity or nematicity emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As shown recently, a strong reorganization of the spectral weight with heavy quasi- particles but without the effects of the fragile topology is expected in other moir´e heterostructures, such as the ABC/hBN trilayer[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascade phenomena origi- nates in the intra moir´e unit cell interaction U between the AAp orbitals within the unit cell, specifically to the local correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' This interaction is not much screened by gates farther than 5 nm[21] therefore the cascades are expected to survive to the proximity of gates above this distance, contrary to other effects controlled by longer range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' When we were completing this manuscript some re- lated works appeared on the arXiv[33–35] We thank conversations with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Rozenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Civelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Miranda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Aguiar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Yazdani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Chubukov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' de Juan and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Bernevig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='C, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' acknowledge funding from PGC2018- 097018-B-I00 (MCIN/AEI/FEDER, EU), M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' PID2021-125343NB-100 (MCIN/AEI/FEDER, EU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' acknowledges support from UBACyT (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 20020170100284BA) and Agencia Nacional de Promoci´on de la Investigaci´on, el Desarrollo Tecnol´ogico y la Innovaci´on (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' PICT-2018-04536).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' acknowledges the French National Research Agency (project TWIST- GRAPH, ANR-21-CE47-0018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Methods We start from a θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='08◦ TBG with interlayer tun- neling ratio between AA and AB w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='78 keep- ing the sin(θ/2) term which slightly breaks particle-hole symmetry[28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We have adapted the eight orbital model per valley and spin from [24] to fit the low en- ergy band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' We keep intra and inter-orbital density-density interactions within a distance equal to 7 moir´e lattice constant, including up to 54 neighboring cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Hund’s coupling, pair-hopping and exchange in- teractions, considerably smaller than the density-density included here[21], are neglected and the intra moir´e unit cell intra and inter-orbital interactions between the AAp orbitals are taken to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Based on the difference between the orbital dependent onsite interactions and bandwidths[21] we distinguish between strongly corre- lated AA p+ and p−, named AAp, and less correlated AA s, AB pz and DW s lc orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The later orbitals are treated at the Hartree level while the DMFT ap- proximation [26, 27] is used on the strongy correlated AAp orbitals, which acquire a frequency dependent self energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In the DMFT calculation, the coupling to the lc orbitals enters via an effective hybridization function which depends on the Hartree onsite potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to the coupling with the correlated AAp orbitals, the lc or- bitals also get a frequency dependent self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A dou- ble step self-consistency loop is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A number of DMFT iterations are run starting from a given set of Hartree onsite potentials and taking care to avoid double counting the effect of the interaction U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' As an outcome of the DMFT, the self energy, the Green’s function and the charge of the AAp orbitals are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to the hybridization between the two types of orbitals, the orbital filling of all the orbitals is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The Hartree onsite shifts are re-calculated with the new fillings and new DMFT iterations are run with the new calculated on- site potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Convergence is reached when the Green’s functions, self-energy and orbital fillings do not change between either DMFT or Hartree iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' To avoid double counting the interaction effectively included in the tight binding model parameters, we substract the onsite potentials obtained at the CNP when the Hartree ap- proximation is used for all the orbitals[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The single- site DMFT calculations are performed at a given chem- ical potential and finite temperature using a continuous time quantum Monte Carlo impurity solver [36] as imple- mented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The density at each chemical poten- tial is obtained to an accuracy ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='01 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Symme- try breaking is prohibited imposing equal fillings, onsite potentials, self-energies and Green’s functions for equiv- alent orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The analytic continuation of the Matsub- ara self-energy required to calculate the spectral weight is performed using the maximum entropy method [38] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' ∗ leni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='bascones@csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='es 8 [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Cao, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Fatemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Demir, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Werner, Continuous-time monte carlo methods for quantum impurity models, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 83, 349 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Haule, Quantum monte carlo impurity solver for clus- ter dynamical mean-field theory and electronic structure calculations with adjustable cluster base, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' B 75, 155113 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Jarrell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 269, 133 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Model details (a) Schematic plot of the triangular AA, honeycomb AB/BA, and kagome DW lattices where the eight orbitals per valley and spin are centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Besides the two correlated AAp orbitals (orange), at the AA regions of the moir´e unit cell another orbital with s character AAs is centered (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The model also includes two pz orbitals centered at the honeycomb lattice formed by the AB/BA regions (green) and three s orbitals at the kagome lattice (purple) formed by domain wall DW regions separating two AB/BA points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Orbital weight of the two correlated AAp (left) and the six less correlated orbitals (right) to the band structure of the eight orbital model used in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (c) Zoom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1a corresponding to the non-interacting flat band obtained from the continuum model (black) for an angle θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='08◦ and its fitting with the eight orbital model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (d) Intra and inter-orbital density-density interactions as a function of the distance between the orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Here λ is the moir´e unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' a 200 100 (meV) 3 100 200 K r M K r M K 8 d AAp-AAp 8=12 6 AAp-AAs AAp-AB/BA AAp-DW AAs-AAs Interactions ( 30 2 AAs-AB/BA 0 AAs-DW 20 AB/BA-AB/BA 2 AB/BA-DW DW-DW 4 10 0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='08 6 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 2 K M K r/△11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Contribution to the spectral weight of the two types of orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Contribution to the density of states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1b of the (a) two strongly correlated AAp orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Same as (a) but for the six less correlated AAs, AB / BApz and DWs orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The lc contribution is reduced at small energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Nevertheless some effects of the resets can be also appreciated in their spectral weight at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The contribution of the AAp orbitals will be more visible in STM measurements at the AA site, while the spectral weight of the less correlated orbitals will have larger weight at the AB position (the AAs has an annular shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to the finite extension of the orbitals, this separation of the orbitals in spatial regions should not be taken too literal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='28 a b 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='06 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='21 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='14 0 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='02 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='07 2 3 3 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='00 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='00 135-90 -45 ¥045 90135 135-90 -45 50459 90135 w (meV) w (meV)12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Density of states and band deformation in the Hartree approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1b but treating all the interactions in the Hartree approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The cascades of spectral weight and the oscillations in the remote band van Hove peaks have disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) and (c) Low energy bands calculated for ν = −2 and ν = 2 in the Hartree approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The band shape has changed with respect to the non-interacting one in (d) as discussed extensively[21, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Nevertheless, the effect of such changes on the spectral weight reorganization in (a) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Oscillations of the van Hove of the remote bands .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Dependence of the remote band van Hove energies as a function of doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The red lines are an approximation to the doping dependent shift of the van Hove positions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='75 b v=-2 a 40 3 20 w (meV) 0 2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='50 40 1 K M K C v=2 1 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='25 20 (meV) 2 0 3 20 3 40 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='00 90 135 K M K 135-90 -45 045 w (meV)135 90 45 (meV) 0 3 45 90 135 4 3 2 1 0 i 2 3 4 V13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Momentum resolved spectral weight for |ν| > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Interacting bandstructure obtained from the DMFT+Hartree calculations for dopings ν < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='27, (b) ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='28 and (c) ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' It can be observed how the remote bands cross the chemical potential contributing significantly to the DOS at the Fermi level and to the doping dependence of the spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The contribution of the remote bands here resembles the one in charge transfer insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Due to this crossing at large dopings the filling of the lc orbitals is less sensitive to the integer dopings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='3 and the oscillations of the van Hove energies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='1 are not so well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Chemical potential as a function of doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In our calculation we can define two different chemical potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The local chemical potential µlocal in (a) enters into the DMFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Its value is primarily controlled by the interaction U and the filling of the AAp orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In (b) The global chemical potential µtotal strongly depends on the long range interactions and account primarily for a global shift of all the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' µtotal has a very large magnitude and an almost linear dependence on the density which closely follows the value obtained when all the interactions, including U, are treated at the Hartree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' In the DMFT+Hartree calculation µtotal has small steps at the integer dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The derivative of µtotal (not shown) is dominated by the linear dependence and the features around the integers are quite noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Temperature dependence of the spectral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Dependence of the density of states on temperatures for dopings close to ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) ν=2 and (b) ν ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='77-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='78 showing a similar behavior as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3 for ν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With increasing temperature the peak shifts towards positive energies, its height decreases and the shape is slightly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The pseudogap around zero or small energies slightly decreases with decreasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (c) and (d) Band structure corresponding to the DOS in at T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='2K and T=24K discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' With increasing temperature, the bands become more incoherent and small band shifts are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='200 V~ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='78 V~-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='0 v=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='97 T=1.' metadata={'source': 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+page_content=' 75 - 75- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='000 K M K K M K K M K300 1000 a b 750 250 500 (meV) 200 (meV) 250 150 0 local total 250 100 U 500 DMFT+Hartree 50 750 All Hartree 1000 0 4 3 2 1 0 1 2 2 3 4 3 2 1 1 3 4 4 014 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Density of states of an only AAp model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (a) Colour plot corresponding to the density of states calculated with DMFT of a Hubbard model containing only the AAp orbitals (four orbital model degenerate in spin) interacting with both intra and inter-orbital interaction U=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 meV, as in the eight orbital model, for fillings n within the range −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='5 < n < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Here half-filling (the equivalent for this model to the CNP) is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' A clear reorganization of the spectral weight with doping extending up to energies of order U is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At each integer the lower (upper) Hubbard band approaches the chemical potential at integer fillings if the system is doped with holes (electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Strong resets in the spectral weight are found associated to the gaps at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At non-integer fillings a large density of states associated to the formation of a heavy quasiparticle, typical of doped Mott insulators, is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' (b) Line cuts of the density of states for selected dopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' Curves are shifted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The typical three peak structure is observed at partial non- integer fillings, being the thin peak at zero energy the quasiparticle peak and the two broad peaks at a few tens meV the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' The Hubbard bands shift with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' At integer values there are large Mott gaps and resets in the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdFPT4oBgHgl3EQfLzS8/content/2301.13024v1.pdf'} +page_content='0 a n=4 0.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..dca0cf9e43c1fb7c98be58ad37942c1525b659b9 --- /dev/null +++ b/rtFST4oBgHgl3EQfQDhb/content/tmp_files/2301.13757v1.pdf.txt @@ -0,0 +1,2145 @@ +Toward Efficient Gradient-Based Value Estimation +Arsalan Sharifnassab 1 +and +Richard Sutton 1 +Abstract +Gradient-based methods for value estimation in +reinforcement learning have favorable stability +properties, but they are typically much slower +than Temporal Difference (TD) learning methods. +We study the root causes of this slowness and +show that Mean Square Bellman Error (MSBE) +is an ill-conditioned loss function in the sense +that its Hessian has large condition-number. To +resolve the adverse effect of poor conditioning of +MSBE on gradient based methods, we propose a +low complexity batch-free proximal method that +approximately follows the Gauss-Newton direc- +tion and is asymptotically robust to parameter- +ization. Our main algorithm, called RANS, is +efficient in the sense that it is significantly faster +than the residual gradient methods while having +almost the same computational complexity, and is +competitive with TD on the classic problems that +we tested. +1. Introduction +Value estimation is a core problem in reinforcement learning +(Sutton & Barto, 2018), and is a key ingredient in several +policy optimization methods, e.g., (Bhatnagar et al., 2009; +Minh et al., 2015; Lillicrap et al., 2015) . The popular class +of value estimation algorithms based on temporal difference +learning via forward bootstrapping; including TD(λ) (Sut- +ton, 1988), Expected Sarsa (van Seijen et al., 2009), and +Q-learning (Watkins & Dayan, 1992) have found substan- +tial empirical success when combined with proper policy +optimization (Minh et al., 2015; Minh et al., 2016; Lilli- +crap et al., 2015). Nevertheless, these algorithms are not +gradient-based optimization methods (Barnard, 1993) and +their convergence cannot be guaranteed for general func- +tion approximation setting (Baird, 1995; Tsitsiklis & Van +Roy, 1997; Brandfonbrener & Bruna, 2019). The stabil- +ity problem of TD learning has inspired other classes of +value estimation algorithms that involve optimizing a loss +1Authors are with the Department of Computing Science, +University of Alberta, Canada. +function through gradient updates. This includes Resid- +ual Gradient (RG) algorithm for minimizing Mean Squared +Bellman Error (MSBE) (Baird, 1995), Gradient-TD algo- +rithms for minimizing projected Bellman error (Sutton et al., +2009; Maei et al., 2010; Maei, 2011; Hackman, 2012), and +their extensions for optimizing a dual formulation of BE2 +(Liu et al., 2015; Macua et al., 2014; Dai et al., 2017). These +algorithms enjoy the general robustness and convergence +properties of Stochastic Gradient Descent (SGD), but are +known to be slower than TD in tabular and linear function +approximation settings (Baird, 1995; Schoknecht & Merke, +2003; Gordon, 1999; Ghiassian & Sutton, 2021) . +In this paper, we investigate the root causes of the slow- +ness problem of gradient-based value estimation by taking a +deeper look into the landscape of MSBE, and propose linear +complexity methods to alleviate these problems. We provide +theoretical results showing that MSBE is an ill-conditioned +loss function in the senses that the condition-number of +its Hessian matrix is typically very large. This explains +slowness of gradient-based value estimation methods, be- +cause gradient descent in general is slow in minimizing +ill-conditioned loss functions. In contrast, algorithms like +Newton and Gauss-Newton methods are invariant to condi- +tioning of the loss. Unfortunately a direct implementation of +these methods requires matrix inversion, which is computa- +tionally costly even if computed incrementally. We propose +a linear complexity incremental algorithm, called Resid- +ual Approximate Gauss-Newton (RAN), that incorporates a +trace to approximate the Gauss-Newton direction and then +updates the weights along that trace. We show that RAN +can be equivalently formulated as a batch-free proximal +algorithm. A weakness of RAN is that it requires double +sampling (Baird, 1995), which limits its use in stochastic en- +vironments. We propose a double-sampling-free extension +of RAN by following similar ideas that underlie GTD-type +methods. The resulting algorithms significantly outperform +RG and GTD2, being orders of magnitudes faster on the +simple classic environments that we tested, while having +almost similar computational complexity to RG and GTD2. +We then turn our focus to a second cause of slowness of +gradient-based value estimation: under function approxi- +mation, sample gradients of MSBE involve large outliers +that carry important information, resulting in large variance +of stochastic updates. Outliers of this type often appear +arXiv:2301.13757v1 [cs.LG] 31 Jan 2023 + +Toward Efficient Gradient-Based Value Estimation +in every episode (usually at pre-terminal transitions), and +are specific to gradient-based value estimation methods (i.e. +such outliers do not appear in TD learning). We propose a +general technique called outlier-splitting, which results in no +information loss as opposed to the standard clipping meth- +ods. Our main value estimation algorithm, called RAN with +outlier-Splitting (RANS), has linear computational com- +plexity and has only one effective hyper-parameter (and +some other hyper-parameters that can be set to their default +values), thanks to its adaptive step-size mechanism. Our +empirical results on a few classic control environments with +neural network function approximation show significant im- +provement over RG, and achieving competitive performance +to TD. +2. Background +We consider a discounted Markov Decision Process (MDP) +defined by the tuple (S, A, R, p, γ), where S is a finite +set of states, A is a finite set of actions, R is a set of re- +wards, p : S × A × S × R → [0, 1] is the environment +dynamics determining the probability of the next state and +immediate reward given a current state and action pair, and +γ ∈ [0, 1] is a discount factor. We fix a stationary policy +π : S × A → [0, 1], and let pπ(s′, a′, r|s, a) = p(St+1 = +s′, Rt+1 = r|St = s, At = a)π(At+1 = a′|St+1 = s′). +We consider an episodic and online setting where a data +stream (S1, A1, R1), (S2, A2, R2), . . . is generated accord- +ing to the policy π. The action-value function qπ : S ×A → +R, at each state s and action a, is the expected discounted +sum of rewards obtained by starting from state s and action +a and following policy π. We then define the value function +vπ : S → R as vπ(s) = Ea∼π(·|s)[qπ(s, a)]. +In value estimation, we aim to obtain an estimate of the +true action-values qπ, usually through a function qw : S × +A → R parameterized by a d-dimensional weight vector w. +Corresponding to qw is a Bellman residual at each state and +action pair (s, a), defined as +δw(s, a) +def= Es′,a′,r∼pπ(·,·,·|s,a) +� +r+γqw(s′, a′)−qw(s, a) +� +. +According to Bellman equations (Sutton & Barto, 2018; +Bertsekas & Tsitsiklis, 1996), qw = qπ if and only if +δw(s, a) = 0 for all (s, a) ∈ S × A. +In this view, +MSBE(w), defined below, serves as a proxy for the quality +of estimates w: +MSBED(w) +def= E(s,a)∼D +� +δw(s, a)2� +, +(1) +where D is some distribution over states and action pairs. +When the distribution is online, we drop the subscript D +and write MSBE(·). For simplicity of notation, we also +write Es,a[·] to denote the expectation with respect to state +and action pairs sampled from the online distribution. In the +same vein, we consider a parameterized estimate vw : S → +R of value function vπ, and let +MSBEV +D(w) +def= Es∼D +� +δw(s)2� +, +(2) +where δw(s) = Ea∼π(·|s)[δw(s, a)], and D is some distribu- +tion over states. +Gradient-based value estimation methods use gradient- +based optimization algorithms to minimize MSBE or other +related objectives such as MSPBE (Sutton et al., 2009). The +first and simplest method in this category is the RG algo- +rithm (Baird, 1995). In this algorithm, to obtain an unbiased +sample estimate of ∇w(δw(St, At)2), we require indepen- +dent samples (St+1, At+1, Rt) and (S′ +t+1, A′ +t+1, R′ +t) from +pπ(·, ·, ·|St, At). For simplicity of notation, at time t, we let +δt +def= Rt + γqw(St+1, At+1) − qw(St, At), +(3) +δ′ +t +def= R′ +t + γqw(S′ +t+1, A′ +t+1) − qw(St, At), +(4) +where wt = w. The RG update is then +w ← w − αδ′ +t∇wδt. +(5) +The requirement for two independent sample transitions at +time t is called double sampling (Sutton & Barto, 2018). +In stochastic environments, double sampling is possible +only if we have a correct model of the world. In other +words, MSBE minimizer is not learnable if an exact model +of the underlying stochastic environment is not available, +which is the case in real-world applications (Sutton & Barto, +2018). A general technique to circumvent double sampling +is using Fenchel duality to obtain an equivalent saddle point +formulation of MSBE (Dai et al., 2017) as +min +w max +ˆδ(·,·) +Es,a +� +δw(s, a) ˆδ(s, a) − 1 +2 +ˆδ(s, a)2 +� +, +(6) +where ˆδ(s, a) is an auxiliary variable that serves as +a proxy of δw(s, a). +In practice, one can consider +a parametric approximation ˆδθ(·, ·) of ˆδ(·, ·), and per- +form gradient updates on the resulting minimax problem +minw maxθ Es,a +� +δw(s, a) ˆδθ(s, a) − 1 +2 ˆδθ(s, a)2� +: +w ← w − αˆδθ(St, At)∇wδt, +θ ← θ + η +� +δt − ˆδθ(St, At) +� +∇θˆδθ(St, At), +(7) +(Sutton et al., 2009; Liu et al., 2020; Dai et al., 2017). In- +tuitively, this is similar to the RG algorithm in (5) except +for using the parametric approximation ˆδθ(St, At) instead +of δ′ +t, and updating ˆδθ(s, a) by SGD on Es,a +� +(ˆδθ(s, a) − +δw(s, a))2� +. The GTD2 algorithm (Sutton et al., 2009) is a +special case of (7) in which qw and ˆδθ are linear approxima- +tions of the form qw(s, a) = φT +s,aw and ˆδθ(s, a) = φT +s,aθ, +for feature vectors φs,a (Liu et al., 2020; Dai et al., 2017). + +Toward Efficient Gradient-Based Value Estimation +3. MSBE loss is ill-conditioned +The condition-number of a symmetric square matrix, H, +is defined as the ratio of its largest to smallest singular +values, maxx:∥x∥=1 |xT Hx|/ miny:∥y∥=1 |yT Hy|. For a +quadratic function f(x) = xT Hx, we define the condition- +number, C(f), of f as the condition-number of its Hessian +matrix H. Intuitively, level sets (or contours) of a con- +vex quadratic function have an elliptical shape, and the +condition-number C(f) equals the squared ratio between +the largest and the smallest diameters of each of these el- +lipsoids (see Fig. 1). We say that f is ill-conditioned if +C(f) is very large. Then, the level sets of an ill-conditioned +quadratic function have the shape of ellipsoids that are thin. +It is known that the convergence rate of the gradient descent +on a quadratic loss f scales with C(f) (Polyak, 1964), which +can be very slow for ill-conditioned loss functions. +In this section, we consider linear function approxima- +tion. In this case, MSBEV +D(·) defined in (2) is a convex +quadratic function. We denote the condition-number of +MSBEV +D(·) under uniform distribution D by C. We let l +be the average episode length, defined as the expected time +until termination when starting from a state, uniformly av- +eraged over all states. We also let h +def= Es∼unif +� +P(St+1 = +s|St = s) +� +be the the average self-loop probability. Note +that h is typically much smaller than 1. +Theorem 3.1. In the tabular case, the following statements +hold for any discount factor γ ∈ [0, 1]: +a) For any MDP and under any policy, we have +C ≥ 1 − γh +4 +min +� +1 +(1 − γ)2 , l2 +� +(8) +where l is the average episode length and h is the +average self-loop probability. +b) For any n ≥ 1, there exists an n-state MDP and a +policy for which C ≥ γ4n2/(1 − γ)2. +The proof is given in Appendix A.1. A similar result also +holds for the condition-number of MSBE defined in (1) +(see Proposition A.1 in Appendix A.2). Theorem 3.1 shows +that MSBE is typically ill-conditioned in the tabular case. +This explains the slow convergence of gradient-based meth- +ods for minimizing MSBE. As an example, the bound in +(8) implies that for γ = .99 and for any MDP and policy +pair with average episode length at least 100 and average +self-loop probability no larger than 0.1, we have C > 2000. +Moreover, Theorem 3.1 (b) implies that for γ = .99, there +is a 100-state Markov chain for which C > 96, 000, 000. +Lower bounds similar to Theorem 3.1 are not possible for +non-tabular linear function approximation. This is because +different feature representations can improve or worsen the +Figure 1. Level sets of MSBE (gray curves) in a 2-state loop en- +vironment with p(s0 → s1) = p(s1 → s0) = 1 for γ = 0.8. +Here, condition number of MSBE is 81, and is equal to the squared +ratio between the diameters (red) of each ellipsoid. The solution +trajectory of RG (blue) for α = .9 and the Gauss-Newton direction +(green) are also depicted. In this environment, C = O(1/(1−γ)2) +(see Theorem 3.1), which rapidly grows for larger γ. +condition-number. To see why, note that in the linear func- +tion approximation case and under uniform state distribution, +MSBEV +unif(w) = wT ΦT (I − γP)T (I − γP)Φw, where +Φ is an n × d matrix, each row of which is a feature vector +of a state; and P is the transition matrix. For the specific +choice Φ = (I − γP)−1 we obtain C = 1, while for the +case that Φ is not full-rank, we have C = ∞. +In general, since underparameterized function approxima- +tion reduces parameters dimension, it usually improves +condition-number. Fig. 2 illustrates dependence of C on +the number of features, d, in an extended Boyan chain en- +vironment (Boyan, 2002) with 200 states and with random +binary features (see Appendix G.1 for details). We ob- +serve that smaller d results in better condition-number; but +this comes at the cost of larger value-error at MSBE min- +imum (the red curve in Fig. 2), where by value-error we +mean Es∼unif +� +Ea∼π(·|s)[∥qw(s, a) − qπ(s, a)∥2] +� +. See Ap- +pendix B for more experiments on condition number under +linear function approximation. +4. A review of the Gauss-Newton method +Consider an expected loss function of the form F(w) = +Ef[f 2(w)], and the associated Hessian matrix HF += +E[∇f∇f T ] + E[f Hf], where Hf denotes Hessian of sam- +ple function f. +The first term on the right hand side, +E[∇f∇f T ], is called the Gauss-Newton matrix and is de- +noted by G. The Gauss-Newton algorithm then updates w +as w ← w − αG−1∇F(w). In the special case that func- +tions f are linear, we have Hf = 0 and thereby HF = G. +In this case, Gauss-Newton and Newton methods become +equivalent. However, the Gauss-Newton algorithm has two +advantages in the non-linear case. Firstly, G−1∇F(w) is + +0.8 +Gauss-Newton direction +0.4 +RG trajectory +0 +-0.4 +Contours of MSBE +-0.8 +-0.8 +-0.4 +0 +0.4 +0.8Toward Efficient Gradient-Based Value Estimation +Figure 2. Condition-number of MSBE (blue) and value-error at +MSBE minimizer (red) versus number of features, in a 200-state +extended Boyan chain with random binary features. +always a descent direction, as opposed to the Newton up- +dates that may climb uphill and converge to local maxima or +saddle points (Nesterov & Polyak, 2006). Secondly, G can +be computed in terms of gradients while H entails second +order derivatives which are not as easily accessible in certain +settings (Nocedal & Wright, 1999). +As opposed to gradient descent which is prohibitively slow +in ill-conditioned problems, Newton and Gauss-Newton +methods are invariant to conditioning of the loss function. +Some recent works proposed using Gauss-Newton method +for value estimation (Gottwald et al., 2021). However, these +algorithms require matrix inversion, which is computation- +ally costly even if computed incrementally via Sherman- +Morrison formula with quadratic complexity (Sherman & +Morrison, 1950). In the next section, we propose a linear- +complexity method for MSBE minimization. +5. Our first algorithm: RAN +The Gauss-Newton direction for minimizing MSBE +is mGN(w) += +G−1 +w ∇MSBE(w)/2, where Gw += +Es,a +� +∇δw(s, a) ∇δw(s, a)T � +is the Gauss-Newton matrix +and MSBE(w) = 2Es,a +� +δw(s, a) ∇δw(s, a) +� +. +Then, +mGN is the minimizer of the following quadratic function: +L(m) +def= 1 +2 Es,a +�� +δw(s, a) − ∇δw(s, a)T m +�2� +. +(9) +This is because for any m, +∇mL(m) = Es,a +�� +∇δw(s, a)T m − δw(s, a) +� +∇δw(s, a) +� += Gwm − ∇MSBE(w)/2, +and therefore ∇mL(mGN) = 0. We follow a two time +scale approach (Bhatnagar et al., 2009; Dabney & Thomas, +2014) to incrementally find an approximate minimizer m +of L and update w along that direction. More concretely, +given a β > 0 and λ ∈ [0, 1], at time t, we update m along +Algorithm 1 +RAN +Parameters: step-sizes α, β, and decay parameter λ +Initialize: m = 0 and w +for t = 1, 2, . . . do +consider δt and δ′ +t defined in (3) and (4), respectively +m ← λm + β δ′ +t ∇δt +m ← m − β(mT ∇δt)∇δt +w ← w − αm +end for +an unbiased sample gradient of βL(m) + (1 − λ)∥m∥2, +m ← λm + β +� +δ′ +t − mT ∇δ′ +t +� +∇δt, +(10) +where δt and δ′ +t are defined in (3) and (4), and (1 − λ)∥m∥2 +is a Levenberg–Marquardt regularizer (Marquardt, 1963)1. +We then update w along m, i.e, w ← w − αm. +Algorithm 1 gives the pseudo code of the RAN algorithm. +For better stability and faster convergence, the update of m +in RAN is of the form +m ← λm + β +� +δ′ +t − mT ∇δt +� +∇δt. +(11) +which is the same as (10) except for using ∇δt instead of +∇δ′ +t. The updates in (11) have lower variance compared +to (10), and additionally ∇δt∇δT +t in (11) is positive semi- +definite, as opposed to the finite sample estimate of the +Gauss-Newton matrix 1 +τ +�τ +t=1 ∇δt∇δ′T +t +in (10). For fixed +w, the update in (11) is in expectation along +− +� +Et[∇δt∇δT +t ] + (1 − λ)I +�−1∇MSBE(w). +(12) +In Appendix C, we provide further intuition for RAN, by +presenting a derivation of Algorithm 1 as a proximal method +with momentum for minimizing MSBE. In this view, m +serves as a momentum of MSBE gradients, to which we add +a correction term equal to the gradient of a penalty function +that aims to regularize the change in δw(s, a) for all state +action pairs (s, a). +Convergence of the RAN algorithm can be shown in two- +time-scale regime where αt, βt → 0, with α diminishing +faster than β (i.e., αt/βt → 0)2. Convergence of such +two-time-scale algorithms is well-studied (Kushner & Yin, +2003; Konda & Tsitsiklis, 1999; Bhatnagar et al., 2009), +under some smoothness and irreducibly conditions. In Ap- +pendix D, we discuss different conditions for convergence +1We have empirically observed that λ = 1 often leads to slow +convergence, because it causes large inertia in m, and therefore +large oscillations in w. The best performance is achieved for +λ ∈ (0.99, 0.9999). +2Note that the two-time-scale view is only for the purpose of +convergence analysis, and in practice we consider fixed or adaptive +step-sizes whose ratio needs not go to zero. + +(log scale) +104 +0.5 +um +0.4 +103 +Condition number of MSBE +r at MSBE +0.3 +100 +0.2 +Error +10 +Value +0.1 +1 +0 +20 +40 +60 +80 +100 120 +140 +160 +180 +Number of featuresToward Efficient Gradient-Based Value Estimation +Figure 3. The Hallway experiment discussed in Section 5. +of Algorithm 1 in the two-time-scale regime. Moreover, in +this regime, RAN is robust to reparameterization: +Proposition 5.1 (Informal). For λ = 1 and asymptotically +small step-sizes α → 0 and α/β → 0, the trajectory of w +in the RAN algorithm is invariant to any differentiable and +bijective non-linear transformation on parameterization. +The formal version of Proposition 5.1 and its proof are given +in Appendix E. +We evaluated the performance of RAN in a simple bench- +mark environment. Consider an environment with n states +and one action, in which each state i = 1, 2 . . . , n transits to +state min(i + 1, n) with probability 1 − ϵ, and transits to a +terminal state with probability ϵ, for some ϵ ∈ [0, 1). This is +a generalization of the Hallway environment (Baird, 1995), +and is known to be a challenging task for the RG algorithm +(Baird, 1995). We tested Algorithm 1 in this environment +with n = 50, ϵ = 0.01, and γ = 0.99 in the tabular setting +(see Appendix G.2 for the details of this experiment). The +learning curves are depicted in Fig. 3. We observe that, in +this experiment, Algorithm 1 is about 30 times faster than +RG, and reaches a convergence rate close to TD(0) (Sutton, +1988; Sutton & Barto, 2018). +6. Double-sampling-free RAN algorithm +In Algorithm 1, we require double sampling to compute δ′ +t. +In this section, we propose a Double-Sampling-Free version +of RAN, called DSF-RAN. Double sampling is easily doable +in deterministic environments (Saleh & Jiang, 2019; Zhang +et al., 2020), in which case δ′ +t can be computed using an +independent sample A′ +t+1 from the policy. However, for +double sampling in stochastic environments, we require a +model to get an independent sample S′ +t+1 of the next state, +which is typically possible only in simulated environments. +To resolve the double sampling issue of RAN in stochastic +environments, we use the technique discussed in Section 2, +which was also used in the GTD2 algorithm. More specif- +Algorithm 2 +DSF-RAN +Parameters: step-sizes α, β, η, and decay parameter λ +Initialize: m = 0, w, θ. +for t = 1, 2, . . . do +∇δt = γ∇wqw(St+1, At+1) − ∇wqw(St, At) +m ← λm + β ˆδθ(St, At) ∇δt +m ← m − β(mT ∇δt)∇δt +w ← w − αm +θ ← θ + η +� +δt − ˆδθ(St, At) +� +∇θˆδθ(St, At) +end for +Figure 4. The Baird’s star experiment discussed in Section 6 +ically, instead of δ′ +t in Algorithm 1, we use a parametric +approximation ˆδθ(St, At) of δw(St, At), parameterized by +θ. Similar to GTD2 (see (7)), we then learn θ through SGD +on Es,a +� +(ˆδθ(s, a)−δw(s, a))2� +. Pseudo code of DSF-RAN +is given in Algorithm 2. +We tested RAN and DNS-RAN algorithms on Baird’s Star +environment (Baird, 1995), that is a Markov chain with six +states, each represented by seven features (see Appendix G.3 +for details of this experiment). The results are illustrated in +Fig. 4. We observe that in this environment, RAN and DNS- +RAN converge about 200 times faster than RG and GTD2 +algorithms, respectively. It is well-known that off-policy +TD(0) is unstable in this environment (Baird, 1995). +7. The problem of outliers +In this section we argue that the gradient of MSBE involves +large outliers and discuss its impact on the RAN algorithm. +For simplicity, temporarily suppose that the set of actions is +a singleton, A = {a}. In the function approximation case, +successive states St and St+1 often have similar representa- +tions. As a result, ∇qw(St, a) and γ∇qw(St+1, a) are often +similar, rendering ∇δt = γ∇qw(St+1, a) − ∇qw(St, a) to +be small (Zhang et al., 2020). This wouldn’t have been prob- +lematic if ∇δt was small for all t, in which case we could +compensate by increasing the step-size. However, ∇δt can + +1 +TD(0) +RAN +RG +0.8 +0.6 +0.4 +0.2 +M +0 +0 +20 +40 +60 +80 +100 +Steps +s(x1000)TD(0) +15 +20 +RG +RAN +Mean sguared value error +10 +GTD2 +15 +DSF-RAN +5 +10 +0 +0 +1000 +2000 +3000 +5 +M +0 +0 +50 +100 +150 +200 +Steps (x1000)Toward Efficient Gradient-Based Value Estimation +occasionally be large, for example when St+1 is a terminal +state in which case ∇δt = −γ∇qw(St, a), or when St+1 is +far from St (e.g., in large jump transitions). Although these +outliers occur with low probability, they carry important +information. For example, the pre-terminal transitions are +important because they pin down the estimated values to +the terminal values. In environments with larger action sets, +if the policy has small entropy, At+1 and At would have +similar representations with high probability, causing ∇δt +to be small. +We now discus how these outliers affect RAN. The up- +dates of m in Algorithm 1 involve a momentum (of +MSBE gradient) term λm + δ′ +t∇δt and a correction term +−β(∇δT +t m)∇δt that aims to slowly modify m towards the +approximate Gauss-Newton direction. However, when ∇δt +is an outlier, β(∇δT +t m)∇δt can grow very large, cause an +overshoot, and completely change the direction of m. In +particular, if β∥∇δt∥2 > 1, then magnitude of the correc- +tion term would be larger than the projection of m on ∇δt, +i.e. +��⟨β(∇δT +t m)∇δt, ∇δt⟩ +�� > +��⟨∇δT +t m⟩ +��, +(13) +which results in an overshoot along ∇δt. Such overshoots +hinder m from tracking the approximate Gauss-Netwon +direction. +To reduce the adverse effect of outliers, one can reduce step- +size β, at the cost of slowed down learning. Another popular +solution is gradient clipping (Zhang et al., 2019). However, +as discussed in the first paragraph of this section, the outliers +in our problem carry important information, which can be +lost via gradient clipping. +8. Outlier-splitting +We now propose outlier-splitting as a general meta- +technique for stochastic optimization, appropriate for the +case that data contains rare sample functions with abnor- +mally large gradients, and these sample functions carry +important information that would be lost in gradient clip- +ping. We first explain the key idea by an example. Consider +minimizing f1 + · · · + fn for smooth functions f1, . . . , fn. +Suppose that f1 is an outlier in the sense that the norm +of its gradient is locally k times larger than the gradient +norms of other functions, for some integer k > 1. The +idea is that instead of applying SGD on f1 + · · · + fn, we +break down f1 into k copies of f1/k and apply SGD on +f1/k + · · · + f1/k + f2 + · · · + fn in a random order. The +latter updates are outlier-free while being equivalent to the +former updates in expectation. We now proceed to a formal +description. +Consider SGD on an objective function F = E[f]. For any +sample sample function f and any point w, we consider a +non-negative measure ξ(f, w); e.g., ξ(f, w) = ∥∇f(w)∥ +Algorithm 3 +Outlier-splitting for online SGD, applied to +loss function F = E[f] +Parameters: step-size β, outlier threshold ρ, trace pa- +rameter λξ, outlier sampling probability σ. +Initialize: ˆξ = 0, w. +for t = 1, 2, . . . do +ˆξ ← λξ ˆξ + (1 − λξ)ξ(ft, w) +¯ξ = ˆξ/(1−λt +ξ) +▷ bias-corrected trace estimate +k = ⌊ξ(ft, w)/(ρ¯ξ)⌋ + 1 +w ← w − (β/k)∇ft(w) +if k > 1 then +Store (f, k, k − 1) in the outlier buffer +end if +With +probability +min(1, σ +∗ +length of outlier bufffer): +Sample (f, k′, j) uniformly form outlier buffer +k′′ = max +� +k′, ⌊ξ(f, w)/(ρ¯ξ)⌋ + 1 +� +w ← w − (β/k′′)∇f(w) +if j > 1 then +Replace (f, k′, j) with (f, k′, j − 1) in the buffer +else if j = 1 then +Remove (f, k′, j) from the outlier buffer +end if +end for +or ∥∇f(w)∥2. Let ¯ξ be a trace of ξ, updated by ¯ξ ← +λξ ¯ξ + (1 − λξ)ξ(ft, wt), where λξ ∈ (0, 1) is a constant +close to 1. We say that ft is an outlier if ξ(ft, wt) ≥ ρ¯ξt, +for some outlier threshold ρ > 1. +The pseudo code of the outlier-splitting method for online +SGD is given in Algorithm 3. At time t of this algorithm, +we let +k = +�ξ(ft, wt) +ρ¯ξt +� ++ 1. +(14) +If ft is an outlier (equivalently k > 1), instead of ft we +pretend to have k copies of ft/k. We use one of these copies +to do a gradient update at time t, and store the remaining +k−1 copies in a buffer to use them for future updates. These +copies are stored in one cell of an outlier-buffer as a tuple +(ft, k, k−1), where k−1 indicates the number of remaining +copies to be used for future updates. In each iteration we +perform one update based on the online sample, and perform +at most one update based on a sample from the buffer. More +concretely, in each iteration t, after applying a gradient +update w ← w−(β/k)∇ft(w), we take a sample (f, kf, j) +from the outlier buffer with some positive probability, and +perform a gradient update w ← w − (β/kf)∇f(w). +We now show that the outlier buffer is stable. The expected +number of copies, k−1, added to the buffer at time t satisfies +E[k − 1] ≤ Et +�ξ(ft, wt) +ρ¯ξt +� +≃ Et [ξ(ft, wt)] +ρE[¯ξt] += 1 +ρ < 1, + +Toward Efficient Gradient-Based Value Estimation +where the inequality is due to (14) and the approximate +equality is because ¯ξt is a long-time average. On the other +hand, as the length of the outlier buffer increases, the proba- +bility of performing a sample update from the buffer goes to +1. In this case, arrival rate to the buffer, 1/ρ, is smaller than +its departure rate, 1; implying stability of the outlier buffer. +9. Our main algorithm: RANS +Our final algorithm, RAN with outlier Splitting (RANS), is +a combination of RAN, outlier-splitting, and adaptive step- +size ideas. In order to improve updates of m, we employ +an adaptive vector step-size β that evolves according to +a mechanism quite similar to RMSProp (Kochenderfer & +Wheeler, 2019), as we discuss next. +Consider a trace vector νt of (∇δt)2 updated according to +νt ← λ′νt−1 + (1 − λ′)(∇δt)2, +where (∇δt)2 is the entrywise square vector of ∇δt, and +λ′ ∈ [0, 1) is a constant. We consider an outlier-measure +ξt = ⟨ 1 +√νt +⊙ ∇δt, ∇δt⟩ +(15) +where 1/√νt is entrywise square root, and ⊙ and ⟨·, ·⟩ de- +note entrywise product and inner product of two vectors, +respectively. We then compute the trace ¯ξ and k as in Sec- +tion 8: ¯ξt ← λ′ ¯ξt + (1 − λ′)ξt and k = +� +ξt/(ρ¯ξt) +� ++ 1. We +finally fix an η ∈ (0, 1) and choose the step-size +βt = η +ρ¯ξt +1 +√νt +. +(16) +The pseudo code of RANS is given in Algorithm 4 in +Appendix F. The algorithm involves applying the outlier- +splitting method on the updates of m in RAN, and using the +adaptive step-size in (16). +We now shows that the outlier-splitting mechanism in RANS +effectively prevents overshoots of type (13) in the updates +of m. Given the above choice of βt, we have +1 +k ⟨βt⊙∇δt, ∇δt⟩ = 1 +k +η +ρ¯ξt +⟨ 1 +√νt +⊙ ∇δt, ∇δt⟩ +≤ ρ¯ξt +ξt +η +ρ¯ξt +⟨ 1 +√νt +⊙ ∇δt, ∇δt⟩ = η, +where the first equality is from the definition of βt in (16), +the inequality is due to the definition of k, and the last +equality follows from the definition of ξt in (15). This +implies that +��⟨1 +k β(∇δT +t m)∇δt, ∇δt⟩ +�� ≤ η +��∇δT +t m +��. +(17) +Therefore overshoots of type (13) do not occur in RANS. +Figure 5. Performance of RANS, TD(0), and RG on classic control +tasks. A single-layer neural network with 64 hidden ReLU units +was used to learn the Q-values, and a softmax disribution on the +Q-values was used as the policy. +The RANS algorithm has hyperparameters α, η, ρ, λ, λ′, +and σ (the outlier sampling probability). Setting η = 0.2 +and ρ = 1.2 are always good choices. Furthermore, our +experiments show that the parameters λ, λ′, and σ can be +set to the default values λ = 0.999, λ′ = 0.9999, and +σ = 0.02 without much performance degradation. In this +case, the RANS algorithm would have essentially one hyper- +parameter α, just like RG and TD algorithms with Adam +optimizer (Kingma & Ba, 2014). The per-iteration com- +putational complexity of RANS is at most twice the RG +algorithm with Adam optimizer. +We tested RANS for control in Acrobot and Cartpole envi- +ronments. We used a single-layer neural network with 64 +hidden units with ReLU activation to learn the action-values, +while choosing actions according to a softmax distribution +on the action-values. Fig. 5 illustrates expected returns ver- +sus number of step for TD(0), RG, and RANS algorithms. +We trained TD(0) and RG using Adam optimizer. Refer +to Appendix G.4 for complementary experimental results +and details of the experiments. The results show that the +RANS algorithm outperforms RG, and its performance is +comparable to TD on these environments. +10. Related works +Poor conditioning of MSBE was previously observed in +(Wang & Ueda, 2021) through study of an example Markov + +Acrobot +-100 +-200 +Return +-300 +TD (Adam) +-400 +RG (Adam) +RANS +-500 +0 +10 +20 +30 +40 +50CartPole +500 +400 +Return +300 +200 +TD (Adam) +100 +RG (Adam) +RANS +0 +0 +20 +40 +60 +80 +100 +Steps (x1000)Toward Efficient Gradient-Based Value Estimation +chains. More specifically, Wang and Ueda (2021) ana- +lyzed a particular n-state Markov chain and showed that +the condition-number of MSBE in this Markov chain scales +with n2. They also showed that the condition-number scales +with 1/(1 − γ)2 in another example Markov chain. In com- +parison, our lower bound in Theorem 3.1 (a) holds for every +Markov chain, and the lower bound in Theorem 3.1 (b) +scales with n2/(1 − γ)2. +A prevalent explanation for slowness of gradient-based +value estimation methods is the so called information flow in +the wrong direction (Baird, 1995). More concretely, each up- +date in RG can be decomposed into a forward bootstrapping +component (or a TD update) and a backward bootstrapping +component (the so called wrong direction of information +flow). A common approach for accelerating the gradient +updates is by suppressing the second component (e.g., via +some sort of combination with TD updates), especially in +early stages of training. The acceleration gained in the +residual algorithm (Baird, 1995), TDC (Sutton et al., 2009), +TDRC, and QRC (Ghiassian et al., 2020) can be understood +from this perspective. In contrast, acceleration gained in our +algorithms does not rely on combinations with TD updates. +Use of Gauss-Newton method for value estimation was ex- +plicitly proposed in (Gottwald et al., 2021; Gottwald & +Shen, 2022), recently. Value estimation algorithms based +on Kalman filter (Choi & Van Roy, 2006; Geist & Pietquin, +2010) are also known to have an equivalent form to online +Gauss-Newton updates (Geist & Pietquin, 2010). Sun and +Bagnell (2015) studied MSBE minimization with Newton +method. However, all of the above methods involve approx- +imating a variant of the Hessian or Gauss-Newton matrices +and solving a system of linear equations in each iteration, +which is computationally costly. Several other algorithms +including least squares TD (Sutton & Barto, 2018) and (De- +vraj & Meyn, 2017) also leverage matrix gain for improved +convergence, under linear function approximation. +In the same spirit, natural gradient methods (Amari, 1998; +Kakade, 2001; Martens, 2020) also enjoy robustness to pa- +rameterization. Dabney and Thomas (2014), Knight and +Lerner (2018), and Achiam et al. (2019) proposed natu- +ral gradients algorithms for value estimation. Dabney and +Thomas (2014) also proposed a low complexity two time +scale implementation that has high-level algorithmic simi- +larities to the RAN algorithm. +In Section 5 and Appendix C we showed that the RAN +algorithm can be perceived as a proximal method. A prox- +imal method for value estimation, called GTD2-MP, was +proposed in (Liu et al., 2020; Mahadevan et al., 2014). +However, these works consider a Bregman divergence +that does not depend on the value estimates. +In fact, +as the step-size goes to zero, update direction of GTD2- +MP tends to the expected GTD2 update direction. Schul- +man et al. +(2015), Sun and Bagnell (2015), and Zhu +and Murray (2022) considered proximal methods with +value dependent penalties of the form E[(vwt+1(St) − +vwt(St))2]. Although the resulting expected update direc- +tion Es[∇vw(s)∇vw(s)T ]−1∇MSBE(w) is robust to pa- +rameterization, it is not robust against poor conditioning. +For example, in the tabular case, this expected update direc- +tion simplifies to ∇MSBE(w), which is the same as RG. +In contrast, in the proximal view of the RAN algorithm, we +used penalties of type E[(δwt+1(St, At) − δwt(St, At))2], +which provides robustness to the conditioning of MSBE, as +discussed in Section 5 and Appendix C. +Karampatziakis and Langford (2010) and Tian and Sutton +(2019) proposed a method, called sliding-step, to reduce +the adverse effect of outliers in certain problems. This +method is pretty similar to the outlier-splitting algorithm, +with the only difference that in the sliding-step method, all +k updates w ← w − ∇ft(w)/k are applied sequentially +and before time t + 1, while the outlier-splitting method +spreads these updates over a long time. Another simple +approach is using momentum to reduce the variance of +updates. However, smoothing large outliers requires large +momentum parameters, in which case the delayed effect of +gradients propagate far into future and become out-dated. +11. Future works and discussion +In this paper, we highlighted causes that underlie slowness +of gradient-based value estimation methods, and proposed +low complexity techniques to resolve them. Our focus was +on the on-policy case, however the proposed algorithms +are easily applicable for off-policy learning when combined +with standard importance sampling techniques. We provided +evidence for the potential of the proposed algorithms via +experiments on a few classic environments. +Other than applying standard techniques (such as batch up- +dates, replay buffers, different forms of step-size adaptation, +etc.) and testing the algorithms on more complex environ- +ments, there are several directions for future research. This +includes adopting the unbiased gradient estimate of (9) in +(10) instead of the biased estimate in (11), and comparing +these methods with other means of solving (9), including +conjugate gradient and low rank approximation of the Gauss- +Newton matrix. Another important direction is further ex- +ploration of the proposed double-sampling-free methods in +stochastic environments with neural network function ap- +proximation. On the theory side, it would be interesting to +study condition-number of MSBE, and in general the shape +of MSBE landscape, under linear and non-linear function +approximation under common feature representations in +asymptotically large environments. + +Toward Efficient Gradient-Based Value Estimation +12. Acknowledgments +The authors want to thank Yi wan, Sina Ghiassian, John +N. Tsitsiklis, and Saber Salehkaleybar for their valuable +feedback in various stages of development of this work. +References +Achiam, J., Knight, E., and Abbeel, P. 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Consider a Markov chain with +termination that has n non-terminal states, and let P be its associated n × n transition matrix. Note that if transitions from a +state can terminate with positive probability, sum over the corresponding row of P will be less than one. Let +A +def= (I − γP)T (I − γP). +(18) +In the tabular setting and under uniform state distribution, we have MSBEV (w) = wT Aw/n. Therefore, the condition- +number C of MSBEV (·) is equal to the condition-number of A. Let λmax and λmin denote the largest and smallest +eigenvalues of A, respectively. It follows that +C = λmax +λmin +. +(19) +Proof of Part (a). We first propose an upper bound for λmin and then a lower bound for λmax. For states i = 1, . . . , n, let +li be the expected number of steps until termination when we start from state i and follow the Markov chain’s transitions. +Then, for any state i = 1, . . . , n, we have +li = 1 + +n +� +j=1 +Pijlj. +(20) +Let l +def= [l1, . . . , ln]T be the vector representation of l1, . . . , ln. Then, (20) can be written in the vector form as +l = 1 + Pl. +(21) +where 1 is the vector of all ones. It then follows that +(I − γP)l = l − γPl = l − γ(l − 1) = (1 − γ)l + γ1, +(22) +where the second equality follows from (21). Let l +def= (l1 +· · · , ln)/n be the mean of l1, . . . , ln. Cauchy-Schwarz inequality +implies that +∥l∥2 +n += 1 +n +n +� +i=1 +l2 +i ≥ +� +1 +n +n +� +i=1 +li +�2 += l2. +(23) +For the smallest eigenvalue of A, we have +λmin ≤ lT Al +∥l∥2 += ∥(I − γP)l∥2 +∥l∥2 += ∥(1 − γ)l + γ1∥2 +∥l∥2 += (I − γ)2∥l∥2 + 2γ(1 − γ)1T l + γ2n +∥l∥2 += (I − γ)2 + 2γ(1 − γ)nl + γ2n +∥l∥2 +≤ (I − γ)2 + 2γ(1 − γ)l + γ2 +l2 += +� +I − γ + γ +l +�2 +, +(24) + +Toward Efficient Gradient-Based Value Estimation +where the first inequality follows from the definition of the smallest eigenvalue of a symmetric matrix, the first equality is +due to the definition of A in (18), the second equality results from (22), the fourth equality is from the definition of l, and +the second inequality follows from (23). +Let trace(A) be the trace of A defined as the sum of diagonal entries of A. It is well-known that the trace of any matrix is +equal to the sum of eigenvalues of that matrix (Strang, 2006). Therefore, for the largest eigenvalue of A, we have +λmax ≥ 1 +ntrace(A) += 1 +n +n +� +i=1 +Aii += 1 +n +n +� +i=1 +n +� +j=1 +(Iji − γPji)2 += 1 +n +n +� +i=1 +� +�(1 − γPii)2 + +� +j̸=i +γ2P 2 +ji +� +� +≥ 1 +n +n +� +i=1 +(1 − γPii)2 +≥ +� +1 +n +n +� +i=1 +(1 − γPii) +�2 += +� +1 − γ +n +n +� +i=1 +Pii +�2 += (1 − γh)2, +(25) +where the first inequality is because trace(A) equals the sum of eigenvalues of A, the first equality is from the definition +of trace, the second equality is due to the definition of A in (18), the third inequality follows from the Cauchy-Schwarz +inequality, and the last equality is from the definition h = �n +i=1 Pii/n in the theorem statement. Plugging (24) and (25) +into (19), we obtain +C = λmax +λmin +≥ (1 − γh)2 +λmin +≥ +(1 − γh)2 +(1 − γ + γ/l)2 ≥ +(1 − γh)2 +2(1 − γ)2 + 2γ2/l2 ≥ (1 − γh)2 +4 +min +� +1 +(1 − γ)2 , l2 +γ2 +� +, +where the first and second inequalities are due to (25) and (24), respectively. This implies (8) and completes the proof of +Part (a) of Theorem 3.1. +Proof of Part (b). Consider an n-state Markov chain with transition matrix +P = +� +�� +0 +· · · +0 +1 +... +... +... +... +0 +· · · +0 +1 +� +�� . +(26) +In what follows, we derive bounds on the largest and smallest eigenvalues of A defined in (18). Let ϵ = γ/(n − 1) and +v = [−ϵ, . . . , −ϵ, 1]T . Then, Pv = 1, and as a result, +vT Av = +��(I − γP)v +��2 = +��v − γ1 +��2 = (n − 1)(γ + ϵ)2 + (1 − γ)2 ≥ (n − 1)(γ + ϵ)2 = n2γ2 +n − 1, +(27) +where the first equality is from the definition of A in (18), and the last equality is due to the definition of ϵ. On the other +hand, +∥v∥2 = (n − 1)ϵ2 + 1 = +γ2 +n − 1 + 1 = n − 1 + γ2 +n − 1 +≤ +n +n − 1. +(28) + +Toward Efficient Gradient-Based Value Estimation +It follows that +λmax ≥ vT Av +∥v∥ +≥ n2γ2/(n − 1) +n/(n − 1) += nγ2, +(29) +where the second inequality is due to (27) and (28). +In order to bound the smallest eigenvalue of A, let x = [γ, . . . , γ, 1]T . Therefore Px = 1 and +(I − γP)x = x − γPx = x − γ1 = [0, . . . , 0, 1 − γ]T . +(30) +It follows that +λmin ≤ xAx +∥x∥2 = ∥(I − γP)x∥2 +∥x∥2 += (1 − γ)2 +∥x∥2 += +(1 − γ)2 +(n − 1)γ2 + 1 ≤ (1 − γ)2 +nγ2 +(31) +where the first equality is from the definition of A in (18), the second equality is due to (30), and the third equality follows +from the definition of x. Plugging (29) and (31) into (19), we obtain +C = λmax +λmin +≥ +nγ2 +(1 − γ)2/(nγ2) = +γ4n2 +(1 − γ)2 . +This completes the proof of Part (b) of Theorem 3.1. +A.2. Condition-number of the MSBE defined in terms of action-values +Theorem 3.1 involves bounds on the condition-number of MSBEV (·) defined in (2). In this appendix, we establish similar +bounds for MSBE(·) defined in (1). +Given an MDP and a policy π, we consider an induced augmented Markov chain that is a Markov chain whose states are the +state-action pairs of the MDP and its transition probabilities are as follows. For any s, s′ ∈ S and a, a′ ∈ A, the probability +of transition from (s, a) to (s′, a′) in the induced augmented Markov chain is +p′ +π(s′, a′|s, a) +def= +� +r +pπ(s′, a′, r|s, a) dr +(32) +where pπ is defined in Section 2. We consider a tabular setting, and denote the condition-number of MSBED(·) under +uniform distribution D on state-action pairs by C′. We let +h′ def= +1 +nm +� +s∈S +� +a∈A +p′ +π(s, a|s, a) +(33) +be the self-loop probability in the induced augmented Markov chain, where n is the number of states and m = |A| is the +number of actions. Also let l′ be the expected number of steps until termination when starting from a uniformly random +state-action pair. The following proposition is the counterpart of Theorem 3.1 for C′. +Proposition A.1. In the tabular case, the following statements hold for any discount factor γ ∈ [0, 1]: +a) For any MDP and any policy π, +C′ ≥ 1 − γh′ +4 +min +� +1 +(1 − γ)2 , l′2 +� +. +(34) +b) For any n, m > 0, there exists an MDP with n states and m actions, and a policy π for which, C′ ≥ γ4(nm)2/(1−γ)2. +Proof. We can perceive the dynamics under any given MDP and policy as an induced augmented Markov chain defined in +(32). Applying the proof of Theorem 3.1 on this induced augmented Markov chain implies Proposition A.1. +B. Experiment on condition number under linear function approximation +We ran an experiment to investigate the growth of condition number, C, in an extended Boyan chain under linear function +approximation. Fig. 6 shows the dependence of C on the size of extended Boyan chain, under standard Boyan feature +vectors (see Appendix G.1 for details). The number of standard Boyan chain features d in this experiments, satisfies +n = 4d − 3, where n is the number of states. We observe that the condition number can grow very large under linear +function approximation even when d/n < 1 (in this case d/n ≃ 1/4). + +Toward Efficient Gradient-Based Value Estimation +Figure 6. Condition-number of MSBE versus number of states in an extended Boyan chain under linear function approximation with +Boyan chain’s standard features (n = 4d − 3). +C. RAN as a proximal algorithm +In this section, we provide further intuition for the RAN algorithm, by showing that Algorithm 1 can be equivalently derived +as a proximal algorithm with momentum for minimizing MSBE. Given an objective function f, a proximal algorithm in its +general form aims to find an approximate solution for a proximal operator of the following form in each iteration +wt+1 ← argmin +w +� +f(w) + Dt(w, wref) +� +, +(35) +where wref is a reference point, usually equal to wt or a trace of past w’s, and Dt is a penalty function (also called +divergence) that encourages wt+1 to stay close to wref. In the special case that Dt is a fixed Bregman divergence, (35) +boils down to the mirror-descent algorithm (Juditsky & Nemirovski, 2011). However, in general, Dt can be a time varying +function and can depend on the local shape of the objective f. +Back to the value estimation problem, for any consecutive state-action pairs (s, a) and (s′, a′), and any pair w and w′ of +weights, we let +δw(s, a, s′, a′) +def= γqw(s′, a′) − qw(s, a), +(36) +and +∆δw,w′(s, a, s′, a′) +def= δw(s, a, s′, a′) − δw′(s, a, s′, a′). +(37) +Consider a divergence measure of the form Dt(w, wt) = c Es,a +� +Es′,a′|s,a +� +∆δw,wt(s, a, s′, a′)2� � +, for some constant +c > 0. Then, the proximal operator in (35) turns into +argmin +w +MSBE(w) + c Es,a +� +Es′,a′|s,a +� +∆δw,wt(s, a, s′, a′)2� � +. +(38) +To obtain a low complexity incremental version of the above proximal updates, we consider doing sample updates along +the gradient of (38) at time t. For this purpose3, we work with wref = wt−1 instead of wref = wt, i.e. we consider the +proximal objective MSBE(w) + c Es,a +� +Es′,a′|s,a +� +∆δw,wt−1(s, a, s′, a′)2� +and its unbiased sample gradient +gt +def= +� +δ′ +t − c ∆δwt,wt−1(St, At, St+1, At+1) +� +∇δt +≃ +� +δ′ +t − c (wt − wt−1)T ∇δt +� +∇δt, +(39) +3Here we avoid using wref = wt because in this case, for any (s, a, s′, a′), ∆δwt,wt(s, a, s′, a′) = 0. This implies that +∇w∆δw,wt(s, a, s′, a′)2�� +w=wt = 0. As such, the penalty would not affect gradient of the proximal objective at w = wt. + +5000 +1SBE +4000 +M +Condition number of +3000 +2000 +1000 +0 +100 +200 +300 +400 +0 +Number of statesToward Efficient Gradient-Based Value Estimation +where the approximate equality is because (wt − wt−1)T ∇δt is a first order approximation of ∆δwt,wt−1. Let ˆgt = +� +δ′ +t − c(wt − wt−1)T ∇δt +� +∇δt and consider the approximate gradient update w ← w − βˆgt. We further employ a +momentum to reduce the variance of these updates. The resulting momentum based algorithm is of the form +mt = λmt−1 + βˆgt, +wt+1 = wt − αmt. +(40) +Let η = 1/α. Since wt − wt−1 = αmt−1, ˆgt simplifies to +ˆgt = +� +δ′ +t − mT +t−1∇δt +� +∇δt. +Plugging this into (40), we obtain updates that are identical to (11). This establishes our earlier claim that the RAN algorithm +can be equivalently formulated as a proximal algorithm with momentum for minimizing MSBE. +As another intuitive perspective for Algorithm 1, we can perceive m as a momentum of MSBE gradients, to which we add a +correction term equal to a sample gradient of the penalty Es,a +� +Es′,a′|s,a +� +∆δw,wt(s, a, s′, a′)2� � +, in each iteration. This +momentum spreads the effect of each MSBE gradient over an O(1/(1 − λ))-long horizon, which provides enough time +for the correction updates to trim the direction of those MSBE gradients. The correction terms are small along directions +that ∇δ is small, allowing MSBE gradients to accumulate along those directions. Conversely, the correction terms are +large along directions that ∇δ is large, preventing m from growing large along those directions. This leads to accelerated +convergence along the directions where ∇δ is small, while preventing instability along directions where ∇δ is large. +D. Convergence of the RAN algorithm +In this appendix, we study conditions for convergence of Algorithm (11) under different choices of λ. Convergence proofs +of two-time-scale approaches are well-studied, and generally involve tedious yet pretty standard statistical arguments. As +such, similar to several other papers (Konda & Tsitsiklis, 1999; Dabney & Thomas, 2014), here we keep our arguments in +a high level, and only discuss the steps of the proof without going into the proof details. Throughout this Appendix, we +consider irreducible and aperiodic Markov chains with finite number of states. We assume that the function approximation +Qw(s, a) is a differentiable function of w with bounded and Lipschitz constant derivatives. By boundedness of derivatives +we mean that there exists a C > 0 such that for consecutive state-action pairs (s, a, s′, a′) and any w, +∥∇δw(s, a, s′, a′) +�� < C, +(41) +where δw(s, a, s′, a′) is defined in (36). Given a distribution D over state and action pairs, for any w ∈ Rd, we let +ˆGw +def= Es,a∼D +� +Es′,a′|s,a +� +∇δw(s, a, s′, a′) ∇δw(s, a, s′, a′)T � � +. +(42) +We study convergence in three different regimes on λ, namely λ = 1, λ = 1 − cβt for some constant c > 0, and for a +constant λ < 1. +Case 1: (λ = 1). We assume that αt and βt are decreasing positive sequneces, satisfying +∞ +� +t=0 +αt = +∞ +� +t=0 +βt = ∞, +∞ +� +t=0 +α2 +t < ∞ +∞ +� +t=0 +β2 +t < ∞, +αt +βt +→ 0. +(43) +We further assume that ˆGw defined in (42) is uniformly positive definite, in the sense that there is an ϵ > 0 such that for any w +and any x ∈ Rd, we have xT ˆGwx ≥ ϵ∥x∥2. In this case, for fixed w, the updates in (11) converge to ˆG−1 +w ∇MSBED(w). +Since αt/βt → 0, w is updated much slower than m. As such, m is updated with much larger step-sizes, perceiving w as +almost stationary, and therefore m converges to an asymptotically small neighborhood of ˆG−1 +w ∇MSBED(w). Since ˆGw is +uniformly positive definite, this m is an absolutely decreasing direction for MSBED(w). Then, standard proof techniques +for stochastic approximation algorithms can be used to establish convergence of w to a stationary point of MSBE. +Case 2: (λ = 1 − cβt, for some constant c > 0). In this case, the update rule (11) boils down to: +m ← λm + βt(δ′ +t − mT ∇δt +� +∇δt += (1 − cβt)m + βt +� +δ′ +t − mT ∇δt +� +∇δt += m − βt +� +(cI + ∇δt∇δT +t )m − δ′ +t∇δt +� += m − βt∇m +�1 +2mT � +cI + ∇wδt∇wδT +t +� +m − δ′ +t∇wδT +t m +� +, +(44) + +Toward Efficient Gradient-Based Value Estimation +where I is the identity matrix. Therefore, in each iteration, m is updated along a sample gradient of the loss function +ˆL(m) = mT � +cI + ˆGw +� +m−∇wMSBE(w)T m, where ˆGw is defined in (42). Thus, assuming (43), m will asymptotically +converge to the minimizer +� +cI + ˆGw +�−1∇MSBE(w) of ˆL. Since cI + ˆGw is uniformly positive definite, this is an +absolutely descent direction for MSBE(w). Then, standard proof techniques for stochastic approximation algorithms can +be used to establish convergence of w to a stationary point of MSBE. +Case 3: +(Constant λ < 1). As opposed to the Cases 1 and 2, here we do not need two-time-scale step-sizes. More +specifically, we assume that αt > 0 is constant and βt is a decreasing positive sequence satisfying �∞ +t=0 βt = ∞ and +�∞ +t=0 β2 +t < ∞. We also assume that w remains bounded which can be enforced either by projection of w on a compact set +or by adequate normalization of αt (see (Konda & Tsitsiklis, 1999) and (Kushner & Yin, 2003)). It follows from (41) that +for any time t, +|δt| = |δwt(St, At, St+1, At+1)| ≤ C′ + C∥wt∥. +(45) +for some constant C′. Then, from (11) we obtain +∥mt∥ = +��λmt−1 + βt +� +δ′ +t − mT +t−1∇δt +� +∇δt +�� +≤ λ∥mt−1∥ + βt +� +|δ′ +t| + ∥mt−1∥ · ∥∇δt∥ +� +∥∇δt∥ +≤ λ∥mt−1∥ + βt +� +|δ′ +t| + C∥mt−1∥ +� +C +≤ λ∥mt−1∥ + βtC +� +C′ + C∥wt∥ + C∥mt−1∥ +� += (λ + βtC2)∥mt−1∥ + βt(C′C + C2∥wt∥), +(46) +where the second and third inequalities follow from (41) nad (45), respectively. When βt is small enough such that +λ + βtC2 < 1, it follows from the boundedness assumption of w that ∥mt∥ = O(βt/(1 − λ)). Therefore, (11) can be +expressed as +mt = λmt−1 + βtδ′ +t∇δt − βt(mT +t−1∇δt)∇δt += λmt−1 + βtδ′ +t∇δt + O +� +β2 +t /(1 − λ) +� += +t +� +τ=0 +� +λτβt−τδ′ +t−τ∇δt−τ +� ++ O +� +β2 +t /(1 − λ)2� +. +Therefore, m is essentially a momentum of sample gradient of MSBE. Then, convergence of w to a stationary point of +MSBE follows from standard techniques for analysing SGD algorithms with momentum. +E. Asymptotic robustness of RAN to parameterization +In this appendix, we present a formal version of Proposition 5.1 and its proof. Here, we assume that λ = 1 and consider +an asymptotically small step-sizes regime with α → 0 and α/β → 0. As discussed in Section 5 and Appendix D, when +α/β → 0, m in the RAN algorithm converges to the approximate Gauss-Newton direction +mGN(w, q) = ˆG−1 +w,q gw,q, +(47) +where +ˆGw,q = Es,a∼D +� +Es′,a′|s,a +�� +γ∇wqw(s′, a′) − ∇wqw(s, a) +� � +γ∇wqw(s′, a′) − ∇wqw(s, a) +�T � � +, +(48) +gw,q = Es,a∼D +� +Es′,a′,r|s,a [r + γqw(s′, a′) − qw(s, a)] Es′,a′|s,a +� +γ∇wqw(s′, a′) − ∇wqw(s, a) +�� +. +(49) +where D is a distribution over state and action pairs. In this case, when α → 0 the trajectory of w approaches the trajectory +of the following Ordinary Differential Equations (ODE): +˙w = −mGN(w, q), +(50) +where ˙w is the derivative of w with respect to time. We refer to (50) as the ODE formulation of two time-scale RAN for the +q function. + +Toward Efficient Gradient-Based Value Estimation +Let u : Rd → Rd be a (possibly non-linear) bijective and differentiable mapping. Suppose that the Jacobian of u(·) is +invertible everywhere. We define ˜q as a reparameterization of the q function by u. More specifically, for any state-action +pair (s, a) and any v ∈ Rd, we let +˜qv(s, a) = qu(v)(s, a). +(51) +Consider the ODE formulation of two time-scale RAN for the ˜q function: +˙v = −mGN(v, ˜q), +(52) +for mGN(w, ˜q) defined in (47). +The following proposition is a formal version of Proposition 5.1. It draws a connection between solution trajectories of (50) +and (52). +Proposition 6.1 (Formal). Let u : Rd → Rd be a (possibly non-linear) bijective and differentiable mapping with invertible +Jacobian, and consider the corresponding reparameterization ˜q of the q function as in (51). Let wt and vt for t ≥ 0 be +solution trajectories of the ODE formulations of two time-scale RAN in (50) and (52), respectively. If w0 = u +� +v0 +� +, then +wt = u(vt), for all t ≥ 0. Moreover, qwt(s, a) = ˜qvt(s, a), for all times t ≥ 0 and all state-action pairs (s, a). +The proposition suggests that the two-time scale RAN algorithm with asymptotically small step-sizes is invariant to any +non-linear bijective reparameterization of the q function. +Proof of Proposition 6.1. For any v ∈ Rd, let Jv be the Jacobian matrix of u(v). Then, for any state-action pair (s, a) and +any v ∈ Rd, +∇v˜qv(s, a) = ∇vqu(v)(s, a) = JT +v ∇wqw(s, a) +�� +w=u(v), +(53) +where the first equality is from the definition of ˜q in (51), and the second equality follows from the chain rule for +differentiation. Then, +ˆGv,˜q = Es,a∼D +� +Es′,a′|s,a +�� +γ∇v˜qv(s′, a′) − ∇v˜qv(s, a) +� � +γ∇v˜qv(s′, a′) − ∇v˜qv(s, a) +�T � � += Es,a∼D +� +Es′,a′|s,a +� +JT +v +� +γ∇wqw(s′, a′) − ∇wqw(s, a) +� � +γ∇wqw(s′, a′) − ∇wqw(s, a) +�T Jv +�� +w=u(v) +� � += JT +v ˆGw,q Jv +�� +w=u(v), +(54) +where the first and the last equalities are due to (48) second equality follows from (53) In the same vein, +gv,˜q = Es,a∼D +� +Es′,a′,r|s,a [r + γ˜qv(s′, a′) − ˜qv(s, a)] Es′,a′|s,a +� +γ∇v˜qv(s′, a′) − ∇v˜qv(s, a) +�� += Es,a∼D +� +Es′,a′,r|s,a [r + γ˜qv(s′, a′) − ˜qv(s, a)] Es′,a′|s,a +� +γJT +v ∇wqw(s′, a′) − JT +v ∇wqw(s, a) +� �� +w=u(v) +� += Es,a∼D +� +Es′,a′,r|s,a [r + γqw(s′, a′) − qw(s, a)] Es′,a′|s,a +� +γJT +v ∇wqw(s′, a′) − JT +v ∇wqw(s, a) +� �� +w=u(v) +� += JT +v gw,q +�� +w=u(v), +(55) +where the first and the last equalities are due to (49), the second equality follows from (53), and the third equality is from +the definition of ˜q in (51). Plugging (54) and (55) into (50) and (52), we obtain +˙v = −mGN(v, ˜q) += − ˆG−1 +v,˜q gv,˜q += − +� +JT +v ˆGw,q J +�−1 gv,˜q +�� +w=u(v) += −J−1 +v +ˆG−1 +w,qJ−T +v +gv,˜q +�� +w=u(v) += −J−1 +v +ˆG−1 +w,q gw,q +�� +w=u(v) += −J−1 +v +mGN(w, q) +�� +w=u(v) += J−1 +v +˙w +�� +w=u(v), +(56) + +Toward Efficient Gradient-Based Value Estimation +Algorithm 4 +RANS +Hyper parameters: stepsize α, η ∈ (0, 1) outlier threshold ρ > 1, momentum and trace parameters λ, λ′ ∈ [0, 1), +outlier sampling probability σ. Good default values for η, ρ, λ, λ′, and p based on our experiments are η = 0.2, ρ = 1.2, +λ = 0.999, λ′ = 0.9999, and σ = 0.02 . +Initialize: m = 0, ˆν = 0, ˆξ = 0, and w. +for t = 1, 2, . . . do +Take two independent samples (St+1, At+1) and (S′ +t+1, A′ +t+1), and consider δt and δ′ +t as in (3) and (4). +ˆνt ← λ′ˆνt−1 + (1 − λ′)(∇δt)2 +▷ (∇δt)2 is the entrywise square vector of ∇δt +νt = ˆνt/(1 − λ′t) +▷ bias-corrected trace of (∇δt)2 +ξt = ⟨(1/√νt) ⊙ ∇δt, ∇δt⟩ +▷ Outlier measure +ˆξ ← λ′ ˆξ + (1 − λ′)ξt +¯ξ = ˆξ/(1 − λ′t +ξ ) +▷ bias-corrected trace of ξ +k = ⌊ξt/(ρ¯ξ)⌋ + 1 +▷ outlier-splitting factor +β = η/(ρ¯ξt +√νt) +m ← λm + +� +δ′ +t − mT ∇δt +� +β ⊙ ∇δt/k +w ← w − αm +if k > 1 then +Store (St, At, St+1, At+1, rt, , S′ +t+1, A′ +t+1, r′ +t, k, k − 1) in the outlier buffer +▷ the last entry in the +tuple indicates the remaining number of future updates based on this sample +end if +With probability min(1, σ∗length of outlier bufffer): +▷ do an update using outlier buffer samples +Sample (Sτ, Aτ, Sτ+1, Aτ+1, rτ, S′ +τ+1, A′ +τ+1, r′ +τ, k′, j) uniformly form the outlier buffer +Let δt and δ′ +t be as in (3) and (4), respectively +ξ = ⟨(1/√νt) ⊙ ∇δτ, ∇δτ⟩ +k′′ = max +� +k′, ⌊ξ/(ρ¯ξ)⌋ + 1 +� +m ← λm + +� +δ′ +τ − mT ∇δ′ +τ +� +β ⊙ ∇δτ/k′′ +w ← w − αm +if j > 1 then +Replace (Sτ, . . . , k′, j) with (Sτ, . . . , k′, j − 1) in the outlier buffer +else if j = 1 then +Remove (Sτ, . . . , k′, j) from the outlier buffer +end if +end for +where the equalities are respectively due to (52), (47), (54), the assumption that the Jacobian J of u is invertible, (55), (47), +and (50). Therefore, if at time t, wt = u(vt), then ˙wt = Jv ˙vt = d +dtu(vt). This together with the assumption w0 = u(v0) +implies that wt = u +� +vt +� +, for all t ≥ 0. It then follows from the definition of ˜q in (51) that qwt(s, a) = ˜qvt(s, a), for all +times t ≥ 0 and all state-action pairs (s, a). This completes the proof of Proposition 1. +F. Pseudo code of RANS +The pseudo code of the RANS algorithm is given in Algorithm 4. Note that in the two time scale regime, where α, η → 0 +and α/η → 0, outlier-splitting would have no effect on the expected direction of m updates. In this case, convergence of the +RANS algorithm follows from similar arguments to Appendix D. +G. Details of experiments +G.1. Experiments of Section 3 and Appendix B +An n-state extended Boyan chain with termination is a Markov chain with termination with states 0, 1, . . . , n − 1 and +transition probabilities: P(1 → 0) = 1 and P(i → i − 1) = P(i → i − 2) = 0.5 for i = 2, 3, . . . , n − 1. Furthermore, +state 0 goes to a terminal state with probability 1. By standard features, we mean feature representations similar to (Boyan, + +Toward Efficient Gradient-Based Value Estimation +2002). More specifically, given a d > 1 and n = 4d − 3, we consider d standard features for the n-state extended +Boyan-chain as follows. In the ith standard feature vector for i = 0, 1, . . . , d − 1, the jth entry is equal to 1 − |j − 4i|/4 for +j = max(0, 4i − 3), . . . , min(d − 1, 4i + 3), and all other entries equal zero. For the special case of n = 13 and d = 4, the +standard features would be the same as the features considered in (Boyan, 2002). +By random binary features we mean an n × d feature matrix Φ with i.i.d. entries that take 0 and 1 values with equal +probability. For evaluating value-errors in Fig. 2 we consider reward 1 for the transition at state 0 (that leads to the terminal +state), and reward 0 for all other transitions. Note that the reward function does not affect condition-number. Each point in +this first experiment is the median of 100 independent runs with different random feature matrices. We use median instead +of mean to eliminate the adverse effect of unbounded values in degenerate cases (e.g., infinite condition-number in the case +of low rank feature matrices). +For both experiments in Fig. 2 and 6 we use discount factor γ = 0.995. Condition-numbers and value-errors in these +experiments are computed with respect to uniform state distribution. +G.2. Experiment of Section 5 +We considered an extension of the Hallway environment with 50 states and one action, in which each state s = 1, 2 . . . , 50 +transits to state min(s + 1, 50) with probability 0.99, and transits to a terminal state with probability 0.01. The experiment +was tabular with γ = 1. All rewards were set equal to zero, in which case the correct q-values are qπ(s, ·) = 0, for all +sates s. All algorithms were initialized with q(s, ·) = 1, for s = 1, . . . , 50. Fig. 3 illustrates learning curves of value-error +�50 +s=1 +� +q(s, 1) − qπ(s, 1) +�2/50 for RAN (Algorithm 1), RG, and TD(0) algorithms. Each point is an average over 100 +independent runs. We optimized the parameters for RG and Algorithm 1. The parameters used in the experiments are as +follows. For RAN, we set α = 0.025, β = 0.4, and λ = 0.9998. For RG and TD(0) we used α = 0.5. +G.3. Experiment of Section 6 +We ran an experiment on Baird’s Star environment (Baird, 1995). We performed off-policy learning with uniform +state distribution. All rewards were set to zero, in which case the correct q-values are qπ(s, ·) = 0, for all sates s. +We used the initial point w = [2, 1, 1, 1, 1, 1, 1] for all algorithms. Fig. 4 illustrates learning curves of value-error +�6 +s=1 +� +qw(s, 1) − qπ(s, 1) +�2/6 for the RAN (Algorithm 1), DSF-RAN (Algorithm 2), RG, GTD2, and TD(0) algorithms. +Each point is an average over 10 independent runs. TD(0) was unstable on this environment, and we chose a very small +step-size α = 10−5 for TD(0). For all other algorithms, we used optimized parameters as follows. For RAN, we set α = 2, +β = 0.15, and λ = 0.995. For DSF-RAN, we set α = 1, β = 0.15, η = 0.3, and λ = 0.995. For RG we used α = 0.3. For +GTD2 we set α = 0.15 and β = 0.3. +G.4. Experiment of Section 9 +We ran an experiment on classic control tasks –Acrobot and Cartpole– to test the performance of the RANS algorithm. We +used a single-layer neural network with 64 hidden ReLU units to learn the action-values, while choosing actions according to +a softmax distribution on the action-values, a ∼ Softmax +� +qw(s, ·) +� +. The network for qw was trained with three algorithms: +TD(0), RG, and RANS (Algorithm 4). Since RANS incorporates adaptive step-sizes, for fair comparison we trained TD(0) +and RG using Adam optimizer. +For each algorithm, we performed training for 100 randomly generated random seeds. For each training seed, in order to +obtain an estimate of expected returns, we took an average over 400 independent environment simulations once every 500 +steps. In Fig. 7 we plot the the average of these estimated expected returns over the 100 training seeds. +In the Cartpole environment, once an algorithm reaches score 500, it will not see any failure for many episodes in a row. In +this case, the agent starts to forget the actions that prevented failure and led it to success, causing the performance to drop +before it can rise again. This phenomenon is known as catastrophic forgetting, and leads to large oscillations in learning +curves. In order to hide the effect of catastrophic forgetting on the learning curves, in Fig. 5 we eliminated the 50 worst +return estimates (corresponding to the 50 worst training seeds) at each point and plotted the mean of top 50 return estimates. +The shades in Fig. 5 and Fig. 7 show the 99 percent confidence intervals over the averaged data. +We did not use replay buffer and batch updates. We used a small quadratic regularizer with coefficient 10−5 on the weights +of the neural network. For all other algorithms, we used optimized parameters that maximize area under the curves, as + +Toward Efficient Gradient-Based Value Estimation +Figure 7. Performance of RANS, TD(0), and RG on classic control tasks. The only difference with Fig.5 is that here, each point is an +average of estimate returns over 100 independent training. See Appendix G.4 for details. +follows. In Acrobot experiment: +• For TD(0), we used softmax coefficient 1 and Adam optimizer with step-size 0.005 while all other parameters were set +to their default values. +• For RG, we used softmax coefficient 16 and Adam optimizer with step-size 0.001 while all other parameters were set +to their default values. +• For RANS, we used softmax coefficient 16 and α = 0.005 and set all other parameters were set to their default values +described in Algorithm 4. +In Cartpole experiment: +• For TD(0), we used softmax coefficient 0.005 and Adam optimizer with step-size 0.3 while all other parameters were +set to their default values. +• For RG, we used softmax coefficient 0.002 and Adam optimizer with step-size 0.3 while all other parameters were set +to their default values. +• For RANS, we used softmax coefficient 8 and α = 0.001 and set all other parameters were set to their default values +described in Algorithm 4. + +Acrobot +-100 +-200 +Return +-300 +TD (Adam) +-400 +RG (Adam) +RANS +-500 +0 +10 +20 +30 +40 +50 +Steps +5(x1000)CartPole +500 +400 +300 +Return +200 +TD (Adam) +100 +RG (Adam) +RANS +0 +0 +20 +40 +60 +80 +100 +Steps (x1000) \ No newline at end of file diff --git a/rtFST4oBgHgl3EQfQDhb/content/tmp_files/load_file.txt b/rtFST4oBgHgl3EQfQDhb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e6bb8f8b731836342c6568b9db671d9b0964079 --- /dev/null +++ b/rtFST4oBgHgl3EQfQDhb/content/tmp_files/load_file.txt @@ -0,0 +1,1245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf,len=1244 +page_content='Toward Efficient Gradient-Based Value Estimation Arsalan Sharifnassab 1 and Richard Sutton 1 Abstract Gradient-based methods for value estimation in reinforcement learning have favorable stability properties, but they are typically much slower than Temporal Difference (TD) learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We study the root causes of this slowness and show that Mean Square Bellman Error (MSBE) is an ill-conditioned loss function in the sense that its Hessian has large condition-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' To resolve the adverse effect of poor conditioning of MSBE on gradient based methods, we propose a low complexity batch-free proximal method that approximately follows the Gauss-Newton direc- tion and is asymptotically robust to parameter- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our main algorithm, called RANS, is efficient in the sense that it is significantly faster than the residual gradient methods while having almost the same computational complexity, and is competitive with TD on the classic problems that we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Introduction Value estimation is a core problem in reinforcement learning (Sutton & Barto, 2018), and is a key ingredient in several policy optimization methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', (Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Minh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Lillicrap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2015) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The popular class of value estimation algorithms based on temporal difference learning via forward bootstrapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' including TD(λ) (Sut- ton, 1988), Expected Sarsa (van Seijen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009), and Q-learning (Watkins & Dayan, 1992) have found substan- tial empirical success when combined with proper policy optimization (Minh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Minh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Lilli- crap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Nevertheless, these algorithms are not gradient-based optimization methods (Barnard, 1993) and their convergence cannot be guaranteed for general func- tion approximation setting (Baird, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Tsitsiklis & Van Roy, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Brandfonbrener & Bruna, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The stabil- ity problem of TD learning has inspired other classes of value estimation algorithms that involve optimizing a loss 1Authors are with the Department of Computing Science, University of Alberta, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' function through gradient updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This includes Resid- ual Gradient (RG) algorithm for minimizing Mean Squared Bellman Error (MSBE) (Baird, 1995), Gradient-TD algo- rithms for minimizing projected Bellman error (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Maei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Maei, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Hackman, 2012), and their extensions for optimizing a dual formulation of BE2 (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Macua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' These algorithms enjoy the general robustness and convergence properties of Stochastic Gradient Descent (SGD), but are known to be slower than TD in tabular and linear function approximation settings (Baird, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Schoknecht & Merke, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Gordon, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Ghiassian & Sutton, 2021) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this paper, we investigate the root causes of the slow- ness problem of gradient-based value estimation by taking a deeper look into the landscape of MSBE, and propose linear complexity methods to alleviate these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We provide theoretical results showing that MSBE is an ill-conditioned loss function in the senses that the condition-number of its Hessian matrix is typically very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This explains slowness of gradient-based value estimation methods, be- cause gradient descent in general is slow in minimizing ill-conditioned loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In contrast, algorithms like Newton and Gauss-Newton methods are invariant to condi- tioning of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Unfortunately a direct implementation of these methods requires matrix inversion, which is computa- tionally costly even if computed incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We propose a linear complexity incremental algorithm, called Resid- ual Approximate Gauss-Newton (RAN), that incorporates a trace to approximate the Gauss-Newton direction and then updates the weights along that trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We show that RAN can be equivalently formulated as a batch-free proximal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A weakness of RAN is that it requires double sampling (Baird, 1995), which limits its use in stochastic en- vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We propose a double-sampling-free extension of RAN by following similar ideas that underlie GTD-type methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The resulting algorithms significantly outperform RG and GTD2, being orders of magnitudes faster on the simple classic environments that we tested, while having almost similar computational complexity to RG and GTD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We then turn our focus to a second cause of slowness of gradient-based value estimation: under function approxi- mation, sample gradients of MSBE involve large outliers that carry important information, resulting in large variance of stochastic updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Outliers of this type often appear arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='13757v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='LG] 31 Jan 2023 Toward Efficient Gradient-Based Value Estimation in every episode (usually at pre-terminal transitions), and are specific to gradient-based value estimation methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' such outliers do not appear in TD learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We propose a general technique called outlier-splitting, which results in no information loss as opposed to the standard clipping meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our main value estimation algorithm, called RAN with outlier-Splitting (RANS), has linear computational com- plexity and has only one effective hyper-parameter (and some other hyper-parameters that can be set to their default values), thanks to its adaptive step-size mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our empirical results on a few classic control environments with neural network function approximation show significant im- provement over RG, and achieving competitive performance to TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Background We consider a discounted Markov Decision Process (MDP) defined by the tuple (S, A, R, p, γ), where S is a finite set of states, A is a finite set of actions, R is a set of re- wards, p : S × A × S × R → [0, 1] is the environment dynamics determining the probability of the next state and immediate reward given a current state and action pair, and γ ∈ [0, 1] is a discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We fix a stationary policy π : S × A → [0, 1], and let pπ(s′, a′, r|s, a) = p(St+1 = s′, Rt+1 = r|St = s, At = a)π(At+1 = a′|St+1 = s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We consider an episodic and online setting where a data stream (S1, A1, R1), (S2, A2, R2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' is generated accord- ing to the policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The action-value function qπ : S ×A → R, at each state s and action a, is the expected discounted sum of rewards obtained by starting from state s and action a and following policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We then define the value function vπ : S → R as vπ(s) = Ea∼π(·|s)[qπ(s, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In value estimation, we aim to obtain an estimate of the true action-values qπ, usually through a function qw : S × A → R parameterized by a d-dimensional weight vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Corresponding to qw is a Bellman residual at each state and action pair (s, a), defined as δw(s, a) def= Es′,a′,r∼pπ(·,·,·|s,a) � r+γqw(s′, a′)−qw(s, a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' According to Bellman equations (Sutton & Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Bertsekas & Tsitsiklis, 1996), qw = qπ if and only if δw(s, a) = 0 for all (s, a) ∈ S × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this view, MSBE(w), defined below, serves as a proxy for the quality of estimates w: MSBED(w) def= E(s,a)∼D � δw(s, a)2� , (1) where D is some distribution over states and action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' When the distribution is online, we drop the subscript D and write MSBE(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For simplicity of notation, we also write Es,a[·] to denote the expectation with respect to state and action pairs sampled from the online distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the same vein, we consider a parameterized estimate vw : S → R of value function vπ, and let MSBEV D(w) def= Es∼D � δw(s)2� , (2) where δw(s) = Ea∼π(·|s)[δw(s, a)], and D is some distribu- tion over states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Gradient-based value estimation methods use gradient- based optimization algorithms to minimize MSBE or other related objectives such as MSPBE (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The first and simplest method in this category is the RG algo- rithm (Baird, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this algorithm, to obtain an unbiased sample estimate of ∇w(δw(St, At)2), we require indepen- dent samples (St+1, At+1, Rt) and (S′ t+1, A′ t+1, R′ t) from pπ(·, ·, ·|St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For simplicity of notation, at time t, we let δt def= Rt + γqw(St+1, At+1) − qw(St, At), (3) δ′ t def= R′ t + γqw(S′ t+1, A′ t+1) − qw(St, At), (4) where wt = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The RG update is then w ← w − αδ′ t∇wδt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (5) The requirement for two independent sample transitions at time t is called double sampling (Sutton & Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In stochastic environments, double sampling is possible only if we have a correct model of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In other words, MSBE minimizer is not learnable if an exact model of the underlying stochastic environment is not available, which is the case in real-world applications (Sutton & Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A general technique to circumvent double sampling is using Fenchel duality to obtain an equivalent saddle point formulation of MSBE (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2017) as min w max ˆδ(·,·) Es,a � δw(s, a) ˆδ(s, a) − 1 2 ˆδ(s, a)2 � , (6) where ˆδ(s, a) is an auxiliary variable that serves as a proxy of δw(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In practice, one can consider a parametric approximation ˆδθ(·, ·) of ˆδ(·, ·), and per- form gradient updates on the resulting minimax problem minw maxθ Es,a � δw(s, a) ˆδθ(s, a) − 1 2 ˆδθ(s, a)2� : w ← w − αˆδθ(St, At)∇wδt, θ ← θ + η � δt − ˆδθ(St, At) � ∇θˆδθ(St, At), (7) (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In- tuitively, this is similar to the RG algorithm in (5) except for using the parametric approximation ˆδθ(St, At) instead of δ′ t, and updating ˆδθ(s, a) by SGD on Es,a � (ˆδθ(s, a) − δw(s, a))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The GTD2 algorithm (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009) is a special case of (7) in which qw and ˆδθ are linear approxima- tions of the form qw(s, a) = φT s,aw and ˆδθ(s, a) = φT s,aθ, for feature vectors φs,a (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Toward Efficient Gradient-Based Value Estimation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' MSBE loss is ill-conditioned The condition-number of a symmetric square matrix, H, is defined as the ratio of its largest to smallest singular values, maxx:∥x∥=1 |xT Hx|/ miny:∥y∥=1 |yT Hy|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For a quadratic function f(x) = xT Hx, we define the condition- number, C(f), of f as the condition-number of its Hessian matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Intuitively, level sets (or contours) of a con- vex quadratic function have an elliptical shape, and the condition-number C(f) equals the squared ratio between the largest and the smallest diameters of each of these el- lipsoids (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We say that f is ill-conditioned if C(f) is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, the level sets of an ill-conditioned quadratic function have the shape of ellipsoids that are thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It is known that the convergence rate of the gradient descent on a quadratic loss f scales with C(f) (Polyak, 1964), which can be very slow for ill-conditioned loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this section, we consider linear function approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, MSBEV D(·) defined in (2) is a convex quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We denote the condition-number of MSBEV D(·) under uniform distribution D by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We let l be the average episode length, defined as the expected time until termination when starting from a state, uniformly av- eraged over all states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We also let h def= Es∼unif � P(St+1 = s|St = s) � be the the average self-loop probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Note that h is typically much smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the tabular case, the following statements hold for any discount factor γ ∈ [0, 1]: a) For any MDP and under any policy, we have C ≥ 1 − γh 4 min � 1 (1 − γ)2 , l2 � (8) where l is the average episode length and h is the average self-loop probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' b) For any n ≥ 1, there exists an n-state MDP and a policy for which C ≥ γ4n2/(1 − γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The proof is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A similar result also holds for the condition-number of MSBE defined in (1) (see Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 shows that MSBE is typically ill-conditioned in the tabular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This explains the slow convergence of gradient-based meth- ods for minimizing MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As an example, the bound in (8) implies that for γ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='99 and for any MDP and policy pair with average episode length at least 100 and average self-loop probability no larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1, we have C > 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Moreover, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 (b) implies that for γ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='99, there is a 100-state Markov chain for which C > 96, 000, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Lower bounds similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 are not possible for non-tabular linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This is because different feature representations can improve or worsen the Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Level sets of MSBE (gray curves) in a 2-state loop en- vironment with p(s0 → s1) = p(s1 → s0) = 1 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Here, condition number of MSBE is 81, and is equal to the squared ratio between the diameters (red) of each ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The solution trajectory of RG (blue) for α = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='9 and the Gauss-Newton direction (green) are also depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this environment, C = O(1/(1−γ)2) (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1), which rapidly grows for larger γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' condition-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' To see why, note that in the linear func- tion approximation case and under uniform state distribution, MSBEV unif(w) = wT ΦT (I − γP)T (I − γP)Φw, where Φ is an n × d matrix, each row of which is a feature vector of a state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' and P is the transition matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For the specific choice Φ = (I − γP)−1 we obtain C = 1, while for the case that Φ is not full-rank, we have C = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In general, since underparameterized function approxima- tion reduces parameters dimension, it usually improves condition-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2 illustrates dependence of C on the number of features, d, in an extended Boyan chain en- vironment (Boyan, 2002) with 200 states and with random binary features (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We ob- serve that smaller d results in better condition-number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' but this comes at the cost of larger value-error at MSBE min- imum (the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2), where by value-error we mean Es∼unif � Ea∼π(·|s)[∥qw(s, a) − qπ(s, a)∥2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' See Ap- pendix B for more experiments on condition number under linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A review of the Gauss-Newton method Consider an expected loss function of the form F(w) = Ef[f 2(w)], and the associated Hessian matrix HF = E[∇f∇f T ] + E[f Hf], where Hf denotes Hessian of sam- ple function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The first term on the right hand side, E[∇f∇f T ], is called the Gauss-Newton matrix and is de- noted by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The Gauss-Newton algorithm then updates w as w ← w − αG−1∇F(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the special case that func- tions f are linear, we have Hf = 0 and thereby HF = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, Gauss-Newton and Newton methods become equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, the Gauss-Newton algorithm has two advantages in the non-linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Firstly, G−1∇F(w) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8 Gauss-Newton direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 RG trajectory 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 Contours of MSBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8Toward Efficient Gradient-Based Value Estimation Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Condition-number of MSBE (blue) and value-error at MSBE minimizer (red) versus number of features, in a 200-state extended Boyan chain with random binary features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' always a descent direction, as opposed to the Newton up- dates that may climb uphill and converge to local maxima or saddle points (Nesterov & Polyak, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Secondly, G can be computed in terms of gradients while H entails second order derivatives which are not as easily accessible in certain settings (Nocedal & Wright, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As opposed to gradient descent which is prohibitively slow in ill-conditioned problems, Newton and Gauss-Newton methods are invariant to conditioning of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Some recent works proposed using Gauss-Newton method for value estimation (Gottwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, these algorithms require matrix inversion, which is computation- ally costly even if computed incrementally via Sherman- Morrison formula with quadratic complexity (Sherman & Morrison, 1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the next section, we propose a linear- complexity method for MSBE minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our first algorithm: RAN The Gauss-Newton direction for minimizing MSBE is mGN(w) = G−1 w ∇MSBE(w)/2, where Gw = Es,a � ∇δw(s, a) ∇δw(s, a)T � is the Gauss-Newton matrix and MSBE(w) = 2Es,a � δw(s, a) ∇δw(s, a) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, mGN is the minimizer of the following quadratic function: L(m) def= 1 2 Es,a �� δw(s, a) − ∇δw(s, a)T m �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (9) This is because for any m, ∇mL(m) = Es,a �� ∇δw(s, a)T m − δw(s, a) � ∇δw(s, a) � = Gwm − ∇MSBE(w)/2, and therefore ∇mL(mGN) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We follow a two time scale approach (Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dabney & Thomas, 2014) to incrementally find an approximate minimizer m of L and update w along that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More concretely, given a β > 0 and λ ∈ [0, 1], at time t, we update m along Algorithm 1 RAN Parameters: step-sizes α, β, and decay parameter λ Initialize: m = 0 and w for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' do consider δt and δ′ t defined in (3) and (4), respectively m ← λm + β δ′ t ∇δt m ← m − β(mT ∇δt)∇δt w ← w − αm end for an unbiased sample gradient of βL(m) + (1 − λ)∥m∥2, m ← λm + β � δ′ t − mT ∇δ′ t � ∇δt, (10) where δt and δ′ t are defined in (3) and (4), and (1 − λ)∥m∥2 is a Levenberg–Marquardt regularizer (Marquardt, 1963)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We then update w along m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='e, w ← w − αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Algorithm 1 gives the pseudo code of the RAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For better stability and faster convergence, the update of m in RAN is of the form m ← λm + β � δ′ t − mT ∇δt � ∇δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (11) which is the same as (10) except for using ∇δt instead of ∇δ′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The updates in (11) have lower variance compared to (10), and additionally ∇δt∇δT t in (11) is positive semi- definite, as opposed to the finite sample estimate of the Gauss-Newton matrix 1 τ �τ t=1 ∇δt∇δ′T t in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For fixed w, the update in (11) is in expectation along − � Et[∇δt∇δT t ] + (1 − λ)I �−1∇MSBE(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (12) In Appendix C, we provide further intuition for RAN, by presenting a derivation of Algorithm 1 as a proximal method with momentum for minimizing MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this view, m serves as a momentum of MSBE gradients, to which we add a correction term equal to the gradient of a penalty function that aims to regularize the change in δw(s, a) for all state action pairs (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Convergence of the RAN algorithm can be shown in two- time-scale regime where αt, βt → 0, with α diminishing faster than β (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', αt/βt → 0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Convergence of such two-time-scale algorithms is well-studied (Kushner & Yin, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Konda & Tsitsiklis, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Bhatnagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009), under some smoothness and irreducibly conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Ap- pendix D, we discuss different conditions for convergence 1We have empirically observed that λ = 1 often leads to slow convergence, because it causes large inertia in m, and therefore large oscillations in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The best performance is achieved for λ ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='9999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2Note that the two-time-scale view is only for the purpose of convergence analysis, and in practice we consider fixed or adaptive step-sizes whose ratio needs not go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (log scale) 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='5 um 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 103 Condition number of MSBE r at MSBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2 Error 10 Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 1 0 20 40 60 80 100 120 140 160 180 Number of featuresToward Efficient Gradient-Based Value Estimation Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The Hallway experiment discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' of Algorithm 1 in the two-time-scale regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Moreover, in this regime, RAN is robust to reparameterization: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For λ = 1 and asymptotically small step-sizes α → 0 and α/β → 0, the trajectory of w in the RAN algorithm is invariant to any differentiable and bijective non-linear transformation on parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The formal version of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 and its proof are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We evaluated the performance of RAN in a simple bench- mark environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider an environment with n states and one action, in which each state i = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , n transits to state min(i + 1, n) with probability 1 − ϵ, and transits to a terminal state with probability ϵ, for some ϵ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This is a generalization of the Hallway environment (Baird, 1995), and is known to be a challenging task for the RG algorithm (Baird, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We tested Algorithm 1 in this environment with n = 50, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='01, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='99 in the tabular setting (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2 for the details of this experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The learning curves are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We observe that, in this experiment, Algorithm 1 is about 30 times faster than RG, and reaches a convergence rate close to TD(0) (Sutton, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Sutton & Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Double-sampling-free RAN algorithm In Algorithm 1, we require double sampling to compute δ′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this section, we propose a Double-Sampling-Free version of RAN, called DSF-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Double sampling is easily doable in deterministic environments (Saleh & Jiang, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020), in which case δ′ t can be computed using an independent sample A′ t+1 from the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, for double sampling in stochastic environments, we require a model to get an independent sample S′ t+1 of the next state, which is typically possible only in simulated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' To resolve the double sampling issue of RAN in stochastic environments, we use the technique discussed in Section 2, which was also used in the GTD2 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More specif- Algorithm 2 DSF-RAN Parameters: step-sizes α, β, η, and decay parameter λ Initialize: m = 0, w, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' do ∇δt = γ∇wqw(St+1, At+1) − ∇wqw(St, At) m ← λm + β ˆδθ(St, At) ∇δt m ← m − β(mT ∇δt)∇δt w ← w − αm θ ← θ + η � δt − ˆδθ(St, At) � ∇θˆδθ(St, At) end for Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The Baird’s star experiment discussed in Section 6 ically, instead of δ′ t in Algorithm 1, we use a parametric approximation ˆδθ(St, At) of δw(St, At), parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Similar to GTD2 (see (7)), we then learn θ through SGD on Es,a � (ˆδθ(s, a)−δw(s, a))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Pseudo code of DSF-RAN is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We tested RAN and DNS-RAN algorithms on Baird’s Star environment (Baird, 1995), that is a Markov chain with six states, each represented by seven features (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3 for details of this experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We observe that in this environment, RAN and DNS- RAN converge about 200 times faster than RG and GTD2 algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It is well-known that off-policy TD(0) is unstable in this environment (Baird, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The problem of outliers In this section we argue that the gradient of MSBE involves large outliers and discuss its impact on the RAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For simplicity, temporarily suppose that the set of actions is a singleton, A = {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the function approximation case, successive states St and St+1 often have similar representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As a result, ∇qw(St, a) and γ∇qw(St+1, a) are often similar, rendering ∇δt = γ∇qw(St+1, a) − ∇qw(St, a) to be small (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This wouldn’t have been prob- lematic if ∇δt was small for all t, in which case we could compensate by increasing the step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, ∇δt can 1 TD(0) RAN RG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2 M 0 0 20 40 60 80 100 Steps s(x1000)TD(0) 15 20 RG RAN Mean sguared value error 10 GTD2 15 DSF-RAN 5 10 0 0 1000 2000 3000 5 M 0 0 50 100 150 200 Steps (x1000)Toward Efficient Gradient-Based Value Estimation occasionally be large, for example when St+1 is a terminal state in which case ∇δt = −γ∇qw(St, a), or when St+1 is far from St (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', in large jump transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Although these outliers occur with low probability, they carry important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For example, the pre-terminal transitions are important because they pin down the estimated values to the terminal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In environments with larger action sets, if the policy has small entropy, At+1 and At would have similar representations with high probability, causing ∇δt to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We now discus how these outliers affect RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The up- dates of m in Algorithm 1 involve a momentum (of MSBE gradient) term λm + δ′ t∇δt and a correction term −β(∇δT t m)∇δt that aims to slowly modify m towards the approximate Gauss-Newton direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, when ∇δt is an outlier, β(∇δT t m)∇δt can grow very large, cause an overshoot, and completely change the direction of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In particular, if β∥∇δt∥2 > 1, then magnitude of the correc- tion term would be larger than the projection of m on ∇δt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ��⟨β(∇δT t m)∇δt, ∇δt⟩ �� > ��⟨∇δT t m⟩ ��, (13) which results in an overshoot along ∇δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Such overshoots hinder m from tracking the approximate Gauss-Netwon direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' To reduce the adverse effect of outliers, one can reduce step- size β, at the cost of slowed down learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Another popular solution is gradient clipping (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, as discussed in the first paragraph of this section, the outliers in our problem carry important information, which can be lost via gradient clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Outlier-splitting We now propose outlier-splitting as a general meta- technique for stochastic optimization, appropriate for the case that data contains rare sample functions with abnor- mally large gradients, and these sample functions carry important information that would be lost in gradient clip- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We first explain the key idea by an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider minimizing f1 + · · · + fn for smooth functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Suppose that f1 is an outlier in the sense that the norm of its gradient is locally k times larger than the gradient norms of other functions, for some integer k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The idea is that instead of applying SGD on f1 + · · · + fn, we break down f1 into k copies of f1/k and apply SGD on f1/k + · · · + f1/k + f2 + · · · + fn in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The latter updates are outlier-free while being equivalent to the former updates in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We now proceed to a formal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider SGD on an objective function F = E[f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For any sample sample function f and any point w, we consider a non-negative measure ξ(f, w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', ξ(f, w) = ∥∇f(w)∥ Algorithm 3 Outlier-splitting for online SGD, applied to loss function F = E[f] Parameters: step-size β, outlier threshold ρ, trace pa- rameter λξ, outlier sampling probability σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Initialize: ˆξ = 0, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' do ˆξ ← λξ ˆξ + (1 − λξ)ξ(ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' w) ¯ξ = ˆξ/(1−λt ξ) ▷ bias-corrected trace estimate k = ⌊ξ(ft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' w)/(ρ¯ξ)⌋ + 1 w ← w − (β/k)∇ft(w) if k > 1 then Store (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k − 1) in the outlier buffer end if With probability min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' σ ∗ length of outlier bufffer): Sample (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' j) uniformly form outlier buffer k′′ = max � k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ⌊ξ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' w)/(ρ¯ξ)⌋ + 1 � w ← w − (β/k′′)∇f(w) if j > 1 then Replace (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' j) with (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' j − 1) in the buffer else if j = 1 then Remove (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' j) from the outlier buffer end if end for or ∥∇f(w)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let ¯ξ be a trace of ξ, updated by ¯ξ ← λξ ¯ξ + (1 − λξ)ξ(ft, wt), where λξ ∈ (0, 1) is a constant close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We say that ft is an outlier if ξ(ft, wt) ≥ ρ¯ξt, for some outlier threshold ρ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The pseudo code of the outlier-splitting method for online SGD is given in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' At time t of this algorithm, we let k = �ξ(ft, wt) ρ¯ξt � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (14) If ft is an outlier (equivalently k > 1), instead of ft we pretend to have k copies of ft/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We use one of these copies to do a gradient update at time t, and store the remaining k−1 copies in a buffer to use them for future updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' These copies are stored in one cell of an outlier-buffer as a tuple (ft, k, k−1), where k−1 indicates the number of remaining copies to be used for future updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In each iteration we perform one update based on the online sample, and perform at most one update based on a sample from the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More concretely, in each iteration t, after applying a gradient update w ← w−(β/k)∇ft(w), we take a sample (f, kf, j) from the outlier buffer with some positive probability, and perform a gradient update w ← w − (β/kf)∇f(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We now show that the outlier buffer is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The expected number of copies, k−1, added to the buffer at time t satisfies E[k − 1] ≤ Et �ξ(ft, wt) ρ¯ξt � ≃ Et [ξ(ft, wt)] ρE[¯ξt] = 1 ρ < 1, Toward Efficient Gradient-Based Value Estimation where the inequality is due to (14) and the approximate equality is because ¯ξt is a long-time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' On the other hand, as the length of the outlier buffer increases, the proba- bility of performing a sample update from the buffer goes to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, arrival rate to the buffer, 1/ρ, is smaller than its departure rate, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' implying stability of the outlier buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our main algorithm: RANS Our final algorithm, RAN with outlier Splitting (RANS), is a combination of RAN, outlier-splitting, and adaptive step- size ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In order to improve updates of m, we employ an adaptive vector step-size β that evolves according to a mechanism quite similar to RMSProp (Kochenderfer & Wheeler, 2019), as we discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider a trace vector νt of (∇δt)2 updated according to νt ← λ′νt−1 + (1 − λ′)(∇δt)2, where (∇δt)2 is the entrywise square vector of ∇δt, and λ′ ∈ [0, 1) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We consider an outlier-measure ξt = ⟨ 1 √νt ⊙ ∇δt, ∇δt⟩ (15) where 1/√νt is entrywise square root, and ⊙ and ⟨·, ·⟩ de- note entrywise product and inner product of two vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We then compute the trace ¯ξ and k as in Sec- tion 8: ¯ξt ← λ′ ¯ξt + (1 − λ′)ξt and k = � ξt/(ρ¯ξt) � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We finally fix an η ∈ (0, 1) and choose the step-size βt = η ρ¯ξt 1 √νt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (16) The pseudo code of RANS is given in Algorithm 4 in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The algorithm involves applying the outlier- splitting method on the updates of m in RAN, and using the adaptive step-size in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We now shows that the outlier-splitting mechanism in RANS effectively prevents overshoots of type (13) in the updates of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Given the above choice of βt, we have 1 k ⟨βt⊙∇δt, ∇δt⟩ = 1 k η ρ¯ξt ⟨ 1 √νt ⊙ ∇δt, ∇δt⟩ ≤ ρ¯ξt ξt η ρ¯ξt ⟨ 1 √νt ⊙ ∇δt, ∇δt⟩ = η, where the first equality is from the definition of βt in (16), the inequality is due to the definition of k, and the last equality follows from the definition of ξt in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This implies that ��⟨1 k β(∇δT t m)∇δt, ∇δt⟩ �� ≤ η ��∇δT t m ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (17) Therefore overshoots of type (13) do not occur in RANS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Performance of RANS, TD(0), and RG on classic control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A single-layer neural network with 64 hidden ReLU units was used to learn the Q-values, and a softmax disribution on the Q-values was used as the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The RANS algorithm has hyperparameters α, η, ρ, λ, λ′, and σ (the outlier sampling probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Setting η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2 are always good choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Furthermore, our experiments show that the parameters λ, λ′, and σ can be set to the default values λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='999, λ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='9999, and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='02 without much performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, the RANS algorithm would have essentially one hyper- parameter α, just like RG and TD algorithms with Adam optimizer (Kingma & Ba, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The per-iteration com- putational complexity of RANS is at most twice the RG algorithm with Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We tested RANS for control in Acrobot and Cartpole envi- ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We used a single-layer neural network with 64 hidden units with ReLU activation to learn the action-values, while choosing actions according to a softmax distribution on the action-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 5 illustrates expected returns ver- sus number of step for TD(0), RG, and RANS algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We trained TD(0) and RG using Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Refer to Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 for complementary experimental results and details of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The results show that the RANS algorithm outperforms RG, and its performance is comparable to TD on these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Related works Poor conditioning of MSBE was previously observed in (Wang & Ueda, 2021) through study of an example Markov Acrobot 100 200 Return 300 TD (Adam) 400 RG (Adam) RANS 500 0 10 20 30 40 50CartPole 500 400 Return 300 200 TD (Adam) 100 RG (Adam) RANS 0 0 20 40 60 80 100 Steps (x1000)Toward Efficient Gradient-Based Value Estimation chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More specifically, Wang and Ueda (2021) ana- lyzed a particular n-state Markov chain and showed that the condition-number of MSBE in this Markov chain scales with n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' They also showed that the condition-number scales with 1/(1 − γ)2 in another example Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In com- parison, our lower bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 (a) holds for every Markov chain, and the lower bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 (b) scales with n2/(1 − γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A prevalent explanation for slowness of gradient-based value estimation methods is the so called information flow in the wrong direction (Baird, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More concretely, each up- date in RG can be decomposed into a forward bootstrapping component (or a TD update) and a backward bootstrapping component (the so called wrong direction of information flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A common approach for accelerating the gradient updates is by suppressing the second component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', via some sort of combination with TD updates), especially in early stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The acceleration gained in the residual algorithm (Baird, 1995), TDC (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2009), TDRC, and QRC (Ghiassian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020) can be understood from this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In contrast, acceleration gained in our algorithms does not rely on combinations with TD updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Use of Gauss-Newton method for value estimation was ex- plicitly proposed in (Gottwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Gottwald & Shen, 2022), recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Value estimation algorithms based on Kalman filter (Choi & Van Roy, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Geist & Pietquin, 2010) are also known to have an equivalent form to online Gauss-Newton updates (Geist & Pietquin, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Sun and Bagnell (2015) studied MSBE minimization with Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, all of the above methods involve approx- imating a variant of the Hessian or Gauss-Newton matrices and solving a system of linear equations in each iteration, which is computationally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Several other algorithms including least squares TD (Sutton & Barto, 2018) and (De- vraj & Meyn, 2017) also leverage matrix gain for improved convergence, under linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the same spirit, natural gradient methods (Amari, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Kakade, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Martens, 2020) also enjoy robustness to pa- rameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dabney and Thomas (2014), Knight and Lerner (2018), and Achiam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (2019) proposed natu- ral gradients algorithms for value estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dabney and Thomas (2014) also proposed a low complexity two time scale implementation that has high-level algorithmic simi- larities to the RAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Section 5 and Appendix C we showed that the RAN algorithm can be perceived as a proximal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A prox- imal method for value estimation, called GTD2-MP, was proposed in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Mahadevan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, these works consider a Bregman divergence that does not depend on the value estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In fact, as the step-size goes to zero, update direction of GTD2- MP tends to the expected GTD2 update direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Schul- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (2015), Sun and Bagnell (2015), and Zhu and Murray (2022) considered proximal methods with value dependent penalties of the form E[(vwt+1(St) − vwt(St))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Although the resulting expected update direc- tion Es[∇vw(s)∇vw(s)T ]−1∇MSBE(w) is robust to pa- rameterization, it is not robust against poor conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For example, in the tabular case, this expected update direc- tion simplifies to ∇MSBE(w), which is the same as RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In contrast, in the proximal view of the RAN algorithm, we used penalties of type E[(δwt+1(St, At) − δwt(St, At))2], which provides robustness to the conditioning of MSBE, as discussed in Section 5 and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Karampatziakis and Langford (2010) and Tian and Sutton (2019) proposed a method, called sliding-step, to reduce the adverse effect of outliers in certain problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This method is pretty similar to the outlier-splitting algorithm, with the only difference that in the sliding-step method, all k updates w ← w − ∇ft(w)/k are applied sequentially and before time t + 1, while the outlier-splitting method spreads these updates over a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Another simple approach is using momentum to reduce the variance of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, smoothing large outliers requires large momentum parameters, in which case the delayed effect of gradients propagate far into future and become out-dated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Future works and discussion In this paper, we highlighted causes that underlie slowness of gradient-based value estimation methods, and proposed low complexity techniques to resolve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Our focus was on the on-policy case, however the proposed algorithms are easily applicable for off-policy learning when combined with standard importance sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We provided evidence for the potential of the proposed algorithms via experiments on a few classic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Other than applying standard techniques (such as batch up- dates, replay buffers, different forms of step-size adaptation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=') and testing the algorithms on more complex environ- ments, there are several directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This includes adopting the unbiased gradient estimate of (9) in (10) instead of the biased estimate in (11), and comparing these methods with other means of solving (9), including conjugate gradient and low rank approximation of the Gauss- Newton matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Another important direction is further ex- ploration of the proposed double-sampling-free methods in stochastic environments with neural network function ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' On the theory side, it would be interesting to study condition-number of MSBE, and in general the shape of MSBE landscape, under linear and non-linear function approximation under common feature representations in asymptotically large environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Toward Efficient Gradient-Based Value Estimation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Acknowledgments The authors want to thank Yi wan, Sina Ghiassian, John N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Tsitsiklis, and Saber Salehkaleybar for their valuable feedback in various stages of development of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' References Achiam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', Knight, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Proceedings of the 19th In- ternational Conference on Autonomous Agents and Mul- tiagent Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 1611–1619, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' and Murray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Gradient descent tem- poral difference-difference learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='04624, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Toward Efficient Gradient-Based Value Estimation Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proof of condition-number bounds In this appendix, we first present the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 that involves bounds on the condition-number of MSBEV (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We then establish similar bounds for MSBE(·) defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 Note that an MDP with a fixed policy boils down to a Markov chain with termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider a Markov chain with termination that has n non-terminal states, and let P be its associated n × n transition matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Note that if transitions from a state can terminate with positive probability, sum over the corresponding row of P will be less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let A def= (I − γP)T (I − γP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (18) In the tabular setting and under uniform state distribution, we have MSBEV (w) = wT Aw/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore, the condition- number C of MSBEV (·) is equal to the condition-number of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let λmax and λmin denote the largest and smallest eigenvalues of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It follows that C = λmax λmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (19) Proof of Part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We first propose an upper bound for λmin and then a lower bound for λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For states i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , n, let li be the expected number of steps until termination when we start from state i and follow the Markov chain’s transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, for any state i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , n, we have li = 1 + n � j=1 Pijlj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (20) Let l def= [l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , ln]T be the vector representation of l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, (20) can be written in the vector form as l = 1 + Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (21) where 1 is the vector of all ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It then follows that (I − γP)l = l − γPl = l − γ(l − 1) = (1 − γ)l + γ1, (22) where the second equality follows from (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let l def= (l1 +· · · , ln)/n be the mean of l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Cauchy-Schwarz inequality implies that ∥l∥2 n = 1 n n � i=1 l2 i ≥ � 1 n n � i=1 li �2 = l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (23) For the smallest eigenvalue of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' we have λmin ≤ lT Al ∥l∥2 = ∥(I − γP)l∥2 ∥l∥2 = ∥(1 − γ)l + γ1∥2 ∥l∥2 = (I − γ)2∥l∥2 + 2γ(1 − γ)1T l + γ2n ∥l∥2 = (I − γ)2 + 2γ(1 − γ)nl + γ2n ∥l∥2 ≤ (I − γ)2 + 2γ(1 − γ)l + γ2 l2 = � I − γ + γ l �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (24) Toward Efficient Gradient-Based Value Estimation where the first inequality follows from the definition of the smallest eigenvalue of a symmetric matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the first equality is due to the definition of A in (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the second equality results from (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the fourth equality is from the definition of l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' and the second inequality follows from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let trace(A) be the trace of A defined as the sum of diagonal entries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It is well-known that the trace of any matrix is equal to the sum of eigenvalues of that matrix (Strang, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' for the largest eigenvalue of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' we have λmax ≥ 1 ntrace(A) = 1 n n � i=1 Aii = 1 n n � i=1 n � j=1 (Iji − γPji)2 = 1 n n � i=1 � �(1 − γPii)2 + � j̸=i γ2P 2 ji � � ≥ 1 n n � i=1 (1 − γPii)2 ≥ � 1 n n � i=1 (1 − γPii) �2 = � 1 − γ n n � i=1 Pii �2 = (1 − γh)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (25) where the first inequality is because trace(A) equals the sum of eigenvalues of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the first equality is from the definition of trace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the second equality is due to the definition of A in (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the third inequality follows from the Cauchy-Schwarz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' and the last equality is from the definition h = �n i=1 Pii/n in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Plugging (24) and (25) into (19), we obtain C = λmax λmin ≥ (1 − γh)2 λmin ≥ (1 − γh)2 (1 − γ + γ/l)2 ≥ (1 − γh)2 2(1 − γ)2 + 2γ2/l2 ≥ (1 − γh)2 4 min � 1 (1 − γ)2 , l2 γ2 � , where the first and second inequalities are due to (25) and (24), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This implies (8) and completes the proof of Part (a) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proof of Part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Consider an n-state Markov chain with transition matrix P = � �� 0 · · 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 0 · · 0 1 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (26) In what follows, we derive bounds on the largest and smallest eigenvalues of A defined in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let ϵ = γ/(n − 1) and v = [−ϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , −ϵ, 1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, Pv = 1, and as a result, vT Av = ��(I − γP)v ��2 = ��v − γ1 ��2 = (n − 1)(γ + ϵ)2 + (1 − γ)2 ≥ (n − 1)(γ + ϵ)2 = n2γ2 n − 1, (27) where the first equality is from the definition of A in (18), and the last equality is due to the definition of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' On the other hand, ∥v∥2 = (n − 1)ϵ2 + 1 = γ2 n − 1 + 1 = n − 1 + γ2 n − 1 ≤ n n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (28) Toward Efficient Gradient-Based Value Estimation It follows that λmax ≥ vT Av ∥v∥ ≥ n2γ2/(n − 1) n/(n − 1) = nγ2, (29) where the second inequality is due to (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In order to bound the smallest eigenvalue of A, let x = [γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , γ, 1]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore Px = 1 and (I − γP)x = x − γPx = x − γ1 = [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , 0, 1 − γ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (30) It follows that λmin ≤ xAx ∥x∥2 = ∥(I − γP)x∥2 ∥x∥2 = (1 − γ)2 ∥x∥2 = (1 − γ)2 (n − 1)γ2 + 1 ≤ (1 − γ)2 nγ2 (31) where the first equality is from the definition of A in (18), the second equality is due to (30), and the third equality follows from the definition of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Plugging (29) and (31) into (19), we obtain C = λmax λmin ≥ nγ2 (1 − γ)2/(nγ2) = γ4n2 (1 − γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This completes the proof of Part (b) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Condition-number of the MSBE defined in terms of action-values Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 involves bounds on the condition-number of MSBEV (·) defined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this appendix, we establish similar bounds for MSBE(·) defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Given an MDP and a policy π, we consider an induced augmented Markov chain that is a Markov chain whose states are the state-action pairs of the MDP and its transition probabilities are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For any s, s′ ∈ S and a, a′ ∈ A, the probability of transition from (s, a) to (s′, a′) in the induced augmented Markov chain is p′ π(s′, a′|s, a) def= � r pπ(s′, a′, r|s, a) dr (32) where pπ is defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We consider a tabular setting, and denote the condition-number of MSBED(·) under uniform distribution D on state-action pairs by C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We let h′ def= 1 nm � s∈S � a∈A p′ π(s, a|s, a) (33) be the self-loop probability in the induced augmented Markov chain, where n is the number of states and m = |A| is the number of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Also let l′ be the expected number of steps until termination when starting from a uniformly random state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The following proposition is the counterpart of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 for C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the tabular case, the following statements hold for any discount factor γ ∈ [0, 1]: a) For any MDP and any policy π, C′ ≥ 1 − γh′ 4 min � 1 (1 − γ)2 , l′2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (34) b) For any n, m > 0, there exists an MDP with n states and m actions, and a policy π for which, C′ ≥ γ4(nm)2/(1−γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We can perceive the dynamics under any given MDP and policy as an induced augmented Markov chain defined in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Applying the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 on this induced augmented Markov chain implies Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Experiment on condition number under linear function approximation We ran an experiment to investigate the growth of condition number, C, in an extended Boyan chain under linear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 6 shows the dependence of C on the size of extended Boyan chain, under standard Boyan feature vectors (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The number of standard Boyan chain features d in this experiments, satisfies n = 4d − 3, where n is the number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We observe that the condition number can grow very large under linear function approximation even when d/n < 1 (in this case d/n ≃ 1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Toward Efficient Gradient-Based Value Estimation Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Condition-number of MSBE versus number of states in an extended Boyan chain under linear function approximation with Boyan chain’s standard features (n = 4d − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' RAN as a proximal algorithm In this section, we provide further intuition for the RAN algorithm, by showing that Algorithm 1 can be equivalently derived as a proximal algorithm with momentum for minimizing MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Given an objective function f, a proximal algorithm in its general form aims to find an approximate solution for a proximal operator of the following form in each iteration wt+1 ← argmin w � f(w) + Dt(w, wref) � , (35) where wref is a reference point, usually equal to wt or a trace of past w’s, and Dt is a penalty function (also called divergence) that encourages wt+1 to stay close to wref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the special case that Dt is a fixed Bregman divergence, (35) boils down to the mirror-descent algorithm (Juditsky & Nemirovski, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' However, in general, Dt can be a time varying function and can depend on the local shape of the objective f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Back to the value estimation problem, for any consecutive state-action pairs (s, a) and (s′, a′), and any pair w and w′ of weights, we let δw(s, a, s′, a′) def= γqw(s′, a′) − qw(s, a), (36) and ∆δw,w′(s, a, s′, a′) def= δw(s, a, s′, a′) − δw′(s, a, s′, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (37) Consider a divergence measure of the form Dt(w, wt) = c Es,a � Es′,a′|s,a � ∆δw,wt(s, a, s′, a′)2� � , for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, the proximal operator in (35) turns into argmin w MSBE(w) + c Es,a � Es′,a′|s,a � ∆δw,wt(s, a, s′, a′)2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (38) To obtain a low complexity incremental version of the above proximal updates, we consider doing sample updates along the gradient of (38) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For this purpose3, we work with wref = wt−1 instead of wref = wt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' we consider the proximal objective MSBE(w) + c Es,a � Es′,a′|s,a � ∆δw,wt−1(s, a, s′, a′)2� and its unbiased sample gradient gt def= � δ′ t − c ∆δwt,wt−1(St, At, St+1, At+1) � ∇δt ≃ � δ′ t − c (wt − wt−1)T ∇δt � ∇δt, (39) 3Here we avoid using wref = wt because in this case, for any (s, a, s′, a′), ∆δwt,wt(s, a, s′, a′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This implies that ∇w∆δw,wt(s, a, s′, a′)2�� w=wt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As such, the penalty would not affect gradient of the proximal objective at w = wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 5000 1SBE 4000 M Condition number of 3000 2000 1000 0 100 200 300 400 0 Number of statesToward Efficient Gradient-Based Value Estimation where the approximate equality is because (wt − wt−1)T ∇δt is a first order approximation of ∆δwt,wt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let ˆgt = � δ′ t − c(wt − wt−1)T ∇δt � ∇δt and consider the approximate gradient update w ← w − βˆgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We further employ a momentum to reduce the variance of these updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The resulting momentum based algorithm is of the form mt = λmt−1 + βˆgt, wt+1 = wt − αmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (40) Let η = 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Since wt − wt−1 = αmt−1, ˆgt simplifies to ˆgt = � δ′ t − mT t−1∇δt � ∇δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Plugging this into (40), we obtain updates that are identical to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This establishes our earlier claim that the RAN algorithm can be equivalently formulated as a proximal algorithm with momentum for minimizing MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As another intuitive perspective for Algorithm 1, we can perceive m as a momentum of MSBE gradients, to which we add a correction term equal to a sample gradient of the penalty Es,a � Es′,a′|s,a � ∆δw,wt(s, a, s′, a′)2� � , in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This momentum spreads the effect of each MSBE gradient over an O(1/(1 − λ))-long horizon, which provides enough time for the correction updates to trim the direction of those MSBE gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The correction terms are small along directions that ∇δ is small, allowing MSBE gradients to accumulate along those directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Conversely, the correction terms are large along directions that ∇δ is large, preventing m from growing large along those directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This leads to accelerated convergence along the directions where ∇δ is small, while preventing instability along directions where ∇δ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Convergence of the RAN algorithm In this appendix, we study conditions for convergence of Algorithm (11) under different choices of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Convergence proofs of two-time-scale approaches are well-studied, and generally involve tedious yet pretty standard statistical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As such, similar to several other papers (Konda & Tsitsiklis, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Dabney & Thomas, 2014), here we keep our arguments in a high level, and only discuss the steps of the proof without going into the proof details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Throughout this Appendix, we consider irreducible and aperiodic Markov chains with finite number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We assume that the function approximation Qw(s, a) is a differentiable function of w with bounded and Lipschitz constant derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' By boundedness of derivatives we mean that there exists a C > 0 such that for consecutive state-action pairs (s, a, s′, a′) and any w, ∥∇δw(s, a, s′, a′) �� < C, (41) where δw(s, a, s′, a′) is defined in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Given a distribution D over state and action pairs, for any w ∈ Rd, we let ˆGw def= Es,a∼D � Es′,a′|s,a � ∇δw(s, a, s′, a′) ∇δw(s, a, s′, a′)T � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (42) We study convergence in three different regimes on λ, namely λ = 1, λ = 1 − cβt for some constant c > 0, and for a constant λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Case 1: (λ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We assume that αt and βt are decreasing positive sequneces, satisfying ∞ � t=0 αt = ∞ � t=0 βt = ∞, ∞ � t=0 α2 t < ∞ ∞ � t=0 β2 t < ∞, αt βt → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (43) We further assume that ˆGw defined in (42) is uniformly positive definite, in the sense that there is an ϵ > 0 such that for any w and any x ∈ Rd, we have xT ˆGwx ≥ ϵ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, for fixed w, the updates in (11) converge to ˆG−1 w ∇MSBED(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Since αt/βt → 0, w is updated much slower than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As such, m is updated with much larger step-sizes, perceiving w as almost stationary, and therefore m converges to an asymptotically small neighborhood of ˆG−1 w ∇MSBED(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Since ˆGw is uniformly positive definite, this m is an absolutely decreasing direction for MSBED(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, standard proof techniques for stochastic approximation algorithms can be used to establish convergence of w to a stationary point of MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Case 2: (λ = 1 − cβt, for some constant c > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, the update rule (11) boils down to: m ← λm + βt(δ′ t − mT ∇δt � ∇δt = (1 − cβt)m + βt � δ′ t − mT ∇δt � ∇δt = m − βt � (cI + ∇δt∇δT t )m − δ′ t∇δt � = m − βt∇m �1 2mT � cI + ∇wδt∇wδT t � m − δ′ t∇wδT t m � , (44) Toward Efficient Gradient-Based Value Estimation where I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore, in each iteration, m is updated along a sample gradient of the loss function ˆL(m) = mT � cI + ˆGw � m−∇wMSBE(w)T m, where ˆGw is defined in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Thus, assuming (43), m will asymptotically converge to the minimizer � cI + ˆGw �−1∇MSBE(w) of ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Since cI + ˆGw is uniformly positive definite, this is an absolutely descent direction for MSBE(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, standard proof techniques for stochastic approximation algorithms can be used to establish convergence of w to a stationary point of MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Case 3: (Constant λ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As opposed to the Cases 1 and 2, here we do not need two-time-scale step-sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More specifically, we assume that αt > 0 is constant and βt is a decreasing positive sequence satisfying �∞ t=0 βt = ∞ and �∞ t=0 β2 t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We also assume that w remains bounded which can be enforced either by projection of w on a compact set or by adequate normalization of αt (see (Konda & Tsitsiklis, 1999) and (Kushner & Yin, 2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It follows from (41) that for any time t, |δt| = |δwt(St, At, St+1, At+1)| ≤ C′ + C∥wt∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (45) for some constant C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, from (11) we obtain ∥mt∥ = ��λmt−1 + βt � δ′ t − mT t−1∇δt � ∇δt �� ≤ λ∥mt−1∥ + βt � |δ′ t| + ∥mt−1∥ · ∥∇δt∥ � ∥∇δt∥ ≤ λ∥mt−1∥ + βt � |δ′ t| + C∥mt−1∥ � C ≤ λ∥mt−1∥ + βtC � C′ + C∥wt∥ + C∥mt−1∥ � = (λ + βtC2)∥mt−1∥ + βt(C′C + C2∥wt∥), (46) where the second and third inequalities follow from (41) nad (45), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' When βt is small enough such that λ + βtC2 < 1, it follows from the boundedness assumption of w that ∥mt∥ = O(βt/(1 − λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore, (11) can be expressed as mt = λmt−1 + βtδ′ t∇δt − βt(mT t−1∇δt)∇δt = λmt−1 + βtδ′ t∇δt + O � β2 t /(1 − λ) � = t � τ=0 � λτβt−τδ′ t−τ∇δt−τ � + O � β2 t /(1 − λ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore, m is essentially a momentum of sample gradient of MSBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, convergence of w to a stationary point of MSBE follows from standard techniques for analysing SGD algorithms with momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Asymptotic robustness of RAN to parameterization In this appendix, we present a formal version of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Here, we assume that λ = 1 and consider an asymptotically small step-sizes regime with α → 0 and α/β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' As discussed in Section 5 and Appendix D, when α/β → 0, m in the RAN algorithm converges to the approximate Gauss-Newton direction mGN(w, q) = ˆG−1 w,q gw,q, (47) where ˆGw,q = Es,a∼D � Es′,a′|s,a �� γ∇wqw(s′, a′) − ∇wqw(s, a) � � γ∇wqw(s′, a′) − ∇wqw(s, a) �T � � , (48) gw,q = Es,a∼D � Es′,a′,r|s,a [r + γqw(s′, a′) − qw(s, a)] Es′,a′|s,a � γ∇wqw(s′, a′) − ∇wqw(s, a) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (49) where D is a distribution over state and action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, when α → 0 the trajectory of w approaches the trajectory of the following Ordinary Differential Equations (ODE): ˙w = −mGN(w, q), (50) where ˙w is the derivative of w with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We refer to (50) as the ODE formulation of two time-scale RAN for the q function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Toward Efficient Gradient-Based Value Estimation Let u : Rd → Rd be a (possibly non-linear) bijective and differentiable mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Suppose that the Jacobian of u(·) is invertible everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We define ˜q as a reparameterization of the q function by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More specifically, for any state-action pair (s, a) and any v ∈ Rd, we let ˜qv(s, a) = qu(v)(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (51) Consider the ODE formulation of two time-scale RAN for the ˜q function: ˙v = −mGN(v, ˜q), (52) for mGN(w, ˜q) defined in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The following proposition is a formal version of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It draws a connection between solution trajectories of (50) and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1 (Formal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let u : Rd → Rd be a (possibly non-linear) bijective and differentiable mapping with invertible Jacobian, and consider the corresponding reparameterization ˜q of the q function as in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Let wt and vt for t ≥ 0 be solution trajectories of the ODE formulations of two time-scale RAN in (50) and (52), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' If w0 = u � v0 � , then wt = u(vt), for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Moreover, qwt(s, a) = ˜qvt(s, a), for all times t ≥ 0 and all state-action pairs (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The proposition suggests that the two-time scale RAN algorithm with asymptotically small step-sizes is invariant to any non-linear bijective reparameterization of the q function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For any v ∈ Rd, let Jv be the Jacobian matrix of u(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then, for any state-action pair (s, a) and any v ∈ Rd, ∇v˜qv(s, a) = ∇vqu(v)(s, a) = JT v ∇wqw(s, a) �� w=u(v), (53) where the first equality is from the definition of ˜q in (51), and the second equality follows from the chain rule for differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ˆGv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a∼D � Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a �� γ∇v˜qv(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ∇v˜qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) � � γ∇v˜qv(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ∇v˜qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) �T � � = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a∼D � Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a � JT v � γ∇wqw(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ∇wqw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) � � γ∇wqw(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ∇wqw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) �T Jv �� w=u(v) � � = JT v ˆGw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='q Jv �� w=u(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (54) where the first and the last equalities are due to (48) second equality follows from (53) In the same vein,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' gv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a∼D � Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='r|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a [r + γ˜qv(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ˜qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a)] Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a � γ∇v˜qv(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ∇v˜qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) �� = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a∼D � Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='r|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a [r + γ˜qv(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − ˜qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a)] Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a � γJT v ∇wqw(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − JT v ∇wqw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) � �� w=u(v) � = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a∼D � Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='r|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a [r + γqw(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − qw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a)] Es′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='a � γJT v ∇wqw(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a′) − JT v ∇wqw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' a) � �� w=u(v) � = JT v gw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='q �� w=u(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (55) where the first and the last equalities are due to (49),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' the second equality follows from (53),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' and the third equality is from the definition of ˜q in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Plugging (54) and (55) into (50) and (52),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' we obtain ˙v = −mGN(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ˜q) = − ˆG−1 v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q gv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q = − � JT v ˆGw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='q J �−1 gv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q �� w=u(v) = −J−1 v ˆG−1 w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='qJ−T v gv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='˜q �� w=u(v) = −J−1 v ˆG−1 w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='q gw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='q �� w=u(v) = −J−1 v mGN(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' q) �� w=u(v) = J−1 v ˙w �� w=u(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' (56) Toward Efficient Gradient-Based Value Estimation Algorithm 4 RANS Hyper parameters: stepsize α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' η ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 1) outlier threshold ρ > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' momentum and trace parameters λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' λ′ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' outlier sampling probability σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Good default values for η, ρ, λ, λ′, and p based on our experiments are η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2, ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='999, λ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='9999, and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Initialize: m = 0, ˆν = 0, ˆξ = 0, and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' do Take two independent samples (St+1, At+1) and (S′ t+1, A′ t+1), and consider δt and δ′ t as in (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ˆνt ← λ′ˆνt−1 + (1 − λ′)(∇δt)2 ▷ (∇δt)2 is the entrywise square vector of ∇δt νt = ˆνt/(1 − λ′t) ▷ bias-corrected trace of (∇δt)2 ξt = ⟨(1/√νt) ⊙ ∇δt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ∇δt⟩ ▷ Outlier measure ˆξ ← λ′ ˆξ + (1 − λ′)ξt ¯ξ = ˆξ/(1 − λ′t ξ ) ▷ bias-corrected trace of ξ k = ⌊ξt/(ρ¯ξ)⌋ + 1 ▷ outlier-splitting factor β = η/(ρ¯ξt √νt) m ← λm + � δ′ t − mT ∇δt � β ⊙ ∇δt/k w ← w − αm if k > 1 then Store (St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' St+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' At+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' S′ t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A′ t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' r′ t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k − 1) in the outlier buffer ▷ the last entry in the tuple indicates the remaining number of future updates based on this sample end if With probability min(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' σ∗length of outlier bufffer): ▷ do an update using outlier buffer samples Sample (Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Aτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Sτ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Aτ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' rτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' S′ τ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' A′ τ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' r′ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' j) uniformly form the outlier buffer Let δt and δ′ t be as in (3) and (4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' respectively ξ = ⟨(1/√νt) ⊙ ∇δτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ∇δτ⟩ k′′ = max � k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' ⌊ξ/(ρ¯ξ)⌋ + 1 � m ← λm + � δ′ τ − mT ∇δ′ τ � β ⊙ ∇δτ/k′′ w ← w − αm if j > 1 then Replace (Sτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , k′, j) with (Sτ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , k′, j − 1) in the outlier buffer else if j = 1 then Remove (Sτ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , k′, j) from the outlier buffer end if end for where the equalities are respectively due to (52), (47), (54), the assumption that the Jacobian J of u is invertible, (55), (47), and (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Therefore, if at time t, wt = u(vt), then ˙wt = Jv ˙vt = d dtu(vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This together with the assumption w0 = u(v0) implies that wt = u � vt � , for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' It then follows from the definition of ˜q in (51) that qwt(s, a) = ˜qvt(s, a), for all times t ≥ 0 and all state-action pairs (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This completes the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Pseudo code of RANS The pseudo code of the RANS algorithm is given in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Note that in the two time scale regime, where α, η → 0 and α/η → 0, outlier-splitting would have no effect on the expected direction of m updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, convergence of the RANS algorithm follows from similar arguments to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Details of experiments G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Experiments of Section 3 and Appendix B An n-state extended Boyan chain with termination is a Markov chain with termination with states 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , n − 1 and transition probabilities: P(1 → 0) = 1 and P(i → i − 1) = P(i → i − 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='5 for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Furthermore, state 0 goes to a terminal state with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' By standard features, we mean feature representations similar to (Boyan, Toward Efficient Gradient-Based Value Estimation 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' More specifically, given a d > 1 and n = 4d − 3, we consider d standard features for the n-state extended Boyan-chain as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the ith standard feature vector for i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , d − 1, the jth entry is equal to 1 − |j − 4i|/4 for j = max(0, 4i − 3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , min(d − 1, 4i + 3), and all other entries equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For the special case of n = 13 and d = 4, the standard features would be the same as the features considered in (Boyan, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' By random binary features we mean an n × d feature matrix Φ with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' entries that take 0 and 1 values with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For evaluating value-errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2 we consider reward 1 for the transition at state 0 (that leads to the terminal state), and reward 0 for all other transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Note that the reward function does not affect condition-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Each point in this first experiment is the median of 100 independent runs with different random feature matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We use median instead of mean to eliminate the adverse effect of unbounded values in degenerate cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=', infinite condition-number in the case of low rank feature matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For both experiments in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 2 and 6 we use discount factor γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Condition-numbers and value-errors in these experiments are computed with respect to uniform state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Experiment of Section 5 We considered an extension of the Hallway environment with 50 states and one action, in which each state s = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , 50 transits to state min(s + 1, 50) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='99, and transits to a terminal state with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The experiment was tabular with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' All rewards were set equal to zero, in which case the correct q-values are qπ(s, ·) = 0, for all sates s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' All algorithms were initialized with q(s, ·) = 1, for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' , 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 3 illustrates learning curves of value-error �50 s=1 � q(s, 1) − qπ(s, 1) �2/50 for RAN (Algorithm 1), RG, and TD(0) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Each point is an average over 100 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We optimized the parameters for RG and Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The parameters used in the experiments are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RAN, we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='025, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='9998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RG and TD(0) we used α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Experiment of Section 6 We ran an experiment on Baird’s Star environment (Baird, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We performed off-policy learning with uniform state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' All rewards were set to zero, in which case the correct q-values are qπ(s, ·) = 0, for all sates s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We used the initial point w = [2, 1, 1, 1, 1, 1, 1] for all algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 4 illustrates learning curves of value-error �6 s=1 � qw(s, 1) − qπ(s, 1) �2/6 for the RAN (Algorithm 1), DSF-RAN (Algorithm 2), RG, GTD2, and TD(0) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Each point is an average over 10 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' TD(0) was unstable on this environment, and we chose a very small step-size α = 10−5 for TD(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For all other algorithms, we used optimized parameters as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RAN, we set α = 2, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='15, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For DSF-RAN, we set α = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='15, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RG we used α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For GTD2 we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='15 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Experiment of Section 9 We ran an experiment on classic control tasks –Acrobot and Cartpole– to test the performance of the RANS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We used a single-layer neural network with 64 hidden ReLU units to learn the action-values, while choosing actions according to a softmax distribution on the action-values, a ∼ Softmax � qw(s, ·) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The network for qw was trained with three algorithms: TD(0), RG, and RANS (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Since RANS incorporates adaptive step-sizes, for fair comparison we trained TD(0) and RG using Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For each algorithm, we performed training for 100 randomly generated random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For each training seed, in order to obtain an estimate of expected returns, we took an average over 400 independent environment simulations once every 500 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 7 we plot the the average of these estimated expected returns over the 100 training seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In the Cartpole environment, once an algorithm reaches score 500, it will not see any failure for many episodes in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In this case, the agent starts to forget the actions that prevented failure and led it to success, causing the performance to drop before it can rise again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' This phenomenon is known as catastrophic forgetting, and leads to large oscillations in learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In order to hide the effect of catastrophic forgetting on the learning curves, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 5 we eliminated the 50 worst return estimates (corresponding to the 50 worst training seeds) at each point and plotted the mean of top 50 return estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The shades in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' 7 show the 99 percent confidence intervals over the averaged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We did not use replay buffer and batch updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' We used a small quadratic regularizer with coefficient 10−5 on the weights of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For all other algorithms, we used optimized parameters that maximize area under the curves, as Toward Efficient Gradient-Based Value Estimation Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Performance of RANS, TD(0), and RG on classic control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' The only difference with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='5 is that here, each point is an average of estimate returns over 100 independent training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' See Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Acrobot experiment: For TD(0), we used softmax coefficient 1 and Adam optimizer with step-size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='005 while all other parameters were set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RG, we used softmax coefficient 16 and Adam optimizer with step-size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='001 while all other parameters were set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RANS, we used softmax coefficient 16 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='005 and set all other parameters were set to their default values described in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' In Cartpole experiment: For TD(0), we used softmax coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='005 and Adam optimizer with step-size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3 while all other parameters were set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RG, we used softmax coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='002 and Adam optimizer with step-size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='3 while all other parameters were set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' For RANS, we used softmax coefficient 8 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content='001 and set all other parameters were set to their default values described in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} +page_content=' Acrobot 100 200 Return 300 TD (Adam) 400 RG (Adam) RANS 500 0 10 20 30 40 50 Steps 5(x1000)CartPole 500 400 300 Return 200 TD (Adam) 100 RG (Adam) RANS 0 0 20 40 60 80 100 Steps (x1000)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtFST4oBgHgl3EQfQDhb/content/2301.13757v1.pdf'} diff --git a/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/2301.02118v1.pdf.txt b/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/2301.02118v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4552a32b07415f8d6fe2c9b0680cec728198d776 --- /dev/null +++ b/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/2301.02118v1.pdf.txt @@ -0,0 +1,1754 @@ +MNRAS 000, 1–12 (2022) +Preprint 2 December 2022 +Compiled using MNRAS LATEX style file v3.0 +Origin of Multifractality in Solar Wind Turbulence: +the Role of Current Sheets +Leonardo F. Gomes,1★ Tiago F. P. Gomes,1 Erico L. Rempel1,2 and Sílvio Gama3 +1Aeronautics Institute of Technology (ITA), 12228-900, São José dos Campos, SP, Brazil +2National Institute for Space Research (INPE), P. O. Box 515, 12227-010, São José dos Campos, SP, Brazil +3Mathematics Center of the Porto University (CMUP), Mathematics Department, Faculty of Sciences, University of Porto, R. Campo Alegre s/n, +4169-007 Porto, Portugal +2 December 2022 +ABSTRACT +In this work, a multifractal framework is proposed to investigate the effects of current sheets in solar wind turbulence. By using +multifractal detrended fluctuation analysis coupled with surrogate methods and volatility, two solar wind magnetic field time +series are investigated, one with current sheets and one without current sheets. Despite the lack of extreme-events intermittent +bursts in the current sheet-free series, both series are shown to be strongly multifractal, although the current sheet-free series +displays an almost linear behavior for the scaling exponent of structure functions. Long-range correlations are shown to be the +main source of multifractality for the series without current sheets, while a combination of heavy-tail distribution and nonlinear +correlations are responsible for multifractality in the series with current sheets. The multifractality in both time series is formally +shown to be associated with an energy-cascade process using the 𝑝-model. +Key words: multifractals – turbulence – data analysis – statistical – solar wind +1 INTRODUCTION +Fractals have been widely employed in nonlinear analysis along the +past decades as a form of representing the complex topological struc- +tures produced by dynamical systems. These topological structures +are subsets of the phase space that may represent chaotic attractors, +stable or unstable manifolds, boundaries between basins of attrac- +tion, etc. Thus, when dynamical systems are investigated through +nonlinear time series analysis, the fractal indices computed from the +time series somehow represent the complexity of the structure of +an underlying set on which the solution lies. Additionally, the dy- +namical structure could be represented either by a monofractal or a +multifractal process. A monofractal process has a scaling law for a +fluctuation function which is a linear function of statistical moments +with a single scaling exponent. A multifractal process has a power- +law scaling which is a nonlinear function of statistical moments with +a range of scaling exponents (Salat et al. 2017). A monofractal scal- +ing is to be expected from dynamical processes behind perfectly +self-similar fractal sets, like deterministically generated Cantor sets +(Cantor 1883), or even from white noise time series (Ihlen 2012); +multifractals, on the other hand, are observed in inhomogeneous sys- +tems, such as strongly intermittent turbulence, where the presence of +strong fluctuations related to coherent structures localized in space +generate a departure from Gaussianity in probability distribution +functions (PDFs) of small-scale structure functions (Carbone et al. +2004), as seen in several analyses of observational magnetohydro- +dynamic data (see, e.g., Marsch & Tu (1998), Burlaga (2001), and +★ E-mail: leofgb@ita.br +Bruno (2019) for reviews on turbulence, intermittency and multifrac- +tal scalings in the solar wind). +A series of recent works have confirmed the complex and multi- +fractal nature of solar wind fluctuations. Chang et al. (2004) studied +the origin of complexity in space plasmas using MHD simulations, +dynamic renormalization group and wavelet analysis, arguing that +the turbulent plasmas in the solar wind and auroral regions are dom- +inated by a combination of propagating modes and nonpropagating +intermittent nonlinear structures, whose interactions with charged +particles may lead to the energization of plasma populations such as +auroral ions. Macek (2007) employed Voyager magnetic field data +in the outer heliosphere and Helios plasma data in the inner helio- +sphere to show that multifractal spectra of intermittent solar wind +fluctuations are consistent with that of the generalized two-scale +weighted Cantor set. Bolzan & Rosa (2012) analyzed magnetic field +data from the ACE satellite and conjectured that the presence of +large scale coherent structures during coronal mass ejections (CME) +decreases the multifractality, when compared with periods after the +CME events. Wavelet-leader multifractal analysis of magnetospheric +dissipations, as measured by the AL index, reveal that the magne- +tosphere is a multi-scale, complex, turbulent system, driven into a +non-equilibrium self-organized state, which may explain the obser- +vations of repeatable and coherent substorm phenomena with under- +lying complex multifractal behavior in the plasma sheet (Valdivia +et al. 2013). The interaction of the solar wind with the Earth’s mag- +netosphere also contributes for multifractality in measurements of +the geomagnetic activity, such as the geomagnetic induced current +(Wirsing & Mili 2020) and the Dst index (Ogunjo et al. 2021), al- +though internal sources of multifractality must also be considered, as +Gopinath (2016) suggests that multifractality of the auroral electrojet +© 2022 The Authors + +2 +L. F. Gomes et al. +index is fairly independent of the solar activity cycle. Wawrzaszek +et al. (2019) characterized multifractality in intermittent turbulence +of heliospheric magnetic field fluctuations from Ulysses spacecraft, +concluding that intermittency/multifractality decreases with helio- +spheric distance, a result that was confirmed by Kiran et al. (2021). +Recent analysis of electron density fluctuations in the E-F valley +region of the ionosphere performed with the multifractal detrended +fluctuation analysis (MF-DFA) method show that irregularities are +multifractal, asymmetric, intermittent and non–homogeneous (Nee- +lakshi et al. 2022). +The direct link between intermittency and multifractality of mag- +netic and velocity field fluctuations in the solar wind was made clear +in Salem et al. (2009). Using data from the Wind spacecraft, they +applied the Haar wavelet transform to filter out intermittency from +the time series and showed that the scaling exponents for the struc- +ture functions behave as a linear function of statistical moments, as +in monofractal processes, therefore attributing multifractality in the +solar wind to intermittency. Gomes et al. (2019) obtained a similar +linear scaling after filtering out the current sheets from Cluster-1 +intermittent magnetic field data, suggesting that the current sheets +are the coherent structures responsible for the nonlinear scaling of +the structure functions in the solar wind. This was confirmed after +inspection of time series of days when current sheets were absent, +that also showed a linear scaling. +A question remained on whether the linear scalings found by Salem +et al. (2009) and Gomes et al. (2019) indeed imply that the filtered +time series are monofractal or not, i.e., is the nonlinearity of the +distribution of scaling exponents of structure functions a general +measure of multifractality or is it just an indication of intermittency, +one among different possible sources of multifractality? One of the +goals of the current work is to answer this question. In this sense, it +is important to stress that the origin of multifractality is not always +related to fat-tailed PDFs, as it may also be caused by different +correlations in small and large fluctuations, such as linear or nonlinear +correlations (Kantelhardt et al. 2002; Wu et al. 2018). The source +of multifractality can be investigated by producing surrogates from +the original time series. Two types of surrogates are useful in this +context (Theiler et al. 1992; Lancaster et al. 2018). First, shuffling the +amplitudes of the original signal breaks all long-range correlations, +while keeping the PDF unchanged. Therefore, if the multifractality is +due to fat-tailed PDFs, it cannot be removed by shuffling the series. +If it is due, solely, to time correlations, the corresponding shuffled +series will be monofractal. If both fat-tailed PDF and linear/nonlinear +correlations are present, the multifractality of the shuffled series +should be smaller than that of the original series (Barunik et al. +2012). The second type of surrogate is produced by randomizing the +phases of the Fourier modes of the original time series, producing a +new series with Gaussian PDF, but preserving the linear correlations +of the original series. If the random phases time series becomes +monofractal, then nonlinear correlations and/or non-Gaussian PDFs +are the source of multifractality. If the multifractality is preserved in +the random phases time series, then linear correlations are its source. +Studies of surrogate time series have been conducted to probe +the origin of multifractality in a wide range of contexts, including +financial markets (Barunik et al. 2012), human gate diseases (Dutta +et al. 2013), near-fault earthquake ground motions (Yang et al. 2015), +solar irradiance fluctuations (Madanchi et al. 2017), air pollutants +(Dong et al. 2017), meteorological time series of air pressure, air +temperature and wind speed (Gos et al. 2021) and rainfall records +(Sarker & Mali 2021). The surrogate method was also employed in +time series of CME linear speed during solar cycle 23 to conclude +that the multifractality is due to both the broad PDF and long range +time correlations (Chattopadhyay et al. 2018). In the present paper, +we use the method to reveal the role of current sheets in the origin +of multifractality in the solar wind. By analyzing two qualitatively +different magnetic field time series from Cluster-1, one filled with +current sheets and another one void of current sheets, we develop +a nonlinear methodology based on the MF-DFA method coupled +with the volatility and surrogate time series. Thus, the contribution +of small- and large-scale magnetic fluctuations can be quantified in +different types of multifractal solar wind series. It is revealed that +when the multifractality is not mainly due to the PDF, the scaling +exponents display an almost linear behavior as a function of the +moments of the structure function, despite the presence of strong +multifractality in the series. In addition, we employ the 𝑝-model +(Halsey et al. 1986; Meneveau & Sreenivasan 1987) to confirm that +the multifractality in both types of solar wind time series can be +attributed to a turbulent energy cascade process. +This paper is organized as follows. In section II, the MF-DFA +methodology is briefly described; in section III, the multifractal anal- +ysis of two solar wind time series is conducted, including their volatil- +ity time series; section IV analyses the surrogate of the original and +volatility time series, to determine if the source of the multifractality +in the solar wind is due to PDF or correlations; section V presents +the scaling exponent analysis of the original and surrogate times se- +ries; section VI describes the 𝑝-model analysis. Finally, section VII +presents the conclusions. +2 MF-DFA +The MF-DFA method is a generalization of the detrended fluctu- +ation analysis (DFA) method for quantifying long-range correla- +tions in non-stationary time series (Kantelhardt et al. 2002). The +method identifies the scaling of 𝑞th-order moments of the time se- +ries (Norouzzadeh et al. 2007). The MF-DFA method consists of five +steps: +(i) The time series 𝑥𝑘 (𝑘 = 1, 2, · · · , 𝑁) is integrated: +𝑌 (𝑖) = +𝑖∑︁ +𝑘=1 +[𝑥𝑘 − ⟨𝑥⟩] , +𝑖 = 1, ..., 𝑁 +(1) +where ⟨𝑥⟩ is the average value of the data set. +(ii) The series𝑌 (𝑖) is divided into 𝑁𝑠 ≡ int(𝑁/𝑠) non-overlapping +segments with equal lengths 𝑠. Since 𝑁 is usually not a multiple of +𝑠, some of the data points in the time series may be left out of the +last segment. To fix this, the procedure is repeated starting from the +opposite end of the time series and going backwards. Consequently, +2𝑁𝑠 segments are obtained. +(iii) The local trend for each 2𝑁𝑠 segments is calculated. Then +the variance is given by +𝐹2(𝑠, 𝜈) = 1 +𝑠 +𝑠 +∑︁ +𝑖=1 +{𝑌 [(𝜈 − 1) 𝑠 + 𝑖] − 𝑦𝜈(𝑖)}2 , +(2) +for each segment indexed by 𝜈 = 1, . . . , 𝑁𝑠 and +𝐹2(𝑠, 𝜈) = 1 +𝑠 +𝑠 +∑︁ +𝑖=1 +{𝑌 [𝑁 − (𝜈 − 𝑁𝑠) 𝑠 + 𝑖] − 𝑦𝜈(𝑖)}2 +(3) +for 𝜈 = 𝑁𝑠 + 1, . . . , 2𝑁𝑠 , where 𝑦𝜈 is the 𝑚-th degree fitting poly- +nomial of each segment 𝜈. This polynomial detrending of order 𝑚 in +MNRAS 000, 1–12 (2022) + +Origin of Multifractality in Solar Wind Turbulence +3 +the 𝑌 profile eliminates trends up to order 𝑚 − 1 in the original time +series and specifies the type of MF-DFA applied. +(iv) The average over all segments is calculated to obtain the 𝑞th- +order fluctuation function: +𝐹𝑞(𝑠) = +( +1 +2𝑁𝑠 +2𝑁𝑠 +∑︁ +𝜈=1 +[𝐹2(𝑠, 𝜈)] +𝑞 +2 +) 1 +𝑞 +, +(4) +where, in general, the 𝑞 parameter can take any real value except +zero. For 𝑞 = 2, the equation returns the DFA method. Steps 2 to 4 +are repeated for different time scales 𝑠. +(v) The scaling behavior of the fluctuation function is defined by +the log-log plot of 𝐹𝑞(𝑠) ×𝑠 for each value of 𝑞. If 𝑥𝑖 have long-range +correlations, for large values of 𝑠, 𝐹𝑞(𝑠) increases as a power-law, +𝐹𝑞(𝑠) ∼ 𝑠ℎ(𝑞). +(5) +The scaling exponents ℎ(𝑞) are the generalized Hurst exponents, +defined as the slope of the log 𝐹𝑞(𝑠) × log(𝑠) graph, where for ℎ(2) +we have the standard Hurst Exponent (Hurst et al. 1965). For positive +values of 𝑞, ℎ(𝑞) describes the scaling behavior of segments with +large fluctuations and for negative values of 𝑞, ℎ(𝑞) describes the +scaling behavior of segments with small fluctuations. For monofrac- +tal series, ℎ(𝑞) is independent of 𝑞, but for multifractal series ℎ(𝑞) +depends on 𝑞. The generalized Hurst exponent is directly related to +the Renyi exponent (Renyi 1976) 𝜏(𝑞) by +𝜏(𝑞) = 𝑞 ℎ(𝑞) − 1 . +(6) +Besides ℎ(𝑞), another way to characterize the multifractality of a +time series is by the singularity spectrum 𝑓 (𝛼), which is related to +𝜏(𝑞) via a Legendre transform, +𝛼 = 𝜏′(𝑞) +and +𝑓 (𝛼) = 𝑞 𝛼 − 𝜏(𝑞) , +(7) +where 𝛼 is the singularity exponent. This 𝑓 (𝛼)×𝛼 relation represents +the multifractal spectrum and has a concave parabolic shape. +From the multifractal spectrum, it is possible to obtain a set of +parameters to characterize each series: (i) the 𝛼 value where 𝑓 (𝛼) is +maximum, 𝛼0; (ii) the 𝛼 width, Δ𝛼 = 𝛼𝑚𝑎𝑥 −𝛼𝑚𝑖𝑛, where 𝛼𝑚𝑖𝑛 and +𝛼𝑚𝑎𝑥 are, respectively, the minimum and maximum values of 𝛼 that +mark the base of the concave parable in the multifractal spectrum (Δ𝛼 +is a measure of multifractal strength); (iii) the asymmetry parameter: +𝐴 = 𝛼𝑚𝑎𝑥 − 𝛼0 +𝛼0 − 𝛼𝑚𝑖𝑛 +, +(8) +where 𝐴 = 1 means the spectrum is symmetric, for 𝐴 > 1 the spec- +trum is right-skewed asymmetric, and for 𝐴 < 1 the spectrum is +left-skewed asymmetric (Shimizu et al. 2002; de Freitas et al. 2016). +A multifractal spectrum with a long right tail has a greater contri- +bution from small fluctuations. By contrast, a multifractal spectrum +with left asymmetry has a greater influence by local fluctuations with +large values (Ihlen 2012). +Another useful multifractal parameter can be extracted from the +𝜏(𝑞) × 𝑞 relation. As can be seen from Eq. (6), 𝜏(𝑞) has a linear +dependence with 𝑞 for monofractal series, where ℎ(𝑞) is constant. +In contrast, for multifractal series, this dependence is nonlinear. The +𝑞-dependency of the Renyi exponent can be quantified by the co- +efficient of determination, 𝑅2. 𝑅2 measures the proportion of the +variance for a dependent variable that is predictable by an inde- +pendent variable in a linear regression model (Barrett 1974). The +coefficient of determination is given by: +𝑅2 = 1 − +Í𝑛 +𝑖=1(𝜏𝑖 − b𝜏𝑖)2 +Í𝑛 +𝑖=1(𝜏𝑖 − ¯𝜏)2 , +(9) +where 𝜏𝑖 = 𝜏(𝑞𝑖) is the observed dependent variable, b𝜏𝑖 is the corre- +sponding predicted value and ¯𝜏 is the mean of the observed data. 𝑅2 +varies from 0 to 1, where in our case 1 represents a perfect fit to the +linear dependence model. In other words, the measure of 𝑅2 for the +𝜏(𝑞) × 𝑞 relation will be closer to 0 for multifractal series and closer +to 1 for monofractal series. +The MF-DFA method has best results if the time series are reason- +ably stationary, i.e., if they have a noise like structure. As suggested +by Eke et al. (2002), it is possible to determine if the time series have +noise like structure by computing a monofractal detrended fluctua- +tion analysis prior to conducting the MF-DFA analysis. Time series +are noise like if their Hurst exponent ℎ(2) is between 0 and 1, and they +are random walk like (nonstationary) if ℎ(2) is above 1. Ihlen (2012) +suggests that time series with ℎ(2) above 1.2 should be differentiated +before application of the MF-DFA analysis. +3 MULTIFRACTAL ANALYSIS OF SOLAR WIND DATA +We analyze solar wind magnetic field data detected with the Flux- +gate Magnetometer (FGM) onboard Cluster-1, with 22 Hz sampling +frequency. Two time series with 24 hours are investigated, one from +2008 March 9 and one from 2016 January 25. To reduce the com- +putational time of the analysis, the data length has been reduced by +using a decimation process. The low-pass Chebychev Type I infinite +impulse response filter was used with a reduction factor 𝑀 = 10, +order 8 and 0.8/𝑀 cut-off frequency. This decimation process is +described in Gomes et al. (2019). +After decimating the time series, we apply the MF-DFA method +with four input parameters: minimum scale 𝑠𝑖, maximum scale 𝑠 𝑓 , +order of fluctuation function 𝑞 and polynomial order 𝑚. The scale +refers to multiple segment sizes of the cumulative series and varies +from a minimum segment size 𝑠𝑖 to a maximum 𝑠 𝑓 . In this work, we +use 𝑠𝑖 = 10 and 𝑠 𝑓 = 𝑁, where 𝑁 is the length of the time series; +𝑞 varies between −20 and 20 with an increment of Δ𝑞 = 0.25, and +𝑚 = 3. This choice of parameters was supported by several tests. The +recommendation for large time series is to use a polynomial trend +order around 𝑚 = 3; 𝑠 𝑓 = 𝑁 was chosen to avoid deformations in the +shape of the multifractal spectra. Meanwhile, for the 𝑞 parameter the +use of values larger than 20 does not change the shape of the spectra +significantly. +3.1 MF-DFA analysis of the |𝐵| time series +Figure 1 shows the solar wind magnetic field time series studied +in this section for days 2008 March 9 and 2016 January 25. In the +upper panel, the time series for 2008 March 9 (red) and its first +order differencing (black) are shown. As it was explained in the +previous section, time-differencing is necessary in this case due to +the high nonstationarity of this series (ℎ(2) = 1.23). Throughout +the remaining of this section, only the differenced time series will +be used for March 9. This time series was characterized by Gomes +et al. (2019) as being permeated by large-scale current sheets. The +green regions in the original time series denote current sheets found +with Li’s method (Li 2008). The lower panel shows the time series +for 2016 January 25, which is characterized by a higher degree of +stationarity and the absence of current sheets (Gomes et al. 2019). +MNRAS 000, 1–12 (2022) + +4 +L. F. Gomes et al. +Due to its higher stationarity (ℎ(2) = 0.96), there is no need to +perform a differencing in this series. +Figure 2 shows different multifractal measures of the two magnetic +field time series. Figure 2(a) shows the multifractal spectra, which +reveal a left asymmetry for the March 09 time series (red) and a +right asymmetry for the January 25 series (blue). The left asymme- +try indicates the stronger contribution to multifractality coming from +large fluctuations associated with values of 𝑞 > 0 in the intermittent +time series of the current sheet-filled time series of March 09; the +right asymmetry found for the current sheet-free time series of Jan- +uary 25 points to the greater contribution of small fluctuations to the +multifractality (Ihlen 2012). The width of the spectrum can be used +as a measure of the degree of multifractality of the series (Shimizu +et al. 2002). Comparing both spectra, it can be seen that they have +almost the same width (Δ𝛼 ≈ 0.541 for March 9 and Δ𝛼 ≈ 0.555 +for January 25), which may be surprising, since the time series of +March 9 is visibly more intermittent, with strong bursts randomly +interspersed in time. In this case, the difference in multifractality can +be better quantified by the Renyi exponent 𝜏(𝑞), shown in Fig. 2(b). +It reveals a nonlinear behavior for both series, but with 𝑅2 ≈ 0.804 +for March 9 and 𝑅2 ≈ 0.986 for January 25, thus, March 9 displays +higher multifractality. +3.2 MF-DFA analysis of the volatility time series +In the previous section, the degree of multifractality, as provided by +the width of the multifractal spectra, could not properly distinguish +between the two time series under investigation, which is unexpected, +given that the original series are not only visually very different, but +one of them is known to be permeated by coherent structures (current +sheets) and the other is not. This is probably because although the +differenced time series of 2008 March 9 is apparently more intermit- +tent than the series of 2016 January 25, most of the abrupt changes +in |𝐵| caused by the current sheets in the March 9 series have a +small amplitude and, therefore, do not produce strong bursts in the +time-differenced series. Such abrupt changes in |𝐵| can be enhanced +by employing the volatility, thus providing a way to investigate the +role of current sheets in the multifractality. In the present section, we +employ the volatility to enhance the distinct features of each series +due to current sheets before repeating the MF-DFA analysis. +The magnetic volatility, vol𝑚𝑎𝑔, can be calculated from the stan- +dard deviations of the log magnetic return Δ𝑟mag(𝑡) in a moving +window of length 𝜔 along 𝑁 sample points (Tsay 2010) +Δ𝑟mag(𝑡) = log + |B(𝑡 + 𝜏)| +|B(𝑡)| + +, +(10) +volmag( 𝑗) = +v +u +u +t +1 +𝜔 − 1 +𝜔+ 𝑗−1 +∑︁ +𝑖=𝑗 +(Δ𝑟mag(𝑖) − 𝜇( 𝑗))2 , +(11) +where 𝜏 is a time-lag, 𝑗 = 1, . . . , 𝑁 − 𝜔 + 1 and 𝜇( 𝑗) is the mean +Δ𝑟mag inside the window (Gomes et al. 2019). Note that since Δ𝑟mag +involves computing a time difference with lag 𝜏, there is no need to +difference the original time series to remove nonstationarities prior +to computation of the volatility. The 𝜔 ant 𝜏 values are estimated +from the Power Spectrum Density (PSD). Figure 3(a) shows the +PSD for the March 9 time series, where the inertial range is the +blue region between the dashed lines. This region was chosen as the +frequency interval where the slope of the fitted line is -5/3, following +Kolmogorov’s K41 theory (Kolmogorov 1941) for fully developed +turbulence (Frisch 1995). The frequency in the middle of the inertial +range marks the scale used to define both 𝜏 and 𝜔. It is also the scale +used in Li’s method to detect the current sheets, shown in Fig. 1. In +this way, we define 𝜏 = 𝜔 = 50𝑠. Figure 3(b) shows the PSD for the +January 25 series. +Figure 4 exhibits the volatility time series for 2008 March 9 (up- +per panel, red) and for 2016 January 25 (lower panel, blue) from the +decimated magnetic field data. Recall that the upper series has many +current sheets while the lower one has none. Note that, unlike the +January 25 series, the March 9 volatility series has several extreme +events. Most of these high peaks are due to the abrupt changes in +the magnetic field that take place when the satellite crosses a current +sheet in the solar wind, as evidenced by the coincidence between +extreme events in the volatility and current sheets detected by Li’s +method (see Fig. 2(a),(b) in Gomes et al. (2019)). As a consequence, +the multifractal spectra obtained from the volatility of both series +are very different, as seen in Fig. 5(a). Now, the spectrum of the +intermittent time series of March 9 is much broader than the one +from January 25. The 𝛼−width is Δ𝛼 = 0.94134 for March 9 and +Δ𝛼 = 0.74921 for January 25. The volatility has enhanced the con- +tribution of the extreme events due to current sheets, thus showing +the signature of coherent structures present in the solar wind that +were partially hidden in the multifractal analysis of the original time +series. The Renyi exponents are shown in Fig. 5(b); once again, the +curve for March 9 is more concave than for January 25, reflecting its +higher level of multifractality. The coefficient of determination for +the Renyi exponents is 𝑅2 = 0.97464 for the volatility of March 9 +and 𝑅2 = 0.98125 for the volatility of January 25. It is clear that the +volatility has highlighted the role of current sheets in the multifractal +singularity spectrum. +4 MF-DFA OF SURROGATE TIME SERIES +According to Madanchi et al. (2017), there are two features in a +time series that can lead to its multifractality: (i) the presence of +heavy-tailed PDFs, as in highly intermittent series, and (ii) the exis- +tence of linear and non-linear correlations. In this section, we try to +identify the origin of the multifractality in the solar wind by means +of two surrogate time series derived from the original |𝐵| data. As +mentioned in the introduction, the shuffled time series is a random +permutation of the original time series in the real space that destroys +all temporal correlations, while keeping the same PDF for the am- +plitudes of |𝐵|. On the other hand, the random phases surrogate is +generated from the Fourier Transform of the original |𝐵| series. A +new Fourier series is generated by shuffling the phases of the Fourier +modes while keeping their power spectrum (Maiwald et al. 2008). +The inverse Fourier transform of this new frequency spectrum is the +random phases surrogate, which keeps the power spectrum and linear +autocorrelation of the original series, but has a Gaussian PDF and +breaks the nonlinear correlations. After generating these two surro- +gates, we repeat the multifractal analysis described in the previous +section; if the shuffled surrogate has a multifractal spectrum which +is considerably narrower than the spectrum of the original series, +it means that time correlations are an important source of multi- +fractality in the original time series. If the random phases surrogate +has a multifractal spectrum which is considerably narrower than the +spectrum of the original series, it means that fat-tailed PDFs and/or +nonlinear correlations are important for the multifractality. Note that +both kinds of multifractality mentioned above can be simultaneously +present in a time series (Norouzzadeh et al. 2007; Madanchi et al. +2017). If both the shuffled and random phases surrogates produce +MNRAS 000, 1–12 (2022) + +Origin of Multifractality in Solar Wind Turbulence +5 +[t] +01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 +-5 +0 +5 +10 +15 +20 +|B| (nT) +2008 March 9 +Current Sheets +Differencing |B| March 9 +01 +03 +05 +07 +09 +11 +13 +15 +17 +19 +21 +23 +Time (hours) +0 +5 +10 +15 +|B| (nT) +2016 January 25 +b +a +Figure 1. Solar wind time series of |𝐵| measured by Cluster-1. (a) For 2008 March 9 (red), containing current sheets (green), and its first order differencing +(black); (b) time series of |𝐵| for 2016 January 25 (blue), without current sheets. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +|B| 2008 March 9 +|B| 2016 January 25 +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-30 +-20 +-10 +0 +10 +20 +(q) +Renyi Exponent +|B| 2008 March 9 +|B| 2016 January 25 +b +Figure 2. (a) Multifractal spectrum of |𝐵| for 2008 March 9 (red), and 2016 +January 25 (blue). (b) Renyi exponents for 2008 March 9 (red), and 2016 +January 25 (blue). +monofractal spectra, then nonlinear correlations (but not fat-tailed +PDFs) are the source of multifractality. In the following subsections, +we perform this analysis for both the |𝐵| and volatility time series of +2008 March 9 and 2016 January 25. +4.1 Magnetic Field time series, 2008 March 9 +Figure 6 shows the differenced time series of |𝐵| for March 9 (red) +with its shuffled (green) and random phases (magenta) surrogates. +Clearly, the shuffled surrogate keeps the extreme events of the dif- +10 -4 +10 -3 +10 -2 +10 -1 +10 0 +f(Hz) +10 -2 +10 0 +10 2 +PSD(nt 2/Hz) +B 2008 March 9 PSD +|B| 2008 March 9 PSD +Regression Points +-5/3 +a +10 -4 +10 -3 +10 -2 +10 -1 +10 0 +f(Hz) +10 -4 +10 -2 +10 0 +10 2 +10 4 +PSD(nt 2/Hz) +B 2016 January 25 PSD +|B| 2016 January 25 PSD +Regression Points +-3/2 +Figure 3. Power spectral density for solar wind magnetic field of (a) 2008 +March 9, and (b) 2016 January 25. The blue region is the inertial range and +the red line is the linear fit for this interval, with a slope equal to -5/3 for +March 9 and slope -3/2 for January 25. +ferenced |𝐵| series, but the same events are absent from the random +phases surrogate. +Figure 7(a) displays the multifractal spectra for the March 9 orig- +inal and surrogate time series. For the shuffled spectrum (green) we +see a small reduction in the width when compared with the original +one (red). This means that there is a contribution from correlations to +multifractality, along with the contribution of the PDF. Considering +the random phases spectrum (magenta), its width reduces drastically +(the Δ𝛼 variation is about 0.32), which points to a significant con- +tribution to multifractality coming from a non-Gaussian PDF and/or +MNRAS 000, 1–12 (2022) + +6 +L. F. Gomes et al. +2 +4 +6 +8 +10 +12 +14 +16 +18 +10 4 +0 +0.4 +0.8 +1.2 +Volatility +Volatility 2008 March 9 +0 +2 +4 +6 +8 +10 +12 +Time (index) +10 4 +0 +0.2 +0.4 +Volatility +Volatility 2016 January 25 +b +a +Figure 4. Volatility of solar wind magnetic field time series for (a) 2008 +March 9, and (b) 2016 January 25. +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +V 2008 March 9 +V 2016 January 25 +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-40 +-30 +-20 +-10 +0 +10 +20 +(q) +Renyi Exponent +V 2008 March 9 +V 2016 January 25 +b +Figure 5. (a) Multifractal spectra for the volatility in 2008 March 9 (red), and +2016 January 25 (blue). (b) Renyi exponents for the volatility in 2008 March +9 (red), and 2016 January 25 (blue). +2 +4 +6 +8 +10 +12 +14 +16 +18 +10 4 +-3 +0 +3 +|B| (nT) +Differencing |B| 2008 March 09 +2 +4 +6 +8 +10 +12 +14 +16 +18 +10 4 +-3 +0 +3 +|B| (nT) +Shuffled +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Time (index) +10 4 +-0.4 +0 +0.4 +|B| (nT) +Random Phases +c +b +a +Figure 6. Differenced time series for 2008 March 9 (red) and the respective +surrogates: shuffled (green), and random phases (magenta). +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +|B| 2008 March 9 +Shuffled +Random Phases +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-15 +-10 +-5 +0 +5 +10 +(q) +Renyi Exponent +|B| 2008 March 9 +Shuffled +Random Phases +b +Figure 7. (a) Multifractal spectrum of |𝐵| for 2008 March 9 (red) and the +respective surrogates: shuffled (green), and random phases (magenta). (b) +Renyi exponents for 2008 March 9 (red) and the respective surrogates: shuffled +(green), and random phases (magenta). +nonlinear correlations. The conclusion from both spectra is that the +PDF has the strongest contribution to multifractality. The contribu- +tion of the PDF is due to the presence of strong intermittent bursts +(extreme events) in the March 9 time series. Since these bursts have +been shown to be related to large current sheets (see Gomes et al. +(2019)), the current sheets can be seen as the origin of most of the +multifractality in this time series. Figure 7(b) confirms this conclu- +sion by showing the Renyi exponent as a function of 𝑞, where the +random phases surrogate has a smaller concavity than the shuffled +surrogate. +4.2 Magnetic field time series, 2016 January 25 +Figure 8 shows the time series for January 25 (blue) with its shuffled +(green) and random phases (magenta) surrogates. Figure 9(a) shows +a significant width reduction in both surrogate spectra in comparison +with the original volatility spectrum (blue). The spectrum of the shuf- +fled series (green) has a width Δ𝛼 = 0.194, indicating a difference +of 0.36 with the spectrum of |𝐵|. Similarly, the spectrum for the ran- +dom phases series has a small width, about Δ𝛼 = 0.32, a difference +of 0.23 with the spectrum of |𝐵|. So, there is strong influence from +long-range correlations as well as non-gaussianity on the January 25 +magnetic field multifractality, but the contribution of the correlations +is preponderant, since the shuffled spectrum is considerably narrower +than the random phases spectrum. +4.3 Volatility time series, 2008 March 9 +We proceed with the analysis of the origin of the multifractality for +March 9 using the volatility, as shown in Fig. 10 for the original (red), +shuffled (green) and random phases (magenta) time series. The cor- +responding multifractal spectra in Fig. 11(a) show a wide parabola +MNRAS 000, 1–12 (2022) + +Origin of Multifractality in Solar Wind Turbulence +7 +[htpb] +2 +4 +6 +8 +10 +12 +10 4 +0 +5 +10 +15 +|B| (nT) +|B| 2016 January 25 +2 +4 +6 +8 +10 +12 +10 4 +0 +5 +10 +|B| (nT) +Shuffled +0 +2 +4 +6 +8 +10 +12 +Time (index) +10 4 +0 +4 +8 +12 +|B| (nT) +Random Phases +c +b +a +Figure 8. Time series for 2016 January 25 (blue) and the respective surrogates: +shuffled (green), and random phases (magenta). +0.4 +0.6 +0.8 +1 +1.2 +1.4 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +|B| 2016 January 25 +Shuffled +Random Phases +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-30 +-20 +-10 +0 +10 +20 +(q) +Renyi Exponent +|B| 2016 January 25 +Shuffled +Random Phases +b +Figure 9. (a) Multifractal spectrum for 2016 January 25 (blue) and the re- +spective surrogates: shuffled (green), and random phases (blue). (b) Renyi +exponents for 2016 January 25 (blue) and the respective surrogates: shuffled +(green), and random phases (magenta). +for the original volatility series (red) and two narrower parabolas +related to its shuffled (green) and random phases (magenta) series. +The random phases spectrum has a width of about Δ𝛼 = 0.39 and the +shuffled spectrum has a width of Δ𝛼 = 0.35. Since both spectra have +approximately the same width, it shows an important feature that was +not so clear from the multifractal spectra of the |𝐵| surrogate series +(Fig. 7), that is, the importance of the nonlinear correlations, which +play a key role, together with the PDF, in the origin of the multifrac- +tality for the March 9 series. Since the volatility is computed with a +lag-time of 𝜏 = 50𝑠, it is better suited for measuring the relevance +of long-range nonlinear correlations than the time-differenced |𝐵| +series. Figure 11(b) confirms that the shuffled and random phases +series have almost linear Renyi exponents, thus, the series are closer +to monofractal. +Figure 10. Time series Volatility for 2008 March 9 (red) and the respective +surrogates: shuffled (green), and random phases (blue). +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +V 2008 March 9 +Shuffled +Random Phases +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-40 +-30 +-20 +-10 +0 +10 +20 +30 +(q) +Renyi Exponent +V 2008 March 9 +Shuffled +Random Phases +b +Figure 11. (a) Multifractal spectrum for the volatility of 2008 March 9 (red) +and the respective surrogates: shuffled (green), and random phases (magenta). +(b) Renyi exponents for the volatility of 2008 March 9 (red) and the respective +surrogates: shuffled (green), and random phases (magenta). +4.4 Volatility time series, 2016 January 25 +Figure 12 shows the volatility time series of the January 25 time +series (blue) and its shuffled (green) and random phases (magenta) +surrogates. Figure 13(a) shows the corresponding multifractal spec- +tra. Once again, the reduction in the width for both surrogate spectra +means that a mutual contribution to multifractality coming from +long-range correlations and non-Gaussianity is present, with a clear +predominance of the long-range correlations effects, since the shuf- +fled spectrum is much narrower than the random phases spectrum. +A quantitative comparison of all the results for the |𝐵| time series +and volatility time series of March 9 and January 25 is provided by +Tables 1 to 3. Table 1 shows 𝑅2 for the Renyi exponent of |𝐵| and its +volatility for March 9 and January 25; Table 2 shows the width of the +multifractal spectra, Δ𝛼; Table 3 shows the asymmetry of the spectra, +𝐴. In general, all spectra for January 25 are right-asymmetric due to +the importance of small scale fluctuations; for March 9, some spectra +MNRAS 000, 1–12 (2022) + +Volatility Volatili +0.5 +0 +Random Phase +0.2 +0.2 +0 +2 +4 +6 +8 +10 +Time (index)12 +14 +16 +18 +X104Volatility '2008 Mar +7 +0.5 +0ch 9 +edD8 +L. F. Gomes et al. +2 +4 +6 +8 +10 +12 +10 4 +0 +0.2 +0.4 +0.6 +Volatility +Volatility 2016 January 25 +2 +4 +6 +8 +10 +12 +10 4 +0 +0.2 +0.4 +Volatility +Shuffled +0 +2 +4 +6 +8 +10 +12 +Time (index) +10 4 +0 +0.2 +0.4 +Volatility +Random Phases +a +b +c +Figure 12. Time series of the volatility for 2016 January 25 (blue) and the +respective surrogates: shuffled (green), and random phases (magenta). +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +Singularity Spectrum +V 2016 January 25 +Shuffled +Random Phases +a +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +q +-40 +-30 +-20 +-10 +0 +10 +20 +(q) +Renyi Exponent +V 2016 January 25 +Shuffled +Random Phases +b +Figure 13. (a) Multifractal spectrum for the volatility of 2016 January 25 +(blue) and the respective surrogates: shuffled (green), and random phases +(magenta). (b) Renyi exponents for the volatility of 2016 January 25 (blue) +and the respective surrogates: shuffled (green), and random phases (magenta). +Table 1. 𝑅2 of the Renyi exponent for magnetic field and volatilities of 2008 +March 9 and 2016 January 25 +March 9 +January 25 +|𝐵| +Volatility +|𝐵| +Volatility +Original +0.80413 +0.97464 +0.98597 +0.98125 +Shuffle +0.97505 +0.96748 +0.99537 +0.99573 +Random Phases +0.98185 +0.99637 +0.99601 +0.99424 +are left-asymmetric due to the importance of large-scale fluctuations, +but the random phases show right asymmetry, since in the random +phases surrogate the effects of non-Gaussian PDFs are destroyed. +Table 2. Width of 𝛼, Δ𝛼, for magnetic field and volatilities of 2008 March 9 +and 2016 January 25. +March 9 +January 25 +|𝐵| +Volatility +|𝐵| +Volatility +Original +0.54112 +0.94134 +0.55568 +0.74921 +Shuffle +0.36663 +0.40332 +0.19468 +0.19873 +Random Phases +0.21802 +0.39299 +0.32181 +0.43088 +Table 3. Spectrum Asymmetry, 𝐴, for magnetic field and volatilities of 2008 +March 9 and 2016 January 25. +March 9 +January 25 +|𝐵| +Volatility +|𝐵| +Volatility +Original +0.49873 +0.63885 +1.10709 +1.31215 +Shuffle +0.51279 +0.53342 +1.30854 +1.18283 +Random Phases +1.33583 +1.41817 +1.45999 +1.47002 +5 ZETA FUNCTION +Another function typically employed in multifractal analyses of time +series is the zeta function. Consider the structure function for |𝐵| +(Frisch 1995): +𝑆𝑝(𝜏) = ⟨[|𝐵(𝑡 + 𝜏)| − |𝐵(𝑡)|] 𝑝⟩ , +(12) +where ⟨·⟩ is the time average, 𝜏 is the time lag and 𝑝 are the statistical +moments for the time series of 𝐵. Assuming scale invariance inside +the inertial range, 𝑆𝑝 follows a power law +𝑆𝑝(𝜏) ∼ 𝜏𝜁 ( 𝑝) , +(13) +where 𝜁(·) is the zeta function or scaling exponent of the structure +function. So, 𝜁(𝑝) is obtained by the slope of the log 𝑆𝑝(𝜏) × log 𝜏 +plot. The importance of this parameter comes from Kolmogorov’s +K41 theory (Kolmogorov 1941) and the IK (Iroshnikov-Kraichnan) +theory (Iroshnikov 1964; Kraichnan 1965) of self-similarity and scale +invariance inside the inertial range for a homogeneous and isotropic +turbulence, where the 𝜁 function was shown to be a linear function +of 𝑝, with 𝜁(𝑝) = 𝑝/3 for K41 and 𝜁(𝑝) = 𝑝/4 for IK. +In Fig. 14(a), the linear K41 theoretical zeta scaling exponent +function is shown by the black dashed line while the IK scaling ex- +ponent is denoted by a dotted line. The top panel (a) also shows the +zeta scaling exponent computed from the time series of |𝐵| for the +intermittent series of March 09 (red line with circles) and for the +current sheet-free series of January 25 (blue line with diamonds). +The zeta function for the March 09 series clearly departs from the +linear behavior, as expected for multifractal intermittent series, but, +surprisingly, the zeta function exhibits an almost linear relation with +𝑝 in the case of January 25, despite the fact that both series have +multifractal spectra with similar widths (see Fig. 2(a)). Thus, one +should be cautious before using the behavior of the scaling exponent +as a definite measure of multifractality, although it is a good mea- +sure of intermittency. To confirm this result, Fig. 14(b) compares +the zeta scaling exponents of the March 09 |𝐵| series (red line with +circles) with the zeta scaling exponents of its random phases series +(magenta line with triangles). Since the random phases series has a +Gaussian PDF, it removes from the original series the intermittent +extreme events responsible for the fat-tailed PDF and the zeta scaling +exponent becomes linear, following the K41 line. This result con- +MNRAS 000, 1–12 (2022) + +Origin of Multifractality in Solar Wind Turbulence +9 +1 +2 +3 +4 +5 +6 +7 +p +0 +0.5 +1 +1.5 +2 +2.5 +(p) +Zeta Function +2008 March 09 +2016 January 25 +K41 +IK +a +Figure 14. (a) Zeta functions for the magnetic field time series for 2008 March +9 (red circles) and 2016 January 25 (blue diamonds). (b) Zeta functions for +|𝐵| 2008 March 9 (red circles) and its Random Phases (magenta triangles). +(c) Zeta functions for |𝐵| 2016 January 25 (blue diamonds) and its Random +Phases (magenta triangles). The dashed lines represent the K41 scaling and +the dotted lines, the IK scaling. +firms the importance of the contribution from a fat-tailed PDF to the +multifractality of the March 09 series. In Fig. 14(c), the same analy- +sis is done for the January 25 series, where both the original series +and its random phases show an IK linear behavior, since none of +the series has fat-tailed PDF, although they have multifractal spectra +(see the blue and magenta spectra in Fig. 9(a)). We conclude from +this that the 𝜁-function is a good measure of multifractality due to +PDF, but misses the contribution of long-range correlations to the +multifractality. +6 𝑃−MODEL +In section 4, we showed that the multifractal spectra of the volatil- +ity of the solar wind are predominantly due to nonlinear and linear +correlations in the time series of January 25 and due to PDF and +nonlinear correlations for the March 9 time series. The presence of +long-range nonlinear correlations in both series is the signature of a +nonlinear dynamical system (possibly with some stochastic compo- +nent) governing the behavior of both series. In the present section, we +employ the 𝑝−model (Halsey et al. 1986; Meneveau & Sreenivasan +1987) to show that both the correlations and the extreme events men- +tioned above are actually a consequence of turbulent energy-cascade +processes with different scaling laws that depend on the presence or +absence of current sheets in the original time series. +The 𝑝−model is a model for non-homogeneous energy-cascading +process in the inertial range of fully-developed turbulence based on +the generalized Cantor set. Consider that the flux of kinetic energy +from eddies of size 𝐿 to smaller eddies is represented by a dissipa- +tion 𝐸𝐿. In the one-dimensional version of the 𝑝−model, 𝐿 is the +length of an interval. Suppose that an eddy of size 𝐿 is unequally di- +0.8 +1 +1.2 +1.4 +1.6 +1.8 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +f( +) +V 2008 March 9 +P Model +0.8 +1 +1.2 +1.4 +1.6 +1.8 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +Singularity Spectrum +V 2016 January 25 +P Model +Figure 15. Left: Multifractal spectrum for the volatility of 2008 March 9 (red +circle) and its 𝑝−model fit (black line with dots). Right: Multifractal spectrum +for the volatility of 2016 January 25 (blue diamond) and its 𝑝−model fit (black +line with dots). +vided into two smaller eddies (i.e., two sub-intervals) of sizes 𝑙1𝐿 and +𝑙2𝐿, where 0 < 𝑙1 < 𝑙2 < 1 are the size factors, with the energy flux +𝐸𝐿 being distributed unto these sub-eddies with different probabili- +ties 𝑝1 and 𝑝2, i.e., the new dissipation values are 𝑝1 𝐸𝐿 and 𝑝2 𝐸𝐿. +In practice, one can start the process with 𝐿 = 𝐸𝐿 = 1. Then, each +new eddy is further sub-divided into two smaller eddies with the same +size factors 𝑙1 and 𝑙2 and probabilities 𝑝1 and 𝑝2. This process may +be repeated until the sub-intervals reach the Kolmogorov dissipation +scale. At each cascading step 𝑛, there will be + 𝑛 +𝑚 + +segments with +length 𝑙𝑚 +1 𝑙𝑛−𝑚 +2 +𝐿 and dissipation 𝑝𝑚 +1 𝑝𝑛−𝑚 +2 +𝐸𝐿, for 𝑚 = 0, 1, . . . , 𝑛. +As shown by Halsey et al. (1986) for the general two-scale Cantor +set, it is possible to obtain the analytic expressions for the singularity +exponent 𝛼 and the singularity spectrum 𝑓 as +𝛼 = ln 𝑝1 + (𝑛/𝑚 − 1) ln 𝑝2 +ln 𝑙1 + (𝑛/𝑚 − 1) ln 𝑙2 +, +(14) +𝑓 = (𝑛/𝑚 − 1) ln(𝑛/𝑚 − 1) − (𝑛/𝑚) ln(𝑛/𝑚) +ln 𝑙1 + (𝑛/𝑚 − 1) ln 𝑙2 +. +(15) +For each 𝑛 and given values of 𝑙1, 𝑙2, 𝑝1 and 𝑝2, the variation of +𝑚 will provide the different values of 𝛼 and 𝑓 for the singularity +spectrum. Since 0 ≤ 𝑚 ≤ 𝑛 and 𝑚 is an integer, larger values +of 𝑛 provide a better definition of the spectrum. For a cascading +process with direct energy dissipation in the inertial range, we have +𝑝1 + 𝑝2 < 1 (Meneveau & Sreenivasan 1987). This means that a new +𝑑𝑝 dissipation parameter must be included, where 𝑑𝑝 = 1 − 𝑝1 − 𝑝2. +Thus, we define 𝑝2 = 1 − 𝑝1 − 𝑑𝑝, as well as 𝑙2 = 1 − 𝑙1, in Eqs. (14) +and (15). +Figure 15 shows the MF-DFA multifractal spectra for the volatil- +ity series of March 9 (red circles) and January 25 (blue diamonds). +The 𝑝−model fits obtained from Eqs. (14) and (15) are also shown +(black line with dots). The values of 𝑝1, 𝑑𝑝 and 𝑙1 were obtained +with a Monte Carlo method that minimized the mean squared error +between the original and fitted spectra. For March 9th, we obtained +𝑝1 = 0.71, 𝑑𝑝 = 0.17 and 𝑙1 = 0.68. For January 25th, we obtained +𝑝1 = 0.51, 𝑑𝑝 = 0.11 and 𝑙1 = 0.66. The agreement between the +observational and theoretical curves confirms that the solar wind +multifractal spectra can be obtained from a turbulence cascade pro- +cess. This is a remarkable result, since the 𝑝−model was specifically +elaborated to represent turbulent cascade processes, and will usually +not be able to approximate the spectra of other processes. +Next, we compare the turbulent time series behind the 𝑝−model +spectra with the observational solar wind volatility time series in +MNRAS 000, 1–12 (2022) + +1.5 +(d) +1 +IBl 2008 March 09 +0.5 +Random Phases +-K41 +...IK. +0 +1 +2 +3 +4 +5 +6 +1 +2 +p1.5 +7 +IBl 2016 January 25 +0.5 +Random Phases +- -K41 +0 +3 +4 +5 +6 +7 +pZeta Function +b +c +2210 +L. F. Gomes et al. +terms of their PSDs. To obtain the 𝑝−model PSDs, we use the prob- +abilities and size factors previously obtained with the Monte Carlo +method. By iterating the generalized two-scale Cantor set model, we +produce two 𝑝−model time series. Figure 16 shows a comparison of +the solar wind volatility time series with the 𝑝−model time series. +The two upper panels depict the solar wind series for March 9 (a) +and the corresponding 𝑝−model (b); the two lower panels depict the +solar wind series for January 25 (a) and the corresponding 𝑝−model +(b). The qualitative similarity between observational and 𝑝−model +time series is apparent in both cases. +A comparison of observed and simulated PSDs is shown in Fig. +17. Figure 17(upper panels) shows the PSDs for the volatility time +series of 2008 March 9 (left) and 2016 January 25 (right). The +blue region between the vertical dashed lines represents the inertial +range and the red line is the linear regression with slope −5/3 for +the March 9 series and −3/2 for the January 25 series. Thus, the +highly intermittent series of March 9 (with current sheets) exhibits a +K41 scaling, whereas the January 25 series (without current sheets) +shows an IK scaling. This fact had been previously established by +Li et al. (2011) and confirmed by Gomes et al. (2019) using PSDs +computed from the time series of |𝐵|. The PSDs computed from the +𝑝−model time series are shown in Fig. 17(lower panels), and they +reveal K41 scaling for the March 9 series and IK scaling for the +January 25 series, just like in the original solar wind series. Note that +in both cases the inertial range can be extended almost throughout +the whole PSDs shown, since our 𝑝-model has small dissipation. +We conclude that a K41 intermittent turbulence cascade is behind +the multifractality of the current sheet-filled time series of March +9 and an IK turbulence cascade is the origin of the multifractality +of the January 25 series. This result is consistent with other time +series analysed by us, that show that current sheets are responsible +for the K41 turbulence multifractality and the absence of current +sheets results in an IK turbulence multifractality in the solar wind +(see Table 4 in Gomes et al. (2019)). +7 CONCLUSIONS +We have presented a new methodology for multifractal analysis of +solar wind magnetic field data, based on MF-DFA, volatility and +surrogate time series. The MF-DFA provides a standard way to gen- +erate the singularity spectrum and the Renyi exponent; the volatility +enhances the extreme events, stressing the differences between series +with current sheets and series without current sheets; the surrogate +time series provide a way to infer the origin of multifractality. Addi- +tionally, the 𝑝-model was used to reproduce the multifractal behavior +of the solar wind series, indicating that a nonlinear turbulence en- +ergy cascade dynamical system is behind the observed dynamics. A +similar framework for multifractal analysis, but without the volatility +and the 𝑝-model, was used by Chattopadhyay et al. (2018) in the +analysis of CME linear speed data in the solar wind. In order to keep +the paper reasonably short, we have limited our presentation to only +two time-series, but we have tested our techniques in other series +and found that the conclusions presented are robust. An example of +analysis with two other time series is included in the supplementary +material (online). Further exploration of the methodology is left for +future works. +Just like in Gomes et al. (2019), we found the volatility to be very +useful to highlight the role of current sheets. In our case, they in- +crease the signature of multifractality due to PDF in the singularity +spectra. The surrogate analysis of both original and volatility series +shows that for time series with current sheets, multifractality is due to +Figure 16. (a) Volatility time series for 2008 March 9 (red) and (b) generated +𝑝−model time series (black) by 10𝑡ℎ interation. (c) Volatility time series for +2016 January 25 (blue) and (d) generated 𝑝−model time series (black) by +15𝑡ℎ interation. +Figure 17. (a) Left: power spectral density for 2008 March 9 volatility. (a) +Right: power spectral density for 2016 January 25 volatility. (b) Left: power +spectral density for generated 𝑝−model time series from 2008 March 9 volatil- +ity. (b) Right: power spectral density for generated 𝑝−model time series from +2016 January 25 volatility. The blue regions mark the inertial range and the +red lines are the linear fits for those intervals. +MNRAS 000, 1–12 (2022) + +Volatility 2008 March 9 +Vol +0.4 +0 +b +P Model (1oth Generation) +@ 0.8 +.0.4 +0.5 +1 +1.5 +2 +2.5 +3 +Time (index) +×104Volatility 2016 January 25 +Volatility +P Model (1i5th Generation) +Amplitude +0.5 +1.5 +2 +2.5 +w +Time (index) +×104Volatility 2008 March 9 PSD +Volatility 2016 January 25 PSD +PSD +TTTTT +PSD +1 +Inertial +Inertial +100 +Range +Range +Slope=-5/3 +Slope=-3/2 +1100 +I +N +10 +L +4 +- +4 +110-4 +1 +1 +1 +1 +10-6 +- +1 +- +Inertial +1 +Inertial +10-6 +Range +Range +10-8 +1 +I +1 +1 +10-8 +HHLL +- +1 +- +10-4 +10-2 +100 +10-2 +100 +f(Hz)P-model V 2008 March 9 PsD +P-model V 2016 ianuary 25 PSD +102 +102 +PSD +PSD +Inertial +Inertial +Range +Range +101 +101 +Slope = -5/3 +Slope = -3/2 +100 +10-1 +10 +10-2 +P10-2 +10-3 +Inertial +10-3 +Range +Inertial +10-4 +Range +10-4 ( +10-3 +10-2 +10-1 +100 +10-3 +10-2 +10-1 +100 +f(Hz)Origin of Multifractality in Solar Wind Turbulence +11 +both intermittency and nonlinear correlations; for time series with- +out current sheets, it is predominantly produced by the long-range +correlations. The 𝑝−model analysis reveals that those are mainly +nonlinear correlations, since the process behind the statistics is a +nonlinear turbulent energy cascade. So, turbulence is the common +source of the multifractality, but current sheets are the source of the +left asymmetry of the singularity spectrum, as well as the nonlinear +scaling exponent for the structure functions. In the absence of current +sheets, the small-amplitude fluctuations are the main source of the +right asymmetry of the singularity spectrum. It is important to stress +that despite being a multifractal process, the current sheet-free series +exhibits an almost linear scaling exponent for the structure functions, +which is sometimes confused with a monofractal process in the liter- +ature. Our results indicate that the Renyi exponent is more sensitive to +multifractality due to correlations than the structure function scaling +exponent (zeta function). +In dealing with separate cases where the presence or absence +of current sheets is considered, we are attacking one of the “nine +outstanding questions of solar wind physics", related by Vaill & +Borovsky (2020), namely, the origin and evolution of the mesoscale +(timescales in the range of minutes up to a few hours) plasma and +magnetic-field structure of the solar wind. These current sheets have +been associated with the border between adjacent flux tubes (Bruno +2019), while also being related to nonlinear turbulent interactions +rather than the presence of advected pre-existing flux-tube structures +(Bowen et al. 2018). In the present work, we do not focus on the origin +of those coherent structures, but measure their weight on the statis- +tics of solar wind fluctuations. We do this not only through Fourier +spectral indices and the scaling of structure functions, as in Salem +et al. (2009), but their contribution to multifractality is explored in +depth through the MF-DFA, volatility and surrogate techniques. As +we said, our results reveal that although the scaling of the structure +functions may be almost linear for series without current sheets, the +singularity spectra may still display broad parabolas, the signature +of highly multifractal signals. Thus, the scaling exponent of struc- +ture functions is adequate to measure multifractality due to PDFs, +but not for multifractality due to long-range correlations, where the +Renyi exponent and singularity spectra should be adopted. Multifrac- +tal series with nearly linear behavior of the scaling exponents were +also reported in Tam et al. (2010) (see their Fig. 4), where the rank- +order multifractal analysis (ROMA) is employed in the description +of auroral zone electric-field fluctuations. +In conclusion, the basic question related to mesoscale plasma tur- +bulence in the solar wind is not whether it is monofractal or mul- +tifractal, but if the source of the ubiquitous multifractality is the +PDF or the long-range correlations. The short answer is that in the +presence of current sheets, the PDF has a strong contribution for +multifractality, but in their absence, it is mainly due to correlations. +It would be interesting to check if the monoscaling of the structure +functions reported in previous solar wind time series, as in Kiyani +et al. (2009, 2013) and Bruno (2019) for turbulence at kinetic scales, +indeed reveal monofractality or if they indicate, in fact, multifractal +series due to correlations and not due to intermittency. +ACKNOWLEDGEMENTS +L.F.G. acknowledges Brazilian agency CAPES for the financial +support; E.L.R. acknowledges Brazilian agencies CAPES (grant +88887.309065/2018-01) and CNPq (Grant 306920/2020-4) for their +financial support, as well as FCT—Fundação para a Ciência e a +Tecnologia (Portugal); S.G. was partially supported by (i) CMUP, +member of LASI, which is financed by national funds through FCT +– Fundação para a Ciência e a Tecnologia, I.P., under the project +with reference UIDB/00144/2020, and (ii) project SNAP NORTE- +01-0145-FEDER-000085, financed by ERDF through NORTE2020 +under Portugal 2020 Partnership Agreement. +DATA AVAILABILITY +The data used for this analysis can be obtained from European Space +Agency (ESA) at the Cluster Science Archive: https://csa.esac. +esa.int/csa-web/ (last access: 2 December 2020, ITA, 2020). +REFERENCES +Barrett J. 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B., et al., 2016, The Astrophysical Journal, 831, 87 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–12 (2022) + diff --git a/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/load_file.txt b/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9761fd3ce5dbcbaaa5ff00e03bad020ab6eab24 --- /dev/null +++ b/tNA0T4oBgHgl3EQfLf8y/content/tmp_files/load_file.txt @@ -0,0 +1,982 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf,len=981 +page_content='MNRAS 000, 1–12 (2022) Preprint 2 December 2022 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='0 Origin of Multifractality in Solar Wind Turbulence: the Role of Current Sheets Leonardo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes,1★ Tiago F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes,1 Erico L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Rempel1,2 and Sílvio Gama3 1Aeronautics Institute of Technology (ITA), 12228-900, São José dos Campos, SP, Brazil 2National Institute for Space Research (INPE), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Box 515, 12227-010, São José dos Campos, SP, Brazil 3Mathematics Center of the Porto University (CMUP), Mathematics Department, Faculty of Sciences, University of Porto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Campo Alegre s/n, 4169-007 Porto, Portugal 2 December 2022 ABSTRACT In this work, a multifractal framework is proposed to investigate the effects of current sheets in solar wind turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' By using multifractal detrended fluctuation analysis coupled with surrogate methods and volatility, two solar wind magnetic field time series are investigated, one with current sheets and one without current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Despite the lack of extreme-events intermittent bursts in the current sheet-free series, both series are shown to be strongly multifractal, although the current sheet-free series displays an almost linear behavior for the scaling exponent of structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Long-range correlations are shown to be the main source of multifractality for the series without current sheets, while a combination of heavy-tail distribution and nonlinear correlations are responsible for multifractality in the series with current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The multifractality in both time series is formally shown to be associated with an energy-cascade process using the 𝑝-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Key words: multifractals – turbulence – data analysis – statistical – solar wind 1 INTRODUCTION Fractals have been widely employed in nonlinear analysis along the past decades as a form of representing the complex topological struc- tures produced by dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' These topological structures are subsets of the phase space that may represent chaotic attractors, stable or unstable manifolds, boundaries between basins of attrac- tion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, when dynamical systems are investigated through nonlinear time series analysis, the fractal indices computed from the time series somehow represent the complexity of the structure of an underlying set on which the solution lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Additionally, the dy- namical structure could be represented either by a monofractal or a multifractal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A monofractal process has a scaling law for a fluctuation function which is a linear function of statistical moments with a single scaling exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A multifractal process has a power- law scaling which is a nonlinear function of statistical moments with a range of scaling exponents (Salat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A monofractal scal- ing is to be expected from dynamical processes behind perfectly self-similar fractal sets, like deterministically generated Cantor sets (Cantor 1883), or even from white noise time series (Ihlen 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' multifractals, on the other hand, are observed in inhomogeneous sys- tems, such as strongly intermittent turbulence, where the presence of strong fluctuations related to coherent structures localized in space generate a departure from Gaussianity in probability distribution functions (PDFs) of small-scale structure functions (Carbone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2004), as seen in several analyses of observational magnetohydro- dynamic data (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', Marsch & Tu (1998), Burlaga (2001), and ★ E-mail: leofgb@ita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='br Bruno (2019) for reviews on turbulence, intermittency and multifrac- tal scalings in the solar wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A series of recent works have confirmed the complex and multi- fractal nature of solar wind fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2004) studied the origin of complexity in space plasmas using MHD simulations, dynamic renormalization group and wavelet analysis, arguing that the turbulent plasmas in the solar wind and auroral regions are dom- inated by a combination of propagating modes and nonpropagating intermittent nonlinear structures, whose interactions with charged particles may lead to the energization of plasma populations such as auroral ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Macek (2007) employed Voyager magnetic field data in the outer heliosphere and Helios plasma data in the inner helio- sphere to show that multifractal spectra of intermittent solar wind fluctuations are consistent with that of the generalized two-scale weighted Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Bolzan & Rosa (2012) analyzed magnetic field data from the ACE satellite and conjectured that the presence of large scale coherent structures during coronal mass ejections (CME) decreases the multifractality, when compared with periods after the CME events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Wavelet-leader multifractal analysis of magnetospheric dissipations, as measured by the AL index, reveal that the magne- tosphere is a multi-scale, complex, turbulent system, driven into a non-equilibrium self-organized state, which may explain the obser- vations of repeatable and coherent substorm phenomena with under- lying complex multifractal behavior in the plasma sheet (Valdivia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The interaction of the solar wind with the Earth’s mag- netosphere also contributes for multifractality in measurements of the geomagnetic activity, such as the geomagnetic induced current (Wirsing & Mili 2020) and the Dst index (Ogunjo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2021), al- though internal sources of multifractality must also be considered, as Gopinath (2016) suggests that multifractality of the auroral electrojet © 2022 The Authors 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' index is fairly independent of the solar activity cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Wawrzaszek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019) characterized multifractality in intermittent turbulence of heliospheric magnetic field fluctuations from Ulysses spacecraft, concluding that intermittency/multifractality decreases with helio- spheric distance, a result that was confirmed by Kiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Recent analysis of electron density fluctuations in the E-F valley region of the ionosphere performed with the multifractal detrended fluctuation analysis (MF-DFA) method show that irregularities are multifractal, asymmetric, intermittent and non–homogeneous (Nee- lakshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The direct link between intermittency and multifractality of mag- netic and velocity field fluctuations in the solar wind was made clear in Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Using data from the Wind spacecraft, they applied the Haar wavelet transform to filter out intermittency from the time series and showed that the scaling exponents for the struc- ture functions behave as a linear function of statistical moments, as in monofractal processes, therefore attributing multifractality in the solar wind to intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019) obtained a similar linear scaling after filtering out the current sheets from Cluster-1 intermittent magnetic field data, suggesting that the current sheets are the coherent structures responsible for the nonlinear scaling of the structure functions in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This was confirmed after inspection of time series of days when current sheets were absent, that also showed a linear scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A question remained on whether the linear scalings found by Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2009) and Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019) indeed imply that the filtered time series are monofractal or not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', is the nonlinearity of the distribution of scaling exponents of structure functions a general measure of multifractality or is it just an indication of intermittency, one among different possible sources of multifractality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' One of the goals of the current work is to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In this sense, it is important to stress that the origin of multifractality is not always related to fat-tailed PDFs, as it may also be caused by different correlations in small and large fluctuations, such as linear or nonlinear correlations (Kantelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The source of multifractality can be investigated by producing surrogates from the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Two types of surrogates are useful in this context (Theiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Lancaster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' First, shuffling the amplitudes of the original signal breaks all long-range correlations, while keeping the PDF unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Therefore, if the multifractality is due to fat-tailed PDFs, it cannot be removed by shuffling the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If it is due, solely, to time correlations, the corresponding shuffled series will be monofractal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If both fat-tailed PDF and linear/nonlinear correlations are present, the multifractality of the shuffled series should be smaller than that of the original series (Barunik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The second type of surrogate is produced by randomizing the phases of the Fourier modes of the original time series, producing a new series with Gaussian PDF, but preserving the linear correlations of the original series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If the random phases time series becomes monofractal, then nonlinear correlations and/or non-Gaussian PDFs are the source of multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If the multifractality is preserved in the random phases time series, then linear correlations are its source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Studies of surrogate time series have been conducted to probe the origin of multifractality in a wide range of contexts, including financial markets (Barunik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2012), human gate diseases (Dutta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2013), near-fault earthquake ground motions (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2015), solar irradiance fluctuations (Madanchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2017), air pollutants (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2017), meteorological time series of air pressure, air temperature and wind speed (Gos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2021) and rainfall records (Sarker & Mali 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The surrogate method was also employed in time series of CME linear speed during solar cycle 23 to conclude that the multifractality is due to both the broad PDF and long range time correlations (Chattopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the present paper, we use the method to reveal the role of current sheets in the origin of multifractality in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' By analyzing two qualitatively different magnetic field time series from Cluster-1, one filled with current sheets and another one void of current sheets, we develop a nonlinear methodology based on the MF-DFA method coupled with the volatility and surrogate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, the contribution of small- and large-scale magnetic fluctuations can be quantified in different types of multifractal solar wind series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It is revealed that when the multifractality is not mainly due to the PDF, the scaling exponents display an almost linear behavior as a function of the moments of the structure function, despite the presence of strong multifractality in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In addition, we employ the 𝑝-model (Halsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Meneveau & Sreenivasan 1987) to confirm that the multifractality in both types of solar wind time series can be attributed to a turbulent energy cascade process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In section II, the MF-DFA methodology is briefly described;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' in section III, the multifractal anal- ysis of two solar wind time series is conducted, including their volatil- ity time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' section IV analyses the surrogate of the original and volatility time series, to determine if the source of the multifractality in the solar wind is due to PDF or correlations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' section V presents the scaling exponent analysis of the original and surrogate times se- ries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' section VI describes the 𝑝-model analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Finally, section VII presents the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2 MF-DFA The MF-DFA method is a generalization of the detrended fluctu- ation analysis (DFA) method for quantifying long-range correla- tions in non-stationary time series (Kantelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The method identifies the scaling of 𝑞th-order moments of the time se- ries (Norouzzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The MF-DFA method consists of five steps: (i) The time series 𝑥𝑘 (𝑘 = 1, 2, · · · , 𝑁) is integrated: 𝑌 (𝑖) = 𝑖∑︁ 𝑘=1 [𝑥𝑘 − ⟨𝑥⟩] , 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', 𝑁 (1) where ⟨𝑥⟩ is the average value of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (ii) The series𝑌 (𝑖) is divided into 𝑁𝑠 ≡ int(𝑁/𝑠) non-overlapping segments with equal lengths 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since 𝑁 is usually not a multiple of 𝑠, some of the data points in the time series may be left out of the last segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' To fix this, the procedure is repeated starting from the opposite end of the time series and going backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Consequently, 2𝑁𝑠 segments are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (iii) The local trend for each 2𝑁𝑠 segments is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Then the variance is given by 𝐹2(𝑠, 𝜈) = 1 𝑠 𝑠 ∑︁ 𝑖=1 {𝑌 [(𝜈 − 1) 𝑠 + 𝑖] − 𝑦𝜈(𝑖)}2 , (2) for each segment indexed by 𝜈 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' , 𝑁𝑠 and 𝐹2(𝑠, 𝜈) = 1 𝑠 𝑠 ∑︁ 𝑖=1 {𝑌 [𝑁 − (𝜈 − 𝑁𝑠) 𝑠 + 𝑖] − 𝑦𝜈(𝑖)}2 (3) for 𝜈 = 𝑁𝑠 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' , 2𝑁𝑠 , where 𝑦𝜈 is the 𝑚-th degree fitting poly- nomial of each segment 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This polynomial detrending of order 𝑚 in MNRAS 000, 1–12 (2022) Origin of Multifractality in Solar Wind Turbulence 3 the 𝑌 profile eliminates trends up to order 𝑚 − 1 in the original time series and specifies the type of MF-DFA applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (iv) The average over all segments is calculated to obtain the 𝑞th- order fluctuation function: 𝐹𝑞(𝑠) = ( 1 2𝑁𝑠 2𝑁𝑠 ∑︁ 𝜈=1 [𝐹2(𝑠, 𝜈)] 𝑞 2 ) 1 𝑞 , (4) where, in general, the 𝑞 parameter can take any real value except zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For 𝑞 = 2, the equation returns the DFA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Steps 2 to 4 are repeated for different time scales 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (v) The scaling behavior of the fluctuation function is defined by the log-log plot of 𝐹𝑞(𝑠) ×𝑠 for each value of 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If 𝑥𝑖 have long-range correlations, for large values of 𝑠, 𝐹𝑞(𝑠) increases as a power-law, 𝐹𝑞(𝑠) ∼ 𝑠ℎ(𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (5) The scaling exponents ℎ(𝑞) are the generalized Hurst exponents, defined as the slope of the log 𝐹𝑞(𝑠) × log(𝑠) graph, where for ℎ(2) we have the standard Hurst Exponent (Hurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For positive values of 𝑞, ℎ(𝑞) describes the scaling behavior of segments with large fluctuations and for negative values of 𝑞, ℎ(𝑞) describes the scaling behavior of segments with small fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For monofrac- tal series, ℎ(𝑞) is independent of 𝑞, but for multifractal series ℎ(𝑞) depends on 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The generalized Hurst exponent is directly related to the Renyi exponent (Renyi 1976) 𝜏(𝑞) by 𝜏(𝑞) = 𝑞 ℎ(𝑞) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (6) Besides ℎ(𝑞), another way to characterize the multifractality of a time series is by the singularity spectrum 𝑓 (𝛼), which is related to 𝜏(𝑞) via a Legendre transform, 𝛼 = 𝜏′(𝑞) and 𝑓 (𝛼) = 𝑞 𝛼 − 𝜏(𝑞) , (7) where 𝛼 is the singularity exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This 𝑓 (𝛼)×𝛼 relation represents the multifractal spectrum and has a concave parabolic shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' From the multifractal spectrum, it is possible to obtain a set of parameters to characterize each series: (i) the 𝛼 value where 𝑓 (𝛼) is maximum, 𝛼0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (ii) the 𝛼 width, Δ𝛼 = 𝛼𝑚𝑎𝑥 −𝛼𝑚𝑖𝑛, where 𝛼𝑚𝑖𝑛 and 𝛼𝑚𝑎𝑥 are, respectively, the minimum and maximum values of 𝛼 that mark the base of the concave parable in the multifractal spectrum (Δ𝛼 is a measure of multifractal strength);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (iii) the asymmetry parameter: 𝐴 = 𝛼𝑚𝑎𝑥 − 𝛼0 𝛼0 − 𝛼𝑚𝑖𝑛 , (8) where 𝐴 = 1 means the spectrum is symmetric, for 𝐴 > 1 the spec- trum is right-skewed asymmetric, and for 𝐴 < 1 the spectrum is left-skewed asymmetric (Shimizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' de Freitas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A multifractal spectrum with a long right tail has a greater contri- bution from small fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' By contrast, a multifractal spectrum with left asymmetry has a greater influence by local fluctuations with large values (Ihlen 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Another useful multifractal parameter can be extracted from the 𝜏(𝑞) × 𝑞 relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (6), 𝜏(𝑞) has a linear dependence with 𝑞 for monofractal series, where ℎ(𝑞) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In contrast, for multifractal series, this dependence is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝑞-dependency of the Renyi exponent can be quantified by the co- efficient of determination, 𝑅2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 𝑅2 measures the proportion of the variance for a dependent variable that is predictable by an inde- pendent variable in a linear regression model (Barrett 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The coefficient of determination is given by: 𝑅2 = 1 − Í𝑛 𝑖=1(𝜏𝑖 − b𝜏𝑖)2 Í𝑛 𝑖=1(𝜏𝑖 − ¯𝜏)2 , (9) where 𝜏𝑖 = 𝜏(𝑞𝑖) is the observed dependent variable, b𝜏𝑖 is the corre- sponding predicted value and ¯𝜏 is the mean of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 𝑅2 varies from 0 to 1, where in our case 1 represents a perfect fit to the linear dependence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In other words, the measure of 𝑅2 for the 𝜏(𝑞) × 𝑞 relation will be closer to 0 for multifractal series and closer to 1 for monofractal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The MF-DFA method has best results if the time series are reason- ably stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', if they have a noise like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As suggested by Eke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2002), it is possible to determine if the time series have noise like structure by computing a monofractal detrended fluctua- tion analysis prior to conducting the MF-DFA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Time series are noise like if their Hurst exponent ℎ(2) is between 0 and 1, and they are random walk like (nonstationary) if ℎ(2) is above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Ihlen (2012) suggests that time series with ℎ(2) above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 should be differentiated before application of the MF-DFA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 3 MULTIFRACTAL ANALYSIS OF SOLAR WIND DATA We analyze solar wind magnetic field data detected with the Flux- gate Magnetometer (FGM) onboard Cluster-1, with 22 Hz sampling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Two time series with 24 hours are investigated, one from 2008 March 9 and one from 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' To reduce the com- putational time of the analysis, the data length has been reduced by using a decimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The low-pass Chebychev Type I infinite impulse response filter was used with a reduction factor 𝑀 = 10, order 8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8/𝑀 cut-off frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This decimation process is described in Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' After decimating the time series, we apply the MF-DFA method with four input parameters: minimum scale 𝑠𝑖, maximum scale 𝑠 𝑓 , order of fluctuation function 𝑞 and polynomial order 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The scale refers to multiple segment sizes of the cumulative series and varies from a minimum segment size 𝑠𝑖 to a maximum 𝑠 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In this work, we use 𝑠𝑖 = 10 and 𝑠 𝑓 = 𝑁, where 𝑁 is the length of the time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 𝑞 varies between −20 and 20 with an increment of Δ𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='25, and 𝑚 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This choice of parameters was supported by several tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The recommendation for large time series is to use a polynomial trend order around 𝑚 = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 𝑠 𝑓 = 𝑁 was chosen to avoid deformations in the shape of the multifractal spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Meanwhile, for the 𝑞 parameter the use of values larger than 20 does not change the shape of the spectra significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='1 MF-DFA analysis of the |𝐵| time series Figure 1 shows the solar wind magnetic field time series studied in this section for days 2008 March 9 and 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the upper panel, the time series for 2008 March 9 (red) and its first order differencing (black) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As it was explained in the previous section, time-differencing is necessary in this case due to the high nonstationarity of this series (ℎ(2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Throughout the remaining of this section, only the differenced time series will be used for March 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This time series was characterized by Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019) as being permeated by large-scale current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The green regions in the original time series denote current sheets found with Li’s method (Li 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The lower panel shows the time series for 2016 January 25, which is characterized by a higher degree of stationarity and the absence of current sheets (Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Due to its higher stationarity (ℎ(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='96), there is no need to perform a differencing in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 2 shows different multifractal measures of the two magnetic field time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 2(a) shows the multifractal spectra, which reveal a left asymmetry for the March 09 time series (red) and a right asymmetry for the January 25 series (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The left asymme- try indicates the stronger contribution to multifractality coming from large fluctuations associated with values of 𝑞 > 0 in the intermittent time series of the current sheet-filled time series of March 09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' the right asymmetry found for the current sheet-free time series of Jan- uary 25 points to the greater contribution of small fluctuations to the multifractality (Ihlen 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The width of the spectrum can be used as a measure of the degree of multifractality of the series (Shimizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Comparing both spectra, it can be seen that they have almost the same width (Δ𝛼 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='541 for March 9 and Δ𝛼 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='555 for January 25), which may be surprising, since the time series of March 9 is visibly more intermittent, with strong bursts randomly interspersed in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In this case, the difference in multifractality can be better quantified by the Renyi exponent 𝜏(𝑞), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It reveals a nonlinear behavior for both series, but with 𝑅2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='804 for March 9 and 𝑅2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='986 for January 25, thus, March 9 displays higher multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 MF-DFA analysis of the volatility time series In the previous section, the degree of multifractality, as provided by the width of the multifractal spectra, could not properly distinguish between the two time series under investigation, which is unexpected, given that the original series are not only visually very different, but one of them is known to be permeated by coherent structures (current sheets) and the other is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This is probably because although the differenced time series of 2008 March 9 is apparently more intermit- tent than the series of 2016 January 25, most of the abrupt changes in |𝐵| caused by the current sheets in the March 9 series have a small amplitude and, therefore, do not produce strong bursts in the time-differenced series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Such abrupt changes in |𝐵| can be enhanced by employing the volatility, thus providing a way to investigate the role of current sheets in the multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the present section, we employ the volatility to enhance the distinct features of each series due to current sheets before repeating the MF-DFA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The magnetic volatility, vol𝑚𝑎𝑔, can be calculated from the stan- dard deviations of the log magnetic return Δ𝑟mag(𝑡) in a moving window of length 𝜔 along 𝑁 sample points (Tsay 2010) Δ𝑟mag(𝑡) = log \x12 |B(𝑡 + 𝜏)| |B(𝑡)| \x13 , (10) volmag( 𝑗) = v u u t 1 𝜔 − 1 𝜔+ 𝑗−1 ∑︁ 𝑖=𝑗 (Δ𝑟mag(𝑖) − 𝜇( 𝑗))2 , (11) where 𝜏 is a time-lag, 𝑗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' , 𝑁 − 𝜔 + 1 and 𝜇( 𝑗) is the mean Δ𝑟mag inside the window (Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Note that since Δ𝑟mag involves computing a time difference with lag 𝜏, there is no need to difference the original time series to remove nonstationarities prior to computation of the volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝜔 ant 𝜏 values are estimated from the Power Spectrum Density (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 3(a) shows the PSD for the March 9 time series, where the inertial range is the blue region between the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This region was chosen as the frequency interval where the slope of the fitted line is -5/3, following Kolmogorov’s K41 theory (Kolmogorov 1941) for fully developed turbulence (Frisch 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The frequency in the middle of the inertial range marks the scale used to define both 𝜏 and 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It is also the scale used in Li’s method to detect the current sheets, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In this way, we define 𝜏 = 𝜔 = 50𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 3(b) shows the PSD for the January 25 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 4 exhibits the volatility time series for 2008 March 9 (up- per panel, red) and for 2016 January 25 (lower panel, blue) from the decimated magnetic field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Recall that the upper series has many current sheets while the lower one has none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Note that, unlike the January 25 series, the March 9 volatility series has several extreme events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Most of these high peaks are due to the abrupt changes in the magnetic field that take place when the satellite crosses a current sheet in the solar wind, as evidenced by the coincidence between extreme events in the volatility and current sheets detected by Li’s method (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2(a),(b) in Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As a consequence, the multifractal spectra obtained from the volatility of both series are very different, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Now, the spectrum of the intermittent time series of March 9 is much broader than the one from January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝛼−width is Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='94134 for March 9 and Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='74921 for January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The volatility has enhanced the con- tribution of the extreme events due to current sheets, thus showing the signature of coherent structures present in the solar wind that were partially hidden in the multifractal analysis of the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The Renyi exponents are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 5(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' once again, the curve for March 9 is more concave than for January 25, reflecting its higher level of multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The coefficient of determination for the Renyi exponents is 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='97464 for the volatility of March 9 and 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='98125 for the volatility of January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It is clear that the volatility has highlighted the role of current sheets in the multifractal singularity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4 MF-DFA OF SURROGATE TIME SERIES According to Madanchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2017), there are two features in a time series that can lead to its multifractality: (i) the presence of heavy-tailed PDFs, as in highly intermittent series, and (ii) the exis- tence of linear and non-linear correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In this section, we try to identify the origin of the multifractality in the solar wind by means of two surrogate time series derived from the original |𝐵| data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As mentioned in the introduction, the shuffled time series is a random permutation of the original time series in the real space that destroys all temporal correlations, while keeping the same PDF for the am- plitudes of |𝐵|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' On the other hand, the random phases surrogate is generated from the Fourier Transform of the original |𝐵| series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A new Fourier series is generated by shuffling the phases of the Fourier modes while keeping their power spectrum (Maiwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The inverse Fourier transform of this new frequency spectrum is the random phases surrogate, which keeps the power spectrum and linear autocorrelation of the original series, but has a Gaussian PDF and breaks the nonlinear correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' After generating these two surro- gates, we repeat the multifractal analysis described in the previous section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' if the shuffled surrogate has a multifractal spectrum which is considerably narrower than the spectrum of the original series, it means that time correlations are an important source of multi- fractality in the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If the random phases surrogate has a multifractal spectrum which is considerably narrower than the spectrum of the original series, it means that fat-tailed PDFs and/or nonlinear correlations are important for the multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Note that both kinds of multifractality mentioned above can be simultaneously present in a time series (Norouzzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Madanchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' If both the shuffled and random phases surrogates produce MNRAS 000, 1–12 (2022) Origin of Multifractality in Solar Wind Turbulence 5 [t] 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 5 0 5 10 15 20 |B| (nT) 2008 March 9 Current Sheets Differencing |B| March 9 01 03 05 07 09 11 13 15 17 19 21 23 Time (hours) 0 5 10 15 |B| (nT) 2016 January 25 b a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Solar wind time series of |𝐵| measured by Cluster-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) For 2008 March 9 (red), containing current sheets (green), and its first order differencing (black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) time series of |𝐵| for 2016 January 25 (blue), without current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum |B| 2008 March 9 |B| 2016 January 25 a 20 15 10 5 0 5 10 15 20 q 30 20 10 0 10 20 (q) Renyi Exponent |B| 2008 March 9 |B| 2016 January 25 b Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectrum of |𝐵| for 2008 March 9 (red), and 2016 January 25 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for 2008 March 9 (red), and 2016 January 25 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' monofractal spectra, then nonlinear correlations (but not fat-tailed PDFs) are the source of multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the following subsections, we perform this analysis for both the |𝐵| and volatility time series of 2008 March 9 and 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='1 Magnetic Field time series, 2008 March 9 Figure 6 shows the differenced time series of |𝐵| for March 9 (red) with its shuffled (green) and random phases (magenta) surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Clearly, the shuffled surrogate keeps the extreme events of the dif- 10 -4 10 -3 10 -2 10 -1 10 0 f(Hz) 10 -2 10 0 10 2 PSD(nt 2/Hz) B 2008 March 9 PSD |B| 2008 March 9 PSD Regression Points 5/3 a 10 -4 10 -3 10 -2 10 -1 10 0 f(Hz) 10 -4 10 -2 10 0 10 2 10 4 PSD(nt 2/Hz) B 2016 January 25 PSD |B| 2016 January 25 PSD Regression Points 3/2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Power spectral density for solar wind magnetic field of (a) 2008 March 9, and (b) 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The blue region is the inertial range and the red line is the linear fit for this interval, with a slope equal to -5/3 for March 9 and slope -3/2 for January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' ferenced |𝐵| series, but the same events are absent from the random phases surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 7(a) displays the multifractal spectra for the March 9 orig- inal and surrogate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For the shuffled spectrum (green) we see a small reduction in the width when compared with the original one (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This means that there is a contribution from correlations to multifractality, along with the contribution of the PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Considering the random phases spectrum (magenta), its width reduces drastically (the Δ𝛼 variation is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='32), which points to a significant con- tribution to multifractality coming from a non-Gaussian PDF and/or MNRAS 000, 1–12 (2022) 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 18 10 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 Volatility Volatility 2008 March 9 0 2 4 6 8 10 12 Time (index) 10 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 Volatility Volatility 2016 January 25 b a Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Volatility of solar wind magnetic field time series for (a) 2008 March 9, and (b) 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum V 2008 March 9 V 2016 January 25 a 20 15 10 5 0 5 10 15 20 q 40 30 20 10 0 10 20 (q) Renyi Exponent V 2008 March 9 V 2016 January 25 b Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectra for the volatility in 2008 March 9 (red), and 2016 January 25 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for the volatility in 2008 March 9 (red), and 2016 January 25 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 18 10 4 3 0 3 |B| (nT) Differencing |B| 2008 March 09 2 4 6 8 10 12 14 16 18 10 4 3 0 3 |B| (nT) Shuffled 0 2 4 6 8 10 12 14 16 18 Time (index) 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 |B| (nT) Random Phases c b a Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Differenced time series for 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum |B| 2008 March 9 Shuffled Random Phases a 20 15 10 5 0 5 10 15 20 q 15 10 5 0 5 10 (q) Renyi Exponent |B| 2008 March 9 Shuffled Random Phases b Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectrum of |𝐵| for 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' nonlinear correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The conclusion from both spectra is that the PDF has the strongest contribution to multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The contribu- tion of the PDF is due to the presence of strong intermittent bursts (extreme events) in the March 9 time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since these bursts have been shown to be related to large current sheets (see Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019)), the current sheets can be seen as the origin of most of the multifractality in this time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 7(b) confirms this conclu- sion by showing the Renyi exponent as a function of 𝑞, where the random phases surrogate has a smaller concavity than the shuffled surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 Magnetic field time series, 2016 January 25 Figure 8 shows the time series for January 25 (blue) with its shuffled (green) and random phases (magenta) surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 9(a) shows a significant width reduction in both surrogate spectra in comparison with the original volatility spectrum (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The spectrum of the shuf- fled series (green) has a width Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='194, indicating a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='36 with the spectrum of |𝐵|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Similarly, the spectrum for the ran- dom phases series has a small width, about Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='32, a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='23 with the spectrum of |𝐵|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' So, there is strong influence from long-range correlations as well as non-gaussianity on the January 25 magnetic field multifractality, but the contribution of the correlations is preponderant, since the shuffled spectrum is considerably narrower than the random phases spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='3 Volatility time series, 2008 March 9 We proceed with the analysis of the origin of the multifractality for March 9 using the volatility, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 10 for the original (red), shuffled (green) and random phases (magenta) time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The cor- responding multifractal spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 11(a) show a wide parabola MNRAS 000, 1–12 (2022) Origin of Multifractality in Solar Wind Turbulence 7 [htpb] 2 4 6 8 10 12 10 4 0 5 10 15 |B| (nT) |B| 2016 January 25 2 4 6 8 10 12 10 4 0 5 10 |B| (nT) Shuffled 0 2 4 6 8 10 12 Time (index) 10 4 0 4 8 12 |B| (nT) Random Phases c b a Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Time series for 2016 January 25 (blue) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum |B| 2016 January 25 Shuffled Random Phases a 20 15 10 5 0 5 10 15 20 q 30 20 10 0 10 20 (q) Renyi Exponent |B| 2016 January 25 Shuffled Random Phases b Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectrum for 2016 January 25 (blue) and the re- spective surrogates: shuffled (green), and random phases (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for 2016 January 25 (blue) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' for the original volatility series (red) and two narrower parabolas related to its shuffled (green) and random phases (magenta) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The random phases spectrum has a width of about Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='39 and the shuffled spectrum has a width of Δ𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since both spectra have approximately the same width, it shows an important feature that was not so clear from the multifractal spectra of the |𝐵| surrogate series (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 7), that is, the importance of the nonlinear correlations, which play a key role, together with the PDF, in the origin of the multifrac- tality for the March 9 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since the volatility is computed with a lag-time of 𝜏 = 50𝑠, it is better suited for measuring the relevance of long-range nonlinear correlations than the time-differenced |𝐵| series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 11(b) confirms that the shuffled and random phases series have almost linear Renyi exponents, thus, the series are closer to monofractal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Time series Volatility for 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum V 2008 March 9 Shuffled Random Phases a 20 15 10 5 0 5 10 15 20 q 40 30 20 10 0 10 20 30 (q) Renyi Exponent V 2008 March 9 Shuffled Random Phases b Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectrum for the volatility of 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for the volatility of 2008 March 9 (red) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 Volatility time series, 2016 January 25 Figure 12 shows the volatility time series of the January 25 time series (blue) and its shuffled (green) and random phases (magenta) surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 13(a) shows the corresponding multifractal spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Once again, the reduction in the width for both surrogate spectra means that a mutual contribution to multifractality coming from long-range correlations and non-Gaussianity is present, with a clear predominance of the long-range correlations effects, since the shuf- fled spectrum is much narrower than the random phases spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A quantitative comparison of all the results for the |𝐵| time series and volatility time series of March 9 and January 25 is provided by Tables 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Table 1 shows 𝑅2 for the Renyi exponent of |𝐵| and its volatility for March 9 and January 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Table 2 shows the width of the multifractal spectra, Δ𝛼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Table 3 shows the asymmetry of the spectra, 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In general, all spectra for January 25 are right-asymmetric due to the importance of small scale fluctuations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' for March 9, some spectra MNRAS 000, 1–12 (2022) Volatility Volatili 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 0 Random Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content="2 0 2 4 6 8 10 Time (index)12 14 16 18 X104Volatility '2008 Mar 7 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 0ch 9 edD8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2 4 6 8 10 12 10 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 Volatility Volatility 2016 January 25 2 4 6 8 10 12 10 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 Volatility Shuffled 0 2 4 6 8 10 12 Time (index) 10 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 Volatility Random Phases a b c Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Time series of the volatility for 2016 January 25 (blue) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) Singularity Spectrum V 2016 January 25 Shuffled Random Phases a 20 15 10 5 0 5 10 15 20 q 40 30 20 10 0 10 20 (q) Renyi Exponent V 2016 January 25 Shuffled Random Phases b Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Multifractal spectrum for the volatility of 2016 January 25 (blue) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Renyi exponents for the volatility of 2016 January 25 (blue) and the respective surrogates: shuffled (green), and random phases (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 𝑅2 of the Renyi exponent for magnetic field and volatilities of 2008 March 9 and 2016 January 25 March 9 January 25 |𝐵| Volatility |𝐵| Volatility Original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='80413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='97464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='98597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='98125 Shuffle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='97505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='96748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='99537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='99573 Random Phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='98185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='99637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='99601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='99424 are left-asymmetric due to the importance of large-scale fluctuations, but the random phases show right asymmetry, since in the random phases surrogate the effects of non-Gaussian PDFs are destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Width of 𝛼, Δ𝛼, for magnetic field and volatilities of 2008 March 9 and 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' March 9 January 25 |𝐵| Volatility |𝐵| Volatility Original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='54112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='94134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='55568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='74921 Shuffle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='36663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='40332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='19468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='19873 Random Phases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='21802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='39299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='32181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='43088 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Spectrum Asymmetry, 𝐴, for magnetic field and volatilities of 2008 March 9 and 2016 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' March 9 January 25 |𝐵| Volatility |𝐵| Volatility Original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='49873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='63885 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='10709 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='31215 Shuffle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='51279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='53342 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='30854 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='18283 Random Phases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='33583 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='41817 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='45999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='47002 5 ZETA FUNCTION Another function typically employed in multifractal analyses of time series is the zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Consider the structure function for |𝐵| (Frisch 1995): 𝑆𝑝(𝜏) = ⟨[|𝐵(𝑡 + 𝜏)| − |𝐵(𝑡)|] 𝑝⟩ , (12) where ⟨·⟩ is the time average, 𝜏 is the time lag and 𝑝 are the statistical moments for the time series of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Assuming scale invariance inside the inertial range, 𝑆𝑝 follows a power law 𝑆𝑝(𝜏) ∼ 𝜏𝜁 ( 𝑝) , (13) where 𝜁(·) is the zeta function or scaling exponent of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' So, 𝜁(𝑝) is obtained by the slope of the log 𝑆𝑝(𝜏) × log 𝜏 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The importance of this parameter comes from Kolmogorov’s K41 theory (Kolmogorov 1941) and the IK (Iroshnikov-Kraichnan) theory (Iroshnikov 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Kraichnan 1965) of self-similarity and scale invariance inside the inertial range for a homogeneous and isotropic turbulence, where the 𝜁 function was shown to be a linear function of 𝑝, with 𝜁(𝑝) = 𝑝/3 for K41 and 𝜁(𝑝) = 𝑝/4 for IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 14(a), the linear K41 theoretical zeta scaling exponent function is shown by the black dashed line while the IK scaling ex- ponent is denoted by a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The top panel (a) also shows the zeta scaling exponent computed from the time series of |𝐵| for the intermittent series of March 09 (red line with circles) and for the current sheet-free series of January 25 (blue line with diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The zeta function for the March 09 series clearly departs from the linear behavior, as expected for multifractal intermittent series, but, surprisingly, the zeta function exhibits an almost linear relation with 𝑝 in the case of January 25, despite the fact that both series have multifractal spectra with similar widths (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, one should be cautious before using the behavior of the scaling exponent as a definite measure of multifractality, although it is a good mea- sure of intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' To confirm this result, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 14(b) compares the zeta scaling exponents of the March 09 |𝐵| series (red line with circles) with the zeta scaling exponents of its random phases series (magenta line with triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since the random phases series has a Gaussian PDF, it removes from the original series the intermittent extreme events responsible for the fat-tailed PDF and the zeta scaling exponent becomes linear, following the K41 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This result con- MNRAS 000, 1–12 (2022) Origin of Multifractality in Solar Wind Turbulence 9 1 2 3 4 5 6 7 p 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 (p) Zeta Function 2008 March 09 2016 January 25 K41 IK a Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Zeta functions for the magnetic field time series for 2008 March 9 (red circles) and 2016 January 25 (blue diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Zeta functions for |𝐵| 2008 March 9 (red circles) and its Random Phases (magenta triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (c) Zeta functions for |𝐵| 2016 January 25 (blue diamonds) and its Random Phases (magenta triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The dashed lines represent the K41 scaling and the dotted lines, the IK scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' firms the importance of the contribution from a fat-tailed PDF to the multifractality of the March 09 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 14(c), the same analy- sis is done for the January 25 series, where both the original series and its random phases show an IK linear behavior, since none of the series has fat-tailed PDF, although they have multifractal spectra (see the blue and magenta spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' We conclude from this that the 𝜁-function is a good measure of multifractality due to PDF, but misses the contribution of long-range correlations to the multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 6 𝑃−MODEL In section 4, we showed that the multifractal spectra of the volatil- ity of the solar wind are predominantly due to nonlinear and linear correlations in the time series of January 25 and due to PDF and nonlinear correlations for the March 9 time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The presence of long-range nonlinear correlations in both series is the signature of a nonlinear dynamical system (possibly with some stochastic compo- nent) governing the behavior of both series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the present section, we employ the 𝑝−model (Halsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Meneveau & Sreenivasan 1987) to show that both the correlations and the extreme events men- tioned above are actually a consequence of turbulent energy-cascade processes with different scaling laws that depend on the presence or absence of current sheets in the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝑝−model is a model for non-homogeneous energy-cascading process in the inertial range of fully-developed turbulence based on the generalized Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Consider that the flux of kinetic energy from eddies of size 𝐿 to smaller eddies is represented by a dissipa- tion 𝐸𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the one-dimensional version of the 𝑝−model, 𝐿 is the length of an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Suppose that an eddy of size 𝐿 is unequally di- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 f( ) V 2008 March 9 P Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 1 Singularity Spectrum V 2016 January 25 P Model Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Left: Multifractal spectrum for the volatility of 2008 March 9 (red circle) and its 𝑝−model fit (black line with dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Right: Multifractal spectrum for the volatility of 2016 January 25 (blue diamond) and its 𝑝−model fit (black line with dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' vided into two smaller eddies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', two sub-intervals) of sizes 𝑙1𝐿 and 𝑙2𝐿, where 0 < 𝑙1 < 𝑙2 < 1 are the size factors, with the energy flux 𝐸𝐿 being distributed unto these sub-eddies with different probabili- ties 𝑝1 and 𝑝2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=', the new dissipation values are 𝑝1 𝐸𝐿 and 𝑝2 𝐸𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In practice, one can start the process with 𝐿 = 𝐸𝐿 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Then, each new eddy is further sub-divided into two smaller eddies with the same size factors 𝑙1 and 𝑙2 and probabilities 𝑝1 and 𝑝2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This process may be repeated until the sub-intervals reach the Kolmogorov dissipation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' At each cascading step 𝑛, there will be \x12 𝑛 𝑚 \x13 segments with length 𝑙𝑚 1 𝑙𝑛−𝑚 2 𝐿 and dissipation 𝑝𝑚 1 𝑝𝑛−𝑚 2 𝐸𝐿, for 𝑚 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' , 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As shown by Halsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (1986) for the general two-scale Cantor set, it is possible to obtain the analytic expressions for the singularity exponent 𝛼 and the singularity spectrum 𝑓 as 𝛼 = ln 𝑝1 + (𝑛/𝑚 − 1) ln 𝑝2 ln 𝑙1 + (𝑛/𝑚 − 1) ln 𝑙2 , (14) 𝑓 = (𝑛/𝑚 − 1) ln(𝑛/𝑚 − 1) − (𝑛/𝑚) ln(𝑛/𝑚) ln 𝑙1 + (𝑛/𝑚 − 1) ln 𝑙2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (15) For each 𝑛 and given values of 𝑙1, 𝑙2, 𝑝1 and 𝑝2, the variation of 𝑚 will provide the different values of 𝛼 and 𝑓 for the singularity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Since 0 ≤ 𝑚 ≤ 𝑛 and 𝑚 is an integer, larger values of 𝑛 provide a better definition of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For a cascading process with direct energy dissipation in the inertial range, we have 𝑝1 + 𝑝2 < 1 (Meneveau & Sreenivasan 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This means that a new 𝑑𝑝 dissipation parameter must be included, where 𝑑𝑝 = 1 − 𝑝1 − 𝑝2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, we define 𝑝2 = 1 − 𝑝1 − 𝑑𝑝, as well as 𝑙2 = 1 − 𝑙1, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (14) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 15 shows the MF-DFA multifractal spectra for the volatil- ity series of March 9 (red circles) and January 25 (blue diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝑝−model fits obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (14) and (15) are also shown (black line with dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The values of 𝑝1, 𝑑𝑝 and 𝑙1 were obtained with a Monte Carlo method that minimized the mean squared error between the original and fitted spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For March 9th, we obtained 𝑝1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='71, 𝑑𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='17 and 𝑙1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' For January 25th, we obtained 𝑝1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='51, 𝑑𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='11 and 𝑙1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The agreement between the observational and theoretical curves confirms that the solar wind multifractal spectra can be obtained from a turbulence cascade pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This is a remarkable result, since the 𝑝−model was specifically elaborated to represent turbulent cascade processes, and will usually not be able to approximate the spectra of other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Next, we compare the turbulent time series behind the 𝑝−model spectra with the observational solar wind volatility time series in MNRAS 000, 1–12 (2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 (d) 1 IBl 2008 March 09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 Random Phases K41 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 0 1 2 3 4 5 6 1 2 p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 7 IBl 2016 January 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 Random Phases -K41 0 3 4 5 6 7 pZeta Function b c 2210 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' terms of their PSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' To obtain the 𝑝−model PSDs, we use the prob- abilities and size factors previously obtained with the Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' By iterating the generalized two-scale Cantor set model, we produce two 𝑝−model time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 16 shows a comparison of the solar wind volatility time series with the 𝑝−model time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The two upper panels depict the solar wind series for March 9 (a) and the corresponding 𝑝−model (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' the two lower panels depict the solar wind series for January 25 (a) and the corresponding 𝑝−model (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The qualitative similarity between observational and 𝑝−model time series is apparent in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A comparison of observed and simulated PSDs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 17(upper panels) shows the PSDs for the volatility time series of 2008 March 9 (left) and 2016 January 25 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The blue region between the vertical dashed lines represents the inertial range and the red line is the linear regression with slope −5/3 for the March 9 series and −3/2 for the January 25 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, the highly intermittent series of March 9 (with current sheets) exhibits a K41 scaling, whereas the January 25 series (without current sheets) shows an IK scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This fact had been previously established by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2011) and confirmed by Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019) using PSDs computed from the time series of |𝐵|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The PSDs computed from the 𝑝−model time series are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 17(lower panels), and they reveal K41 scaling for the March 9 series and IK scaling for the January 25 series, just like in the original solar wind series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Note that in both cases the inertial range can be extended almost throughout the whole PSDs shown, since our 𝑝-model has small dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' We conclude that a K41 intermittent turbulence cascade is behind the multifractality of the current sheet-filled time series of March 9 and an IK turbulence cascade is the origin of the multifractality of the January 25 series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' This result is consistent with other time series analysed by us, that show that current sheets are responsible for the K41 turbulence multifractality and the absence of current sheets results in an IK turbulence multifractality in the solar wind (see Table 4 in Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 7 CONCLUSIONS We have presented a new methodology for multifractal analysis of solar wind magnetic field data, based on MF-DFA, volatility and surrogate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The MF-DFA provides a standard way to gen- erate the singularity spectrum and the Renyi exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' the volatility enhances the extreme events, stressing the differences between series with current sheets and series without current sheets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' the surrogate time series provide a way to infer the origin of multifractality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Addi- tionally, the 𝑝-model was used to reproduce the multifractal behavior of the solar wind series, indicating that a nonlinear turbulence en- ergy cascade dynamical system is behind the observed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' A similar framework for multifractal analysis, but without the volatility and the 𝑝-model, was used by Chattopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2018) in the analysis of CME linear speed data in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In order to keep the paper reasonably short, we have limited our presentation to only two time-series, but we have tested our techniques in other series and found that the conclusions presented are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' An example of analysis with two other time series is included in the supplementary material (online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Further exploration of the methodology is left for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Just like in Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2019), we found the volatility to be very useful to highlight the role of current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In our case, they in- crease the signature of multifractality due to PDF in the singularity spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The surrogate analysis of both original and volatility series shows that for time series with current sheets, multifractality is due to Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Volatility time series for 2008 March 9 (red) and (b) generated 𝑝−model time series (black) by 10𝑡ℎ interation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (c) Volatility time series for 2016 January 25 (blue) and (d) generated 𝑝−model time series (black) by 15𝑡ℎ interation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Left: power spectral density for 2008 March 9 volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (a) Right: power spectral density for 2016 January 25 volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Left: power spectral density for generated 𝑝−model time series from 2008 March 9 volatil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (b) Right: power spectral density for generated 𝑝−model time series from 2016 January 25 volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The blue regions mark the inertial range and the red lines are the linear fits for those intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Volatility 2008 March 9 Vol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0 b P Model (1oth Generation) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 3 Time (index) ×104Volatility 2016 January 25 Volatility P Model (1i5th Generation) Amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='f(Hz)Origin of Multifractality in Solar Wind Turbulence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='both intermittency and nonlinear correlations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' for time series with- out current sheets, it is predominantly produced by the long-range correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The 𝑝−model analysis reveals that those are mainly nonlinear correlations, since the process behind the statistics is a nonlinear turbulent energy cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' So, turbulence is the common source of the multifractality, but current sheets are the source of the left asymmetry of the singularity spectrum, as well as the nonlinear scaling exponent for the structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the absence of current sheets, the small-amplitude fluctuations are the main source of the right asymmetry of the singularity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It is important to stress that despite being a multifractal process, the current sheet-free series exhibits an almost linear scaling exponent for the structure functions, which is sometimes confused with a monofractal process in the liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Our results indicate that the Renyi exponent is more sensitive to multifractality due to correlations than the structure function scaling exponent (zeta function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In dealing with separate cases where the presence or absence of current sheets is considered, we are attacking one of the “nine outstanding questions of solar wind physics", related by Vaill & Borovsky (2020), namely, the origin and evolution of the mesoscale (timescales in the range of minutes up to a few hours) plasma and magnetic-field structure of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' These current sheets have been associated with the border between adjacent flux tubes (Bruno 2019), while also being related to nonlinear turbulent interactions rather than the presence of advected pre-existing flux-tube structures (Bowen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In the present work, we do not focus on the origin of those coherent structures, but measure their weight on the statis- tics of solar wind fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' We do this not only through Fourier spectral indices and the scaling of structure functions, as in Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2009), but their contribution to multifractality is explored in depth through the MF-DFA, volatility and surrogate techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' As we said, our results reveal that although the scaling of the structure functions may be almost linear for series without current sheets, the singularity spectra may still display broad parabolas, the signature of highly multifractal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Thus, the scaling exponent of struc- ture functions is adequate to measure multifractality due to PDFs, but not for multifractality due to long-range correlations, where the Renyi exponent and singularity spectra should be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' Multifrac- tal series with nearly linear behavior of the scaling exponents were also reported in Tam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2010) (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' 4), where the rank- order multifractal analysis (ROMA) is employed in the description of auroral zone electric-field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' In conclusion, the basic question related to mesoscale plasma tur- bulence in the solar wind is not whether it is monofractal or mul- tifractal, but if the source of the ubiquitous multifractality is the PDF or the long-range correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' The short answer is that in the presence of current sheets, the PDF has a strong contribution for multifractality, but in their absence, it is mainly due to correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' It would be interesting to check if the monoscaling of the structure functions reported in previous solar wind time series, as in Kiyani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' (2009, 2013) and Bruno (2019) for turbulence at kinetic scales, indeed reveal monofractality or if they indicate, in fact, multifractal series due to correlations and not due to intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' ACKNOWLEDGEMENTS L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' acknowledges Brazilian agency CAPES for the financial support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' acknowledges Brazilian agencies CAPES (grant 88887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='309065/2018-01) and CNPq (Grant 306920/2020-4) for their financial support, as well as FCT—Fundação para a Ciência e a Tecnologia (Portugal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} +page_content=' was partially 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1–12 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNA0T4oBgHgl3EQfLf8y/content/2301.02118v1.pdf'} diff --git a/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/2301.02570v1.pdf.txt b/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/2301.02570v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1c8b47b1fe4c095f90c4dd44383bc284226849f --- /dev/null +++ b/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/2301.02570v1.pdf.txt @@ -0,0 +1,1286 @@ +arXiv:2301.02570v1 [math.GR] 6 Jan 2023 +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S +DIMENSION +KEVIN I. PITERMAN* +Abstract. Given a finite group 퐺 and a prime 푝, let A푝(퐺) be the poset of nontrivial elementary +abelian 푝-subgroups of 퐺. The group 퐺 satisfies the Quillen dimension property at 푝 if A푝(퐺) +has non-zero homology in the maximal possible degree, which is the 푝-rank of 퐺 minus 1. For +example, D. Quillen showed that solvable groups with trivial 푝-core satisfy this property, and later, +M. Aschbacher and S.D. Smith provided a list of all 푝-extensions of simple groups that may fail +this property if 푝 is odd. In particular, a group 퐺 with this property satisfies Quillen’s conjecture: +퐺 has trivial 푝-core and the poset A푝(퐺) is not contractible. +In this article, we focus on the prime 푝 = 2 and prove that the 2-extensions of the exceptional +finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with +only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and +usually reducing the problem to the same question in small linear groups, where we establish this +property via counting arguments. As a corollary, we reduce the list of possible components in a +minimal counterexample to Quillen’s conjecture at 푝 = 2. +1. Introduction +Since the early 70s, there has been a growing interest in the 푝-subgroup posets and their +connections with finite group theory, the classification of the finite simple groups, finite geometries, +group cohomology and representation theory. The poset S푝(퐺) of nontrivial 푝-subgroups of a +group 퐺 was introduced by Kenneth Brown in [2]. In that paper, Brown worked with the Euler +characteristic χ(퐺) of groups 퐺 satisfying certain finiteness conditions andestablishedconnections +between the 푝-fractional part of χ(퐺) and the 푝-subgroup structure of 퐺. One of the consequences +of his results is the commonly known “Homological Sylow theorem”, which states that the Euler +characteristic of S푝(퐺) is 1 modulo |퐺| 푝, the order of a Sylow 푝-subgroup of 퐺. +Some years later, Daniel Quillen introduced the poset A푝(퐺) of nontrivial elementary abelian +푝-subgroups of a finite group 퐺 and exhibited several applications of the topological properties +of these posets [21]. Indeed, the study of elementary abelian 푝-subgroups goes back to Quillen’s +earlier work on the Bredon cohomology of 퐺-spaces and his proof of the Atiyah-Swan conjecture, +that relates the Krull dimension of a ring to the dimension of A푝(퐺) (see [20]). +In [21], Quillen showed that S푝(퐺) and A푝(퐺) are (퐺-equivariantly) homotopy equivalent, +and provided a new proof of Brown’s result. In fact, when 퐺 is the set of rational points of +a semisimple algebraic group over a finite field of characteristic 푝, these posets are homotopy +equivalent to the building of 퐺 and, hence, they have the homotopy type of a wedge of spheres of +dimension 푙 − 1, where 푙 is the rank of the underlying algebraic group. Furthermore, in that case, +the homology � +퐻∗(A푝(퐺)) affords the classical Steinberg module for 퐺. +2010 Mathematics Subject Classification. 20G41, 20D20, 20D30, 05E18. +Key words and phrases. 푝-subgroups, exceptional groups of Lie type, Quillen’s conjecture. +*Supported by a postdoctoral fellowship of the Alexander von Humboldt Foundation. +1 + +2 +KEVIN I. PITERMAN +Quillen also exhibited other connections between intrinsic algebraic properties of 퐺 and the +topology of these posets. For instance, he showed that A푝(퐺) is disconnected if and only if 퐺 +contains a strongly 푝-embedded subgroup. Recall that the classification of the groups with this +property is indeed one of the many important steps towards the classification of the finite simple +groups (see for example Section 7.6 of [8]). +On the other hand, Quillen proved that if 퐺 has a fixed point on A푝(퐺) (or, equivalently on +S푝(퐺)), then these posets are contractible. Note that 퐺 has a fixed point if and only if its 푝-core +푂 푝(퐺) is nontrivial. In view of this and further evidence, Quillen conjectured that the reciprocal +to this statement should hold. That is, if A푝(퐺) is contractible then there is a fixed point, or, +equivalently, 푂 푝(퐺) ≠ 1 (see Conjecture 2.9 of [21]). In other words, Quillen’s conjecture asserts +that A푝(퐺) is contractible if and only if 푂 푝(퐺) ≠ 1. +A significant part of Quillen’s article is devoted to proving the solvable case of this conjecture. +In [21] it is shown that for a 푝-nilpotent group 퐺 with abelian Sylow 푝-subgroups and 푂 푝(퐺) = +1, A푝(퐺) is homotopy equivalent to a nontrivial wedge of spheres of the maximal possible +dimension, which is 푚 푝(퐺) − 1, the 푝-rank of 퐺 minus 1. Then, if 퐺 is any solvable group with +푂 푝(퐺) = 1, 퐺 contains a 푝-nilpotent subgroup 푂 푝′(퐺)퐴, with 퐴 ∈ A푝(퐺) of maximal 푝-rank +and 푂 푝(푂 푝′(퐺)퐴) = 1, and thus � +퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0. +Later, Michael Aschbacher and Stephen D. Smith formalised this property and gave a name +to it: an arbitrary group 퐺 with � +퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0 is said to satisfy the Quillen dimension +property at 푝, or (QD)푝 for short (see [1]). Therefore, a solvable group 퐺 with 푂 푝(퐺) = 1 +satisfies (QD)푝 and thus Quillen’s conjecture. Furthermore, it was shown that 푝-solvable groups +satisfy this property by using Quillen’s techniques and, in addition, the CFSG (see [5, 23]). These +results also suggest that a stronger statement of the conjecture may hold: if 푂 푝(퐺) = 1 then +� +퐻∗(A푝(퐺); Q) ≠ 0. Therefore, from now on, by Quillen’s conjecture we will be referring to this +stronger version. +It is not hard to see that not every group 퐺 with 푂 푝(퐺) = 1 satisfies (QD)푝. For example, we +mentioned that finite groups of Lie type in characteristic 푝 satisfy the conjecture, but since the Lie +rank is usually strictly smaller than the 푝-rank, they fail (QD)푝. This has led to the development of +new methods to prove Quillen’s conjecture. One of the most notorious advances in the conjecture +was achieved by Aschbacher-Smith in [1]. There, they established Quillen’s conjecture for a group +퐺 if 푝 > 5 and in addition, roughly, all the 푝-extensions of finite unitary groups PSU푛(푞), with +푞 odd and 푝 | 푞 + 1, satisfy (QD)푝 (see Main Theorem of [1] for the precise statement). Here, +a 푝-extension of a group 퐿 is a split extension of 퐿 by an elementary abelian 푝-subgroup of +Out(퐿). In [1] it is not shown that the group 퐺 satisfies (QD)푝. Instead, they proved that if every +푝-extension of a fixed component of 퐺 satisfies (QD)푝, then � +퐻∗(A푝(퐺); Q) ≠ 0 if 푂 푝(퐺) = 1 +(under suitable inductive hypotheses). This result restricts the possibilities of the components of +a minimal counterexample to Quillen’s conjecture: every component has a 푝-extension failing +(QD)푝. In view of this result and the classification of the finite simple groups, Aschbacher and +Smith described for 푝 ≥ 3, all the possible 푝-extensions of simple groups which may potentially +fail (QD)푝. This is the (QD)-List, Theorem 3.1, of [1]. Moreover, it is conjectured in [1] that +the unitary groups PSU푛(푞) with 푞 odd and 푝 | 푞 + 1 should not appear in this list, and so the +extra hypothesis on the unitary groups in the main result of [1] could be omitted. Nevertheless, +this problem remains open (see [19] for recent results in this direction). +In the last few years, there have been further developments in the Quillen conjecture [15, 16, 17, +18]. Recently, in [18], new tools for the study of the conjecture have been provided. For example, + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +3 +it is shown that the Aschbacher-Smith general approach to the conjecture can be extended to +every prime 푝 by reducing reliance on results of [1] stated only for odd primes and invoking the +Classification. In particular, Theorem 1.1 of [18] shows that the Main Theorem of [1] extends +to 푝 ≥ 3, keeping the additional constraint on the unitary groups. On the other hand, for 푝 = 2, +one important obstruction for this extension is the lack of a (QD)-List for this prime. Roughly, +Corollary 1.8 of [18] concludes that a minimal counterexample to Quillen’s conjecture contains a +component of Lie type in characteristic 푟 ≠ 3, and every component of 퐺 has a 2-extension failing +(QD)2. +In view of these results on Quillen’s conjecture, in this article, we focus on showing that +the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy +(QD)2, with a small number of exceptions. This improves the conclusions of [18] on Quillen’s +conjecture for 푝 = 2, and allows us to conclude then that exceptional groups of Lie type in +odd characteristic different from 3 cannot be components of a minimal counterexample to the +conjecture (see Corollary 1.2 below). +The main result of this article is the following theorem, whose proof is given in different +propositions in Section 5. +Theorem 1.1. Let 퐿 be an exceptional finite simple group of Lie type in odd characteristic. That +is, 퐿 = 3퐷4(푞), 퐹4(푞), 퐺2(푞), 2퐺2(푞)′, 퐸6(푞), 2퐸6(푞), 퐸7(푞) or 퐸8(푞), with 푞 odd. Then every +2-extension of 퐿 satisfies the Quillen dimension property at 푝 = 2, except possibly in the following +cases: +3퐷4(9) extended with field automorphisms; 퐹4(3); 퐹4(9) extended with field automorphisms; +2-extensions of 퐺2(3); 퐺2(9) extended with field automorphisms; 2퐺2(3)′; +퐸8(3); 퐸8(9) extended with field automorphisms. +Indeed, the extensions of 퐺2 (3), 퐺2(9) and 2퐺2(3)′ mentioned above do fail (QD)2 by Example +5.3 and Proposition 5.1. +To achieve the conclusions of Theorem 1.1, in most cases we exhibit a maximal subgroup 푀 +of a 2-extension 퐿퐵 of 퐿 such that 푚2(푀) = 푚2(퐿퐵) and 푀 satisfies (QD)2. Since there is an +inclusion � +퐻푚2(퐿퐵)−1(A2(푀)) ↩→ � +퐻푚2(퐿퐵)−1(A2(퐿퐵)) in the top-degree homology groups, this +establishes (QD)2 for 퐿퐵 (see Lemma 3.3). In some cases, the subgroup 푀 arises from suitable +parabolic subgroups. More concretely, when it is possible, we pick 푃 to be a maximal parabolic +subgroup of 퐿 which is stabilised by 퐵 and such that 푀 := 푃퐵 realises the 2-rank of 퐿퐵. Then +we get a 2-nilpotent configuration 푈퐴, where 푈 is the unipotent radical of 푃, 퐴 is an elementary +abelian 2-subgroup realising the 2-rank of 푃퐵, and 푂2(푈퐴) = 퐶퐴(푈) = 1 by one of the corollaries +of the Borel-Tits theorem. Hence, by Quillen’s results on the solvable case, 푈퐴 satisfies (QD)2, +and thus also 푀 and 퐿퐵. +When the choice of such parabolic 푃 is not possible, we pick one of the maximal rank subgroups +of 퐿. Here, the components of the maximal subgroup 푀 are usually smaller exceptional groups, +low-dimensional linear group 퐴1(푞) and 퐴2(푞) or unitary groups 퐴 +2 (푞). Therefore, we first prove +that the 2-extensions of simple linear and unitary groups in dimensions 2 and 3 satisfy (QD)2. +Although there is a large literature on maximal subgroups of exceptional groups of Lie type, +we will only need the results from [3, 10, 11, 12, 13]. +Finally, from Theorem 1.1 and the results of [18] for 푝 = 2, we can conclude: +Corollary 1.2. Let 퐺 be a minimal counterexample to Quillen’s conjecture for 푝 = 2. Then 퐺 +contains a component of Lie type in characteristic 푟 ≠ 3. Moreover, every such component fails + +4 +KEVIN I. PITERMAN +(QD)2 in some 2-extension and belongs to one of the following families: +PSL푛(2푎)(푛 ≥ 3), 퐷푛(2푎)(푛 ≥ 4), 퐸6(2푎), +PSL± +푛(푞)(푛 ≥ 4), 퐵푛(푞)(푛 ≥ 2), 퐶푛(푞)(푛 ≥ 3), 퐷± +푛(푞)(푛 ≥ 4), +where 푞 = 푟푎 and 푟 > 3. +The 2-extensions of PSL2(푞), PSL3(푞) and PSU3(푞) satisfy (QD)2 by Propositions 4.2, 4.5 +and 4.6 respectively, with exceptions when 푞 = 3, 5, 9. Nevertheless, the results of [18] eliminate +these possibilities from a minimal counterexample. +Further results on the Quillen dimension property at 푝 = 2 for the classical groups could be +pursued by combining the methods presented in this article with the results of [5, 6]. +The paper is organised as follows. In Section 2, we set the notation and conventions that we +will need to work with the finite groups of Lie type. We also provide some useful properties to +work out the 푝-extension and compute 푝-ranks. In Section 3 we gather previous results on the +Quillen dimension property and related tools that will help us establish this property. Then in +Section 4 we establish (QD)2 for some 2-extensions of linear groups and recall the structure of +the centralisers of graph automorphisms, following Table 4.5.1 of [8]. In Section 5 we prove each +case of Theorem 1.1. +All groups considered in this article are finite. We suppress the notation for the homology +coefficients, and we assume that they are always taken over Q. The interested reader may note that +our results can be extended to homology with coefficients in other rings. Finally, we emphasise that +we adopt the language and conventions of [8]. This is particularly important when we name the +different types of automorphisms of groups of Lie type. Computer calculations were performed +with GAP [7]. +Acknowledgements. The author thanks Stephen D. Smith for many helpful discussions con- +cerning the algebraic properties of groups of Lie type. He also thanks Volkmar Welker for his +suggestions on a preliminary version of the article. +2. Preliminaries +We assume that the reader is familiar with the construction of the finite groups of Lie type as +fixed points of Steinberg endomorphisms, and the basic properties concerning root systems of +reductive algebraic groups. We will follow the language of [8], which also contains the required +background on finite groups of Lie type. In this section, we will only recall some notations and +names, and state results that will be used later. +We denote by C푛, D푛, Sym푛 and Alt푛 the cyclic group of order 푛, the dihedral group of order +푛, the symmetric group on 푛 points and the alternating group on 푛 points. +If 퐺 is a group, then Aut(퐺), Inn(퐺) and Out(퐺) denote the automorphism group, the group of +inner automorphisms and the outer automorphism group of 퐺 respectively. We denote by 푍(퐺) +the centre of 퐺. We usually write 퐺 : 퐻, or simply 퐺퐻, for a split extension of 퐺 by 퐻. When an +extension of 퐺 by 퐻 may not split, we denote it by 퐺.퐻. By an element 푔 (resp. a subgroup 퐵) of +퐺 inducing outer automorphisms on 퐿 ≤ 퐺 we mean that 푔 embeds into Aut(퐿) \ Inn(퐿) (resp. +퐵 embeds in Aut(퐿) with 퐵 ∩ Inn(퐿) = 1). Finally, 퐻 ◦푚 퐾 denotes a central product of 퐻 and 퐾 +by a central cyclic subgroup of order 푚. That is, 퐻 ◦푚 퐾 = (퐻 × 퐾)/C푚 where C푚 embeds into +both 푍(퐻) and 푍(퐾). + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +5 +We will usually use the notation 푛 in a group extension to denote a cyclic group of order 푛, and +푛푚 a direct product of 푚 copies of cyclic groups of order 푛. A number between brackets [푛] in +the structure description of an extension means some group of order 푛. +In this article, we are mainly interested in extensions by elementary abelian groups. Below we +recall the definition of 푝-extension given in the introduction and introduce some useful notation. +Definition 2.1. Let 퐿 be a finite group and 푝 a prime number. A 푝-extension of 퐿 is a split +extension 퐿퐵 of 퐿 by an elementary abelian 푝-group 퐵 inducing outer automorphisms on 퐿. +If 퐿 ≤ 퐺, we denote by O퐺(퐿) the poset of elements 퐵 ∈ A푝(퐺) such that 퐵 ∩ 퐿퐶퐺(퐿) = 1. +Thus O퐺(퐿) is the set of 퐵 ∈ A푝(퐺) inducing outer automorphisms on 퐿. We write O2(퐿) for +OAut(퐿) (퐿) at 푝 = 2. We also let ˆO퐺(퐿) = O퐺(퐿) ∪ {1} and ˆO2(퐿) = O2(퐿) ∪ {1}. +Definition 2.2. For a prime number 푝, we say that a group 퐺 satisfies the Quillen dimension +property at 푝 if A푝(퐺) has non-zero homology in dimension 푚 푝(퐺) − 1, where 푚 푝(퐺) denotes +the 푝-rank of 퐺: +(QD)푝 +� +퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0. +A remarkable study of the Quillen dimension property for odd primes 푝 was carried out in +Theorem 3.1 of [1]. This theorem contains a list of the potential 푝-extensions of simple groups +that might fail (QD)푝, for 푝 ≥ 3. In particular, this list contains the 푝-extensions of unitary +groups PSU푛(푞) with 푞 odd and 푝 | 푞 + 1. However, Conjecture 4.1 of [1] basically claims that +these groups should not belong to this list. In fact, it is shown there that if 푛 < 푞(푞 − 1) then these +푝-extensions satisfy (QD)푝. Nevertheless, this problem remains open. +The aim of this article is to achieve some progress on a similar list for the prime 푝 = 2. +Therefore, we will focus on showing that 2-extensions of certain simple groups satisfy (QD)2. To +that end, we introduce the following convenient definition. +Definition 2.3. A group 퐿 satisfies (E-(QD)) if every 2-extension of 퐿 satisfies (QD)2: +(E-(QD)) +For every 퐵 ∈ ˆO2(퐿), 퐿퐵 satisfies (QD)2. +In order to establish (QD)푝 for 푝-extensions, it is crucial to be able to compute 푝-ranks of +extensions. The following result, extracted from Lemma 4.2 in [16], will be a useful tool to +compute 푝-ranks of extensions. +Lemma 2.4 (푝-rank of extensions). Let 퐺 = 푁.퐾 be an extension of finite groups, and let 푝 be a +prime number. Then +푚 푝(퐺) = max +퐴∈S +�푚 푝(퐶푁 (퐴)) + 푚 푝(퐴)�, +where S = {퐴 ∈ A푝(퐺) ∪ {1} : 퐴 ∩ 푁 = 1}. In particular, 푚 푝(퐺) ≤ 푚 푝(푁) + 푚 푝(퐾). +We will implicitly use this result at many points of the proofs. Note that, in order to apply +this lemma, we should be able to compute centralisers of elementary abelian 2-subgroups, usually +inducing outer automorphisms. We will often proceed as follows: if 퐿퐵 is a 2-extension of 퐿, +then take a suitable decomposition 퐵 = 퐵0 ⊕ 퐵1, with |퐵1| = 2. Suppose that we can inductively +compute the 2-rank of 퐿퐵0. Then, by Lemma 2.4, we have +(2.1) +푚2(퐿퐵) = max +� +푚2(퐿퐵0), 1 + 푚2(퐶퐿퐵0(푡)) : 푡 ∈ 퐿퐵 \ 퐿퐵0 is an involution +� +. +Moreover, this computation depends only on the conjugacy classes of the involutions 푡, and, in +most of the cases that we are interested in, such classes are completely classified. + +6 +KEVIN I. PITERMAN +Now we recall, rather informally, the names of the different types of automorphisms of a simple +group of Lie type 퐾 defined over a field of odd characteristic, following Definition 2.5.13 of [8]. +We refer to [8] for the full details. Let 푡 ∈ Aut(퐾) be an involution and 퐾∗ = Inndiag(퐾). Then +we have the following names for 푡: +(1) inner-diagonal if 푡 ∈ 퐾∗; +(2) inner if 푡 ∈ Inn(퐾); +(3) diagonal if 푡 ∈ 퐾∗ \ Inn(퐾); +(4) field automorphism if 푡 ∈ Aut(퐾) \ 퐾∗ is Aut(퐾)-conjugated to a field automorphism of +the ground field and 퐾 is not 2퐴푛(푞), 2퐷푛(푞) or 2퐸6(푞); +(5) graph if 푡 ∈ Aut(퐾) \ 퐾∗, roughly, is Aut(퐾)-conjugated to an involution arising as an +automorphism of the underlying Dynkin diagram (except for 퐾 = 퐺2(푞)), or else from a +field automorphism in cases 2퐴푛(푞), 2퐷푛(푞) and 2퐸6(푞); and +(6) graph-field automorphism if it can be expressed as a product 푔 푓 of a graph involution 푔 +and a field automorphism 푓 , or else 퐾 = 퐺2(푞) and 푡 arises from a Aut(퐾)-conjugated of +an involution automorphism of the underlying Dynkin diagram. +It follows from Proposition 4.9.1 of [8] that the centralisers of field involutions 푡 verify that +푚2(퐶퐾 (푡)) = 푚2(퐾) and 푚2(퐶퐾 ∗(푡)) = 푚2(퐾∗). +By Eq. +(2.1), we see that 푚2(퐾 ⟨푡⟩) = +푚2(퐾) + 1. Below we reproduce a simplified version of this proposition. +Proposition 2.5. Let 퐾 = 푑Σ(푞) be a group of Lie type in adjoint version in characteristic 푟, and +let 푥 be a field or graph-field automorphism of prime order 푝. Set 퐾푥 = 푂푟′(퐶퐾 (푥)). Then the +following hold: +(1) If 푥 is a field automorphism then 퐾푥 � 푑Σ(푞1/푝). +(2) If 푥 is a graph-field automorphism then 푑 = 1, 푝 = 2 or 3, and 퐾푥 � 푝Σ(푞1/푝). +(3) 퐾푥 is adjoint and 퐶Inndiag(퐾) (푥) � Inndiag(퐾푥). +(4) Fields (resp. graph-field) automorphism are all Inndiag(퐾)-conjugated, except for graph- +fields for 퐾 = 퐷4(푞) and 푝 = 3. +The previous proposition does not determine, a priori, the structure of 퐶퐾 (푥), but just of +the centraliser taken over the inner-diagonal automorphism group. Since we are interested in +computing 푚2(퐶퐾 (푥)), it will be crucial for us to decide when a diagonal involution can centralise +a field or graph-field automorphism 푥. We recall below Lemma 12.8 of [9, Ch. 17], which +provides a partial solution to this problem. +Lemma2.6. Let 퐾 � PSL2(푞), PΩ2푛+1(푞), PSp2푛(푞) or 퐸7(푞), where 푞 is apower of an odd prime +푟. Let 휙 be a field automorphism of order 2, and let 퐾휙 = 푂푟′(퐶퐾 (휙)). Then Inndiag(퐾휙) = +퐶Inndiag(퐾) (휙) = 퐶Inn(퐾) (휙). In particular, 휙 does not commute with diagonal involutions of +Inndiag(퐾). +We will mainly work with Table 4.5.1 of [8] to compute the 2-ranks of extensions by diagonal +and graph involutions, mostly for the groups of type 퐴± +푚(푞) and the exceptional groups. In the +next paragraph, we briefly and informally describe how to read such table. See [8, pp. 171-182] +for a complete and accurate description of Table 4.5.1. +This table records the 퐾∗-conjugacy classes of inner-diagonal and graph involutions 푡 of a finite +group of Lie type 퐾 in adjoint version, and the structure of their centralisers when taken over +퐾∗ = Inndiag(퐾). The centraliser of an involution 푡 is denoted by 퐶∗ = 퐶퐾 ∗(푡). The first column +of Table 4.5.1 denotes the family for which the involutions are listed (퐴푛, 퐵푛, 퐶푛, etc.) The second + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +7 +column indicates the restrictions for these classes to exist, while the third column is a label for the +conjugacy class of that involution. For the purposes of this article, we will not need to interpret +the fourth column. In the fifth column, it is indicated when such classes are of inner type (denoted +by 1), diagonal type (denoted by 푑) or graph type (several notations like 푔, 푔′). The notation 1/푑 +indicates that it is inner if the condition inside the parentheses at the right holds, and it is diagonal +otherwise. From the sixth column to the end, the structure of the centraliser 퐶∗ is described. +Roughly, 퐶∗ is an extension of a central product of groups of Lie type 퐿∗ = 푂푟′(퐶∗) (column +six), whose versions are specified in the column “version” and whose centres can be recovered +from the column 푍(퐿∗). An extra part centralising this product can be computed from the column +퐶퐶∗◦(퐿∗). Here 퐶∗◦ = 퐿∗푇∗ is the connected-centraliser, where 푇∗ is a certain 푟′-subgroup arising +from a torus 푇 normalised by 푡 and inducing inner-diagonal automorphisms on 퐿∗. From the +columns 퐿∗, version, 푍(퐿∗) and 퐶퐶∗◦(퐿∗), one can compute the “inner-part” of 퐶∗◦. Finally, from +the last two columns we can recover the outer automorphisms of 퐿∗ arising in 퐶∗◦ (in general +of diagonal type) and the remaining part of 퐶∗/퐶∗◦, which is often an involution acting on the +components of 퐿∗ (as field or graph automorphism, or by switching two components) and on the +central part 퐶퐶∗(퐿∗) (which is usually cyclic and the involution acts by inversion). To recover the +action of the last column, the symbols 푖, ↔, 휙, 훾, 1 mean, respectively, an action by inversion, a +swap of two components, a field automorphism of order 2, a graph automorphism of order 2, and +an inner action on a component or trivial action on 퐶퐶∗◦ (퐿∗). +3. Tools to achieve (QD)푝 +In this section, we provide tools and collect results that will help us to establish (QD)2 on +certain 2-extensions. Many of these tools were introduced and exploded by Aschbacher-Smith to +determine the (QD)-list in [1]. +The following proposition is an easy consequence of the Künneth formula for the join of spaces +and the fact that A푝(퐻 × 퐾) ≃ A푝(퐻) ∗ A푝(퐾) (see [21, Prop. 2.6]). +Proposition 3.1. If 푝 is a prime and 퐻, 퐾 satisfy (QD)푝, then 퐻 × 퐾 satisfies (QD)푝. +The following lemma corresponds to Lemmas 0.11 and 0.12 of [1]. +Lemma 3.2. Let 푁 ≤ 퐺 be such that 푁 ≤ 푂 푝′(퐺). Then there is an inclusion +� +퐻∗(A푝(퐺/푁)) ⊆ � +퐻∗(A푝(퐺)). +In particular, 푚2(퐺) = 푚2(퐺/푁), and if 퐺/푁 satisfies (QD)푝 then so does 퐺. +If 푁 ≤ 푍(퐺), then indeed A푝(퐺) ≡ A푝(퐺/푁). +The following observation is an easy consequence of the inclusion between the homology +groups of top-degree. +Lemma 3.3. Let 퐻 ≤ 퐺 be such that 푚 푝(퐻) = 푚 푝(퐺). If 퐻 satisfies (QD)푝, then so does 퐺. +Next, we recall one of the essential results on the Quillen dimension property. +Theorem 3.4 (Quillen). If 퐺 is a solvable group with 푂 푝(퐺) = 1, then 퐺 satisfies (QD)푝. +This theorem settles the solvable case of Quillen’s conjecture (see [21, Thm. 12.1]). Later, +it was extended to the family of 푝-solvable groups by using the CFSG if 푝 is odd. We refer to +Chapter 8 of [23] for further details on Quillen’s conjecture and the Quillen dimension property. + +8 +KEVIN I. PITERMAN +In view of Theorem 3.4 and the Inclusion Lemma 3.3, it is convenient to look for solvable +subgroups of 퐺 with maximal 푝-rank. +Some standard solvable subgroups in a group of Lie +type 퐿 arise by taking extensions of unipotent radicals by elementary abelian subgroups of their +normalisers. +These extensions lie then inside parabolic subgroups. +The following result on +parabolic subgroups will help us to achieve (E-(QD)) for arbitrary groups of Lie type (cf. Step v +at p.506 of [1]). +Lemma 3.5. Let 퐿 be a simple group of Lie type, and 푝 a prime not dividing the characteristic of +퐿. Suppose that 퐿퐵 is a 푝-extension of 퐿 and that there exists a 퐵-invariant parabolic subgroup +푃 ≤ 퐿 such that 푚 푝(퐿퐵) = 푚 푝(푃퐵). Then 퐿퐵 satisfies (QD)푝. +Proof. Let 푅 := 푂푟 (푃), where 푟 is the characteristic of the ground field. Then, as a consequence +of the Borel-Tits theorem, 퐶Aut(퐿) (푅) ≤ 푅 (see Corollary 3.1.4 of [8]). In particular, if 푇 ≤ 푃퐵 +realises the 푝-rank of 푃퐵, then 푇 normalises 푅, and 퐶푇 (푅) ≤ 푅 ∩ 푇 = 1. This means that 푇 +is faithful on 푅, i.e. 푂 푝(푅푇) = 1, and 푚 푝(푅푇) = 푚 푝(푃퐵) = 푚 푝(퐿퐵). Then 푅푇 is a solvable +group with trivial 푝-core, and by Theorem 3.4, 푅푇 satisfies (QD)푝. By Lemma 3.3, 퐿퐵 satisfies +(QD)푝. +□ +Lemma 3.6. Let 퐿 be a simple group of Lie type defined in odd characteristic. Suppose that 푃 +is a proper parabolic subgroup of 퐿 containing a Sylow 2-subgroup of 퐿 (that is, |퐿 : 푃| is odd). +Then 퐿 and the extension of 퐿 by a field automorphism of order 2 satisfy (QD)2. +Proof. Let 퐿 and 푃 be as in the hypotheses of the lemma. Since 푃 has odd index in 퐿, it contains +a Sylow 2-subgroup of 퐿. Therefore, 푚2(푃) = 푚2(퐿) and by Lemma 3.5, 퐿 satisfies (QD)2. +Next, let 퐵 ∈ ˆO2(퐿) be cyclic inducing field automorphisms. By passing through algebraic +groups and root systems, it can be shown that 퐵 normalises some conjugate of 푃, which we may +assume is 푃 itself. Thus, after conjugation, we suppose that 퐵 ≤ 푁Aut(퐿) (푃). Note that a Sylow +2-subgroup of 푃퐵 is a Sylow 2-subgroup of 퐿퐵, so 푚2(푃퐵) = 푚2(퐿퐵). By Lemma 3.5, 퐿퐵 +satisfies (QD)2. +□ +We close this section with a few more results on low 푝-ranks. The following lemma follows +from the 푝-rank 2 case of Quillen’s conjecture. See [21, Prop. 2.10]. +Lemma 3.7. If A푝(퐺) is connected, 푚 푝(퐺) = 2 and 푂 푝(퐺) = 1, then 퐺 satisfies (QD)푝. +It will be convenient to recall the classification of groups with a strongly 2-embedded subgroup, +that is, those groups with disconnected 2-subgroup poset. See [8, Thm. 7.6.1] and [21, Prop. 5.2]. +Theorem 3.8. Let 푝 = 2 and 퐺 be a finite group. Then A2(퐺) is disconnected if and only if +푂2(퐺) = 1 and one of the following holds: +(1) 푚2(퐺) = 1; +(2) Ω1(퐺)/푂 푝′(Ω1(퐺))) � PSL2(2푛), PSU3(2푛) or Sz(22푛−1) for some 푛 ≥ 2. +In particular, from the isomorphisms among the simple groups, we see that +Alt5 � PSL2(5) � PSL2(22), 2퐺2(3)′ � PSL2(23), +are included in the list of item (2). +Indeed, sometimes in low dimensions, we will be able to conclude (QD)푝 by computing the +sign of the Euler characteristic of A푝(퐺). Therefore, we will use the following well-known +expression of this invariant. + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +9 +Proposition 3.9. The reduced Euler characteristic of A푝(퐺) is: +�χ(A푝(퐺)) = +� +퐸 ∈A푝 (퐺)/퐺∪{1} +(−1)푚푝 (퐸)−1푝( +푚푝 (퐸) +2 +)|퐺 : 푁퐺(퐸)|. +Finally, the next lemma will help us to produce non-zero homology by inductively looking into +the homology of the Quillen poset of a certain normal subgroup and centralisers of outer elements +acting on it. The main reference for this lemma is [22]. +Lemma 3.10. Let 퐺 be a finite group and 푝 a prime number. Suppose that 퐿 ⊴ 퐺 is a normal +subgroup such that O퐺(퐿) consists only of cyclic subgroups. Then we have a long sequence +. . . → � +퐻푚+1(A푝(퐺)) → +� +퐵∈O퐺 (퐿) +� +퐻푚(A푝(퐶퐿(퐵))) +푖∗→ � +퐻푚(A푝(퐿)) +푗∗→ � +퐻푚(A푝(퐺)) → . . . +where 푖∗ and 푗∗ are the natural maps induced by the inclusions A푝(퐶퐿(퐵)) ⊆ A푝(퐿) and +A푝(퐿) ⊆ A푝(퐺), respectively. +In particular, the following hold: +(1) Let 푋 be the union of the subposets A푝(퐶푁 (퐵)) for 퐵 ∈ O퐺(퐿). We have indeed a +factorisation +(3.1) +� +퐵∈O퐺 (퐿) � +퐻푚(A푝(퐶퐿(퐵))) +푖∗ +� +푖′∗ +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +� +퐻푚(A푝(퐿)) +� +퐻푚(푋) +푘∗ +�r +r +r +r +r +r +r +r +r +r +where also 푖′ +∗ and 푘∗ are induced by the inclusions A푝(퐶퐿(퐵)) ⊆ 푋 and 푋 ⊆ A푝(퐿), +respectively. +(2) 푚 푝(퐺) ≤ 푚 푝(퐿) + 1. +(3) If � +퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) = 0 for all 퐵 ∈ O퐺(퐿), then 퐻푚푝 (퐿) (A푝(퐺)) = 0. +(4) We have a bound +dim 퐻푚푝 (퐿) (A푝(퐺)) ≥ +� +퐵∈O퐺 (퐿) +dim � +퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) − dim � +퐻푚푝 (퐿)−1(푋) +≥ +� +퐵∈O퐺 (퐿) +dim � +퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) − dim � +퐻푚푝 (퐿)−1(A푝(퐿)). +(5) If 푚 푝(퐺) = 푚 푝(퐿) + 1 and 퐺 fails (QD)푝, then, for 푚 = 푚 푝(퐿) − 1, we get inclusions +� +퐵∈O퐺 (퐿) +� +퐻푚(A푝(퐶퐿(퐵))) ↩→ � +퐻푚(푋) ↩→ � +퐻푚(A푝(퐿)). +Proof. The long exact sequence arises from the main result of [22]. Then Eq. (3.1) in item (1) is +an immediate consequence of this sequence. Item (2) holds by Lemma 2.4. Items (3-5) follow by +looking into the last terms of the long exact sequence, at 푚 = 푚 푝(퐿). +□ +4. Some linear groups satisfy (QD)2 +In this section, we prove that the linear groups PSL2(푞) and PSL3(푞) satisfy (E-(QD)) for every +푞, with a few exceptions for 푞 = 3, 5, 9. These cases will serve as basic cases for the exceptional +groups, where we will occasionally find linear groups as direct factors in some of their maximal +subgroups. +From [8, Prop. 4.10.5], we recall the 2-ranks of the small dimensional linear groups: + +10 +KEVIN I. PITERMAN +Proposition 4.1. If 푞 is a power of an odd prime and 푛 = 2, 3, then PSL± +푛(푞) and PGL± +푛(푞) have +2-rank 2. +We begin by studying the linear group of dimension 2. +Proposition 4.2. Let 퐿 � PSL2(푞) with 푞 odd and 푞 ≠ 3. Then every 2-extension 퐿퐵 of 퐿 +satisfies (QD)2, with the following exceptions: +(1) 퐿 � PSL2(5), 퐵 = 1; +(2) 퐿 � PSL2(9), 퐵 induces field automorphisms of order 2. +Moreover, every 2-extension of Inndiag(퐿) � PGL2(푞) satisfies (QD)2, except in case (2). +Proof. We consider the possible 2-extensions of 퐿. In any case, we know that 퐿 is simple and that +Out(퐿) = C2 × C푎, where C2 � Outdiag(퐿) and C푎 is the group of field automorphisms of F푞. +Suppose that 휙 is an order 2-field automorphisms of F푞 (if it exists), and that 푑 ∈ Inndiag(퐿) \ 퐿 +is a diagonal involution. Then the 2-extensions of 퐿 are given in Table 1. This table follows since +2-extension 퐿퐵 +퐶퐿(퐵) +푚2(퐿퐵) +퐵 = 1 +퐿 +2 +퐵 = ⟨휙⟩ +PGL2(푞1/2) +3 +퐵 = ⟨푑⟩ +D푞+휖 +2 +Table 1. 2-extensions of PSL2(푞), 푞 ≥ 5 odd. Here 푞 ≡ 휖 (mod 4), 휖 ∈ {1, −1}. +every involution of Aut(PSL2(푞)) − PGL2(푞) is a field automorphism. Recall also that field and +diagonal automorphisms of order 2 do not commute by Lemma 2.6 The structure of the centraliser +for 푑 follows from the first row of Table 4.5.1 of [8]. Finally, observe that 퐿 ⟨푑⟩ = Inndiag(퐿) and +푚2(Inndiag(퐿) ⟨휙⟩) = 3 since 푚2(퐿) = 푚2(Inndiag(퐿)) = 2. +We prove that each 2-extension of 퐿 satisfies (QD)2 by computing the Euler characteristic. +First, 2-extensions 퐿퐵 and Inndiag(퐿) ⟨휙⟩ have connected A2-poset by Theorem 3.8, except for +퐿 = PSL2(5), 퐵 = 1. Therefore, by Lemma 3.7, 퐿 and Inndiag(퐿) satisfy (QD)2, except for +퐿 = PSL2(5). Note that A2(PSL2(5)) = A2(Alt5) = A2(PSL2(4)) is homotopically discrete with +5 points, and the 2-extension PGL2(5) � Sym5 does satisfy (QD)2. This yields the conclusions +of the statement for the case 푞 = 5. +Next we show (QD)2 for the 2-extensions 퐿 ⟨휙⟩ and Inndiag(퐿) ⟨휙⟩, both of 2-rank 3 by Lemma +3.3. Thus, it is enough to show that 퐿 ⟨휙⟩ satisfies (QD)2. In order to do this, we compute the +dimensions of 퐻1(A2(퐿)) and 퐻1(A2(퐶퐿(휙))). +Since in this situation 푞 is a square, 푞 ≠ 5. Second, if 푞 = 25, 퐶퐿(휙) = PGL2(5). Hence, in +any case, the dimension of these degree 1 homology groups can be computed from the reduced +Euler characteristic of the underlying A2-poset. Here we use the formula given in Proposition 3.9. +Thus, for 퐾 = 퐿 or 퐶퐿(휙), +dim 퐻1(A2(퐾)) = −�χ(A2(퐾)) += 1 − # of involutions in 퐾 + 2 · # of 4-subgroups of 퐾. +(4.1) +In Table 2 we describe these numbers: +Proof of Table 2. The number of involutions and 4-subgroups of PSL2(푞) follows from Dickson’s +classification of the subgroups of PSL2(푞) (see also Theorem 6.5.1 of [8]). + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +11 +Group +Number of involutions +Number of 4-subgroups +PSL2(푞) +푞(푞+휖 ) +2 +푞(푞2−1) +24 +PGL2(푞) +푞2 +푞(푞2−1) +6 +Table 2. Here 푞 ≡ 휖 (mod 4), 휖 ∈ {1, −1}. +The number of involutions of PGL2(푞) follows since there is a unique conjugacy class of +diagonal involutions 푑 by Table 4.5.1 of [8]. Thus, the number of elements in such conjugacy +class is equal to 푞(푞−휖 ) +2 +, which gives 푞2 after adding the number of involutions in PSL2(푞). +Finally, to compute the number of four-subgroups of PGL2(푞) we proceed as follows: each +four-subgroup of PGL2(푞) is either contained in PSL2(푞) or else it contains a unique involution +of PSL2(푞) and 2 diagonal involutions. Therefore, for a given diagonal involution 푑, there is a +one-to-one correspondence between 4-subgroups containing 푑 and involutions in 퐶퐿(푑) � D푞+휖 . +This shows that each diagonal involution is contained in (푞 + 휖)/2 4-subgroups. Since we have +푞(푞−휖 ) +2 +diagonal involutions, the total number of 4-subgroups in PGL2(푞) containing diagonal +involutions is +푞(푞 − 휖) +2 +· (푞 + 휖) +2 +· 1 +2 = 푞(푞2 − 1) +8 +. +Thus the total number of 4-subgroups in PGL2(푞) is +푞(푞2 − 1) +24 ++ 푞(푞2 − 1) +8 += 푞(푞2 − 1) +6 +. +This completes the proof of Table 2. +□ +Indeed, by Table 2, we get concrete values for the dimensions of the degree 1 homology groups +of A2(PSL2(푞)) and A2(PGL2(푞)): +(4.2) +dim 퐻1(A2(PSL2(푞))) = −�χ(A2(PSL2(푞))) = 1 +12 (푞 − 휖)(푞2 − (6 − 휖)푞 − 휖12), +(4.3) +dim 퐻1(A2(PGL2(푞))) = −�χ(A2(PGL2(푞))) = 1 +3 (푞 − 3)(푞2 − 1). +Now we need to describe the number of field automorphisms in PSL2(푞) ⟨휙⟩ and in PGL2(푞) ⟨휙⟩. +Recall that the fieldautomorphisms of PSL2(푞) ⟨휙⟩ are all PGL2(푞)-conjugated, withcentraliser +퐶PGL2(푞) (휙) = 퐶PSL2(푞) (휙). Thus, the number of field automorphisms of order 2 in PSL2(푞) ⟨휙⟩ +is exactly +| PGL2(푞)| +|퐶PSL2(푞) (푞)| = 푞(푞2 − 1) +푞1/2(푞 − 1) = 푞1/2(푞 + 1). +This gives 푞1/2(푞 + 1) involutions in PSL2(푞) ⟨휙⟩ \ PSL2(푞). Let 퐿 = PSL2(푞), 퐵 = ⟨휙⟩. By +Lemma 3.10, the values in Table 2 and formula (4.1), we conclude that: +dim 퐻2(A2(퐿퐵)) ≥ 푞1/2(푞 + 1) dim 퐻1(A2(PGL2(푞1/2))) − dim 퐻1(A2(PSL2(푞))) += 푞1/2(푞 + 1) 1 +3 (푞1/2 − 3)(푞 − 1) − 1 +12 (푞 − 1)(푞2 − 5푞 − 12) += 1 +4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4). +Note that 푞 ≡ 1 (mod 4). The above number is positive for all 푞 ≥ 13, which is our case since 푞 +is an even power of an odd prime and 푞 ≠ 9 by hypothesis. We conclude that 퐿퐵 = PSL2(푞) ⟨휙⟩ + +12 +KEVIN I. PITERMAN +satisfies (QD)2. Then also PGL2(푞) ⟨휙⟩ satisfies (QD)2. Moreover, +dim 퐻2(A2(PGL2(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSL2(푞) ⟨휙⟩)) +≥ 1 +4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4). +(4.4) +We have shown that every possible 2-extension of PSL2(푞) and PGL2(푞) satisfies (QD)2, +except for the cases described in the statement of the theorem. +□ +We note that the excepted cases in Proposition 4.2 actually fail (QD)2. Indeed, PSL2(5) fails +(QD)2 since it has 2-rank 2 and A2(PSL2(5)) = A2(PSL2(4)) is homotopically discrete. The +following example provides the details that show that PSL2(9) ⟨휙⟩ and PGL2(9) ⟨휙⟩ fail (QD)2, +where 휙 is a field automorphism of order 2. +Example 4.3. Let 퐿 = PSL2(9) and let 퐴 = Aut(퐿). Then 퐴/퐿 � C2 × C2, so every 2-extension +of 퐿 is a nontrivial normal subgroup of 퐴. This gives 3 possible 2-extensions of 퐿, but not 4. Let +휙 be a field automorphism of 퐿 and 푑 a diagonal automorphism of 퐿, both of order 2. Then the +possible 2-extensions of 퐿 are: +(1) 퐿, with 2-rank 2, satisfies (QD)2 with 퐻1(A2(퐿)) of rank 16; +(2) 퐿 ⟨휙⟩, with 2-rank 3, fails (QD)2 since 퐶퐿(휙) � Sym4, which has nontrivial 2-core +푂2(퐶퐿(휙)) � C2 × C2 ≠ 1; +(3) 퐿 ⟨푑⟩ = PGL2(9), with 2-rank 2, satisfies (QD)2 with 퐻1(A2(퐿) ⟨푑⟩) of rank 160 and +퐶퐿(푑) � D10. +Note that Aut(퐿) has 2-rank 3 and does not satisfy (QD)2, and it is not a 2-extension of 퐿 since +diagonal and field automorphisms do not commute in Aut(퐿). Also PGL2(9) ⟨휙⟩ fails (QD)2 +since 퐶PGL2(9) (휙) = 퐶퐿(휙) has nontrivial 2-core. +There is also a remaining almost simple group 푁 with 퐿 < 푁 < Aut(퐿), not contained +in the previous cases. +This is the extension 푁 = PSL2(9).2 � Alt6 .2, and it satisfies that +A2(푁) = A2(퐿). Therefore, although this group 푁 is not a 2-extension of 퐿, it is a “non-split +2-extension”, and it does satisfy (QD)2. +Finally, these computations show that A2(퐿) ↩→ A2(Aut(퐿)) induces an inclusion in homology, +and hence a non-zero map. By the main result of [17], PSL2(9) is not a component of a minimal +counterexample to Quillen’s conjecture. +Our next aim is to show that 2-extensions of PSL3(푞) satisfy (QD)2, with only a few exceptions. +We will needthe followinglemmawhichrecords the values of the Euler characteristic of the Quillen +poset of some linear groups and the unitary groups in dimension 3. +Lemma 4.4. For 퐿 = PSL푛(푞) and 푛 odd, we have +�χ(A2(퐿)) = �χ(A2(PGL푛(푞))) = (−1)푛 +푛 +푛−1 +� +푖=1 +(푞푖 − 1) 푓푛(푞), +where 푓푛(푞) denotes a polynomial as described in [25]. For instance, 푓3(푞) = 푞3 + 3푞2 + 3푞 + 3. +Moreover, since A2(퐿) is Cohen-Macaulay of dimension 푛 − 2, the above Euler characteristic +computes the dimension of 퐻푛−2(A2(퐿)). +If 퐿 = PSU3(푞), then +�χ(A2(퐿)) = �χ(A2(PGU3(푞))) = 1 +3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3). + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +13 +Proof. The value of the Euler characteristic for PGL푛(푞) follows from Proposition 4.1 and The- +orem 4.4 of [25] (note that there is a typo in the formula of Theorem 4.4, and the product over +푖 should be up to 푟 − 1). +Also, since 푛 is odd, by Proposition 7.5 of [19], A2(PSL푛(푞)) = +A2(PGL푛(푞)) = A2(GL푛(푞))>푍 where 푍 is the cyclic subgroup of order 2 of 푍(GL푛(푞)). By +[21] (see also [25]), A2(PSL푛(푞)) is Cohen-Macaulay of dimension 푛 − 2. +The formula for PGU3(푞) follows from Example 7.6 of [19]. +□ +Next, we show that the 2-extensions of PSU3(푞) satisfy (QD)2, except for 푞 = 3. These cases +will be important during our analysis for PSL3(푞), especially when working with 2-extensions by +graph-field automorphisms. +Proposition 4.5. Let 퐿 = PSU3(푞). Then 퐿 satisfies (E-(QD)) if 푞 ≠ 3. Moreover, let 휙 be a +graph automorphism of order 2 of 퐿. Then we have +dim 퐻2(A2( PGU3(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSU3(푞) ⟨휙⟩)) +≥ 1 +3 (푞2 − 1)(푞 + 1) +�푞2(푞2 − 푞 + 1) +(3, 푞 + 1) +(푞 − 3) − (푞3 − 3푞2 + 3푞 − 3) +� +, +which is a positive polynomial for 푞 > 3. +Finally, for 푞 = 3, PSU3(3) satisfies (QD)2 but +PSU3(3) ⟨휙⟩ fails (QD)2. +Proof. We have that A2(퐿) is connected by Theorem 3.8, and 푚2(퐿) = 2 by Proposition 4.1. +Thus 퐿 satisfies (QD)2 by Lemma 3.7. Moreover, by Lemma 4.4, +(4.5) +dim 퐻1(A2(퐿)) = −�χ(A2(퐿)) = 1 +3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3). +Next, the only possible nontrivial 2-extension of 퐿 is by a graph automorphism 휙 of order 2 +(which indeed arises from the field automorphism 푥 ↦→ 푥푞). Let 퐿1 = 퐿 ⟨휙⟩ be such extension. +By Table 4.5.1 of [8], +퐶PGU3(푞) (휙) � Inndiag(Ω3(푞)) = PGL2(푞). +This implies that 퐶퐿(휙) = PGL2(푞). Moreover, there is a unique PGU3(푞)-conjugacy class of +graph automorphisms, and such elements act by inversion on Outdiag(퐿) = (3, 푞 + 1). Thus the +conjugacy class of 휙 in Out(퐿) has size (3, 푞+1), and this gives rise to exactly (3, 푞+1) extensions +퐿 ⟨휙′⟩ ≤ Aut(퐿) of 퐿 by a conjugate 휙′ of 휙, and these extensions are Aut(퐿)-conjugated. We +conclude then that the number of graph automorphisms contained in 퐿1 is +푛푔 := +| PGU3(푞)| +| PGL2(푞)|(3, 푞 + 1) = 푞2(푞3 + 1) +(3, 푞 + 1) . +Finally, by Lemma 3.10, we conclude that +dim 퐻2(A2( PGU3(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSU3(푞) ⟨휙⟩)) +≥ 푛푔 dim 퐻1(A2(PGL2(푞))) − dim 퐻1(A2(PSU3(푞))) += 푞2(푞3 + 1) +(3, 푞 + 1) +1 +3 (푞 − 3)(푞2 − 1) − 1 +3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3) += 1 +3 (푞2 − 1)(푞 + 1) +�푞2(푞2 − 푞 + 1) +(3, 푞 + 1) +(푞 − 3) − (푞3 − 3푞2 + 3푞 − 3) +� +. +This polynomial is positive for all 푞 > 3. Therefore, 퐿1 satisfies (QD)2 if 푞 ≠ 3. +When 푞 = 3, 퐶퐿(휙) = PGL2(3) has nontrivial 2-core, so 퐻1(A2(퐶퐿(휙))) = 0, and by Lemma +3.10, 퐻2(A2(퐿1)) = 0. +□ + +14 +KEVIN I. PITERMAN +Now we have the necessary background to prove that PSL3(푞) satisfies (E-(QD)), except for a +small number of cases. +Proposition 4.6. Let 퐿 = PSL푛(푞) with 푛, 푞 odd. The following assertions hold: +(1) 퐿 and 퐿 extended by a field involution satisfy (QD)2. +(2) If 푛 = 3, then every 2-extension of 퐿 satisfies (QD)2, with the following exceptions that +fail (QD)2: +• 퐿 = PSL3(3) extended by a graph automorphism, and +• 퐿 = PSL3(9) extended by a group generated by a field involution and a graph +automorphism. +Proof. Let 퐿 = PSL푛(푞), with 푛 odd, and consider the stabiliser 푃 of a 1-dimensional sub- +space of the underlying module 푉 = F푛 +푞. +Then 푃 is a parabolic subgroup with structure +푃 � [푞푛−1]퐿푃, where 퐿푃, a Levi complement for 푃, has structure SL푛−1(푞) ◦(푛,푞−1) C푞−1. +Thus |퐿푃| = | GL푛−1(푞)|/(푛, 푞 − 1) and the index of 푃 in 퐿 is: +|퐿 : 푃| = +푞푛(푛−1)/2 �푛 +푖=2(푞푖 − 1) +푞푛−1 · 푞(푛−1)(푛−2)/2 �푛−1 +푖=1 (푞푖 − 1) += 푞푛 − 1 +푞 − 1 = 푞푛−1 + 푞푛−2 + . . . + 푞 + 1. +Since 푛 is odd, the index of 푃 in PSL푛(푞) is odd. By Lemma 3.6, 퐿 = PSL푛(푞) and 퐿 extended +by a field involution satisfy (QD)2. This proves item (1). +Before moving to the case 푛 = 3, we list all the possible 2-extensions of 퐿. Denote by 휙, 훾 and +훿 a field automorphism of order 2, a graph automorphism and a graph-field automorphism of 퐿, +respectively, such that [휙, 훾] = 1 and 훿 = 휙훾. Let also 퐿∗ = PGL푛(푞). Then the 2-extensions of +퐿 are: +(i) 퐿; +(ii) 퐿 ⟨휙⟩, with 퐶퐿∗(휙) � PGL푛(푞1/2) by Proposition 2.5; +(iii) 퐿 ⟨훾⟩, with 퐶퐿(훾) � Inndiag(Ω푛(푞)) by Table 4.5.1 of [8]; +(iv) 퐿 ⟨훿⟩, with 퐶퐿∗(훿) � PGU푛(푞1/2) by Proposition 2.5; +(v) 퐿 ⟨휙, 훾⟩, with 퐶퐿(휙, 훾) � Inndiag(Ω푛(푞1/2)) by (iii) and Proposition 2.5. +Now suppose that 푛 = 3, that is 퐿 = PSL3(푞). We know that the extensions of cases (i) and +(ii) above satisfy (QD)2 by the parabolic argument. So it remains to show that the 2-extensions +by graph, graph-field and both graph and field automorphisms, satisfy (QD)2. To that end, we +compute the dimensions of the top-degree homology groups, similar to what we did for PSL2(푞) +in the proof of Proposition 4.2. +First, recall that we have the following number of involutions of each type. Let 퐵 = ⟨휙, 훾⟩. +푛 푓 := # field involutions in 퐿 ⟨휙⟩ = # field involutions in 퐿퐵 += +| PGL3(푞)| +| PGL3(푞1/2)|(3, 푞1/2 + 1) , +푛푔 := # graph involutions in 퐿 ⟨훾⟩ = # graph involutions in 퐿퐵 += +| PGL3(푞)| +| PGL2(푞)|(3, 푞 − 1) , +푛푔 푓 := # graph-field involutions in 퐿 ⟨훿⟩ = # graph-field involutions in 퐿퐵 += +| PGL3(푞)| +| PGU3(푞1/2)|(3, 푞1/2 − 1) . + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +15 +To compute these numbers, we have used the structure of the centraliser in each case, the fact that +there is a unique 퐿∗-conjugacy class for each type of involution, and the structure of Out(퐿) = +(3, 푞 − 1) : ⟨휙, 훾⟩ (cf. Theorem 2.5.12 of [8]). +Let 푡 be a field, graph or graph-field involution of 퐿, and let 퐿1 = 퐿 ⟨푡⟩. Then the number +푛1 of involutions in 퐿1 \ 퐿 is 푛 푓 , 푛푔 or 푛푔 푓 , accordingly to the type of 푡. +Note also that +푚2(퐿1) = 푚2(퐿) + 1 = 3. +By Lemma 3.10, +(4.6) +dim 퐻2(A2(퐿1)) ≥ 푛1 · dim 퐻1(A2(퐶퐿(푡))) − dim 퐻1(A2(퐿)). +We compute 푑(푡) := dim 퐻1(A2(퐶퐿(푡))) in each case, by using Lemma 4.4 and Eq. +(4.3). +Note that Ω1(퐶퐿(휙)) = PSL3(푞1/2) by item (ii) above. Also 퐶퐿(훾) = PGL2(푞) by the classical +isomorphism Inndiag(Ω3(푞)) � PGL2(푞). By Lemma 4.4, we have: +푑(휙) = dim 퐻1(A2(PSL3(푞1/2))) = 1 +3 (푞1/2 − 1)(푞 − 1)(푞3/2 + 3푞 + 3푞1/2 + 3), +푑(훾) = dim 퐻1(A2(PGL2(푞))) = 1 +3 (푞 − 3)(푞2 − 1), +푑(훿) = dim 퐻1(A2(PGU3(푞1/2))) = 1 +3 (푞3 − 2푞5/2 − 푞2 + 2푞3/2 − 3푞 + 3). +Let 푑 := dim 퐻1(A2(퐿)). By Eq. (4.2), this dimension is +푑 = 1 +12 (푞 − 휖)(푞2 − (6 − 휖)푞 − 휖12), +with 푞 ≡ 휖 (mod 4) and 휖 ∈ {±1}. +Now it is routine to verify that 푛1푑(푡) > 푑 if 푡 = 훾 or 푡 = 훿, if and only if (푡, 푞) ≠ (훾, 3). Indeed, +for 푞 = 3, 퐶퐿(훾) = PGL2(3) � Sym4 has non-trivial 2-core, so 푑(훾) = 0 and in consequence, +퐻2(퐿 ⟨훾⟩) = 0. This shows that 퐿 ⟨훾⟩ fails (QD)2 if 푞 = 3. Therefore, a 2-extension of 퐿 by a +field, graph or graph-field involution satisfies (QD)2 if and only if 푞 ≠ 3 when 퐿 is extended by a +graph involution. +It remains to show that 퐿퐵 = 퐿 ⟨휙, 훾⟩ verifies (QD)2. For this case, we take 퐿 푓 = 퐿 ⟨휙⟩, +퐿2 = 퐿퐵 and consider the long exact sequence of Lemma 3.10 at 푚 = 2 there (since 푚2(퐿2) = 4). +That is, we need to show that 퐻3(A2(퐿2)) ≠ 0. +Note that the set of involutions 푡 ∈ 퐿2 \ 퐿1 is exactly the set of all graph and graph-field +automorphisms of the extension 퐿2 = 퐿퐵. +Let 푑푔 := dim 퐻2(A2(PGL2(푞) ⟨휙⟩)), 푑푔 푓 +:= +dim 퐻2(A2(PGU3(푞1/2) ⟨휙⟩)) and 푑 푓 := dim 퐻2(A2(퐿 푓 )). Therefore, by Lemma 3.10, +(4.7) +dim 퐻3(A2(퐿2)) ≥ 푛푔푑푔 + 푛푔 푓 푑푔 푓 − 푑 푓 . +We show that the right-hand side of this equation is positive if 푞 ≠ 9 by providing proper bounds +of the dimensions 푑푔, 푑푔 푓 and 푑 푓 . +By Eq. (4.4), +(4.8) +푑푔 = dim 퐻2(A2(PGL2(푞) ⟨휙⟩)) ≥ 1 +4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4). +Next, by Proposition 4.5, +(4.9) +푑푔 푓 ≥ 1 +3 (푞 − 1)(푞1/2 + 1) +�푞(푞 − 푞1/2 + 1) +(3, 푞1/2 + 1) +(푞1/2 − 3) − (푞3/2 − 3푞 + 3푞1/2 − 3) +� +, +which is positive for all 푞 > 9. + +16 +KEVIN I. PITERMAN +Finally, we need to bound 푑 푓 from above. Indeed, by Lemma 3.10 at 푚 = 2, we have +푑 푓 = dim 퐻2(A2(퐿 푓 )) = dim 퐻2(A2(PSL3(푞) ⟨휙⟩)) +≤ 푛 푓 dim 퐻1(A2(PSL3(푞1/2))) += 푞3/2(푞 + 1)(푞3/2 + 1) +(3, 푞1/2 + 1) +. +Now we check with the given bounds that 푛푔푑푔 + 푛푔 푓 푑푔 푓 − 푑 푓 is positive if and only if 푞 > 9. +Indeed, if 푞 = 9, similar arguments show 퐻3(A2(퐿퐵)) = 0 since 푑푔 = 0 by Example 4.3 and +푑푔 푓 = 0 by Proposition 4.5. +We conclude that every 2-extension of PSL3(푞) satisfies (QD)2, except for PSL3(3) extended +by a graph automorphism and for PSL3(9) extended by field and graph automorphisms, which +actually fail (QD)2. +□ +5. The Quillen dimension property on exceptional groups of Lie type +We use the results of the preceding sections to show that, with only finite exceptions, the +2-extensions of the exceptional groups of Lie type satisfy (QD)2. For that purpose, it will be +convenient to recall first which 2-extension can arise in each case. Table 3 records the 2-ranks of +the exceptional groups of Lie type in adjoint version and the structure of the outer automorphism +group. From this, we can compute the possible 2-extensions in each case. Recall that we follow +the terminology of [8]. The 2-ranks were extracted from [4] and [8, Prop. 4.10.5]. +Group +2-rank +Outdiag +Out/Outdiag +3퐷4(푞) +3 +1 +3Φ +퐺2(푞) +3 +1 +ΦΓ, where |ΦΓ : Γ| = 2 if 푞 = 3푎, and Γ = 1 otherwise +2퐺2(푞) +3 +1 +Φ (odd order) +퐹4(푞) +5 +1 +Φ +퐸6(푞) +6 +(3, 푞 − 1) +Φ × Γ, Γ � C2 +2퐸6(푞) +6 +(3, 푞 + 1) +2Φ +퐸7(푞) +8 +2 +Φ +퐸8(푞) +9 +1 +Φ +Table 3. Out/Outdiag is cyclic unless specified; Φ = Aut(F푞) � C푎, where +푞 = 푟푎, 푟 is an odd prime, and the usual conventions for the twisted types hold. +Also, Γ is a set of graph automorphisms. +5.1. Cases 퐺2(푞) and 2퐺2(푞). We start by proving that the Ree groups 2퐺2(푞) satisfy (QD)2 if +and only if 푞 ≠ 3. Note that, by Table 3 for example, 2퐺2(푞) has no non-trivial 2-extension. +Proposition 5.1. Let 퐿 be the Ree group 2퐺2(푞), where 푞 is a power of 3 by an odd positive +integer. Then the following hold: +(1) 퐿 has no non-trivial 2-extensions. +(2) A Sylow 2-subgroup of 퐿 is an elementary abelian group of order 8, so 푚2(퐿) = 3. +(3) 2-subgroups of equal order of 퐿 are conjugated. +(4) 퐿 satisfies (QD)2 if and only if 푞 ≠ 3. Moreover, if 푞 > 3 then +(5.1) +dim 퐻2(A2(퐿)) ≥ �χ(A2(퐿)) = 1 +21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21) > 0. + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +17 +(5) For 푞 = 3, A2(퐿) = A2(PSL2(8)) is homotopy equivalent to a discrete space of 8 points. +Proof. Items (1-3) are well-known facts about the Ree groups and can be found in [24]. +If 퐿 = 2퐺2(3), then 퐿′ = PSL2(8) has index 3 in 퐿, and A2(퐿) � A2(PSL2(8)) is homotopy +equivalent to a discrete space with 8 points. Since 푚2(퐿) = 3, we conclude that 퐿 fails (QD)2 for +푞 = 3. This proves item (5) and the “only if” part of item (4). +Now suppose that 푞 ≠ 3 and 퐿 = 2퐺2(푞). Since A2(퐿) has dimension 2 by item (2), we +show that its second homology group is non-zero. +To that end, it is enough to see that its +Euler characteristic is positive since A2(퐿) is connected for 푞 ≠ 3 by Theorem 3.8. Indeed, +�χ(A2(퐿)) = dim 퐻2(A2(퐿)) − dim 퐻1(A2(퐿)) ≤ dim 퐻2(A2(퐿)). +We invoke Theorem C of [10] to describe the normalisers of 2-subgroups: the centraliser of an +involution is 2×PSL2(푞), the normaliser of a four-subgroup is (22 ×D 푞+1 +2 ) : 3, and the normaliser +of a Sylow 2-subgroup is 23 : 7 : 3. From this information, items (2,3) and Proposition 3.9, we +can compute the Euler characteristic of A2(퐿): +�χ(A2(퐿)) = −1 + +|퐿| +2| PSL2(푞)| − 2 +|퐿| +6(푞 + 1) + 8 |퐿| +168 += −1 + 푞3(푞3 + 1)(푞 − 1) +� +1 +푞(푞2 − 1) − +1 +3(푞 + 1) + 1 +21 +� += 1 +21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21). +Since the polynomial 푞5 − 8푞4 + 15푞3 + 21 is positive for every prime power 푞 ≠ 4, we conclude +that 퐻2(A2(퐿)) ≠ 0. In consequence, 퐿 satisfies (QD)2 if 푞 ≠ 3. This completes the proof of +item (4), and hence of this proposition. +□ +For the case 퐺2(푞), we refer the reader to the classification of maximal subgroups of 퐺2(푞) by +P. Kleidman [10]. We will follow the terminology of that article. +Proposition 5.2. Let 퐿 = 퐺2(푞), with 푞 odd. Then every 2-extension of 퐿 satisfies (QD)2, except +possibly for the 2-extensions of 퐺2(3) and the 2-extension of 퐺2(9) by a field involution. +Proof. Let 퐿 = 퐺2(푞). We prove first that 퐺2(푞) and its extension by a field automorphism of +order 2 satisfy (QD)2, by exhibiting a maximal subgroup of the same rank that satisfies (QD)2. +By Theorem A in [10], 퐺2(푞) contains a subgroup 퐾+ = SL3(푞) : 2. Let 퐿+ = 퐹∗(퐾+) � +SL3(푞) and 푍 = 푍(퐿+). Then 퐿0 := 퐿+/푍 = PSL3(푞) and 퐻0 := 퐾+/푍 = 퐿0 ⟨훾⟩, where 훾 +induces a graph automorphism on 퐿0 (see Proposition 2.2 and its proof in [10]). By Proposition +4.6, 퐿0 satisfies (QD)2 if 푞 ≠ 3, so 퐻0 satisfies (QD)2. +On the other hand, 푚2(퐿) = 3 by Table 3, and also 푚2(퐿0) = 3 by the proof of Proposition 4.6. +Recall from Lemma 3.2 that +� +퐻∗(A2(퐻0)) = � +퐻∗(A2(퐾+/푍)) ⊆ � +퐻∗(A2(퐾+)). +In particular, we get the following inclusions between the top-degree homology groups +� +퐻2(A2(퐻0)) ⊆ � +퐻2(A2(퐾+)) ⊆ � +퐻2(A2(퐿)), +which show that 퐿 satisfies (QD)2 if 푞 ≠ 3. +Next, a nontrivial 2-extension of 퐿 = 퐺2(푞) can only be given by field automorphisms of +order 2 if 푞 is not a power of 3. +Moreover, by the construction of the subgroup 퐾+ given +in [10], field automorphisms of 퐺2(푞) induce field automorphisms on (a suitable conjugate + +18 +KEVIN I. PITERMAN +of) 퐾+, and hence on the quotient 퐻0. Thus, for 퐵 ∈ O2(퐿) inducing field automorphisms, +we may take 퐾+ fixed by 퐵, and then 퐾+퐵 � SL3(푞) : (2 × 퐵) after a suitable choice of +conjugates (recall that Out(SL3(푞)) = (3, 푞 − 1) : (Aut(F푞) × Γ), where Γ = 2 is a group of +graph automorphisms). +Similar as before, we have a split extension 퐾+퐵/푍 = 퐿0퐵′, where +퐵′ = ⟨훾⟩ × 퐵 ∈ O(퐿0). By Proposition 4.6, 퐿0퐵′ satisfies (QD)2 if 푞 ≠ 9. Analogously to +the previous case, 푚2(퐿0퐵′) = 4 = 푚2(퐿) = 푚2(퐾+퐵), and we get an inclusion in the degree 3 +homology groups, showing that 퐾+퐵 and 퐿퐵 satisfy (QD)2. Therefore, an extension of 퐿 by a +field automorphism of order 2 satisfies (QD)2 if 푞 ≠ 9. +It remains to analyse the case 푞 = 3푎. By Table 4.5.1 of [8] (see also Theorem 2.5.12 of +[8]), only field or graph-field automorphisms can arise in Aut(퐿). We have shown above that the +extension of 퐿 by a field automorphism of order 2 satisfies (QD)2 if 푞 ≠ 9. Thus we need to prove +that if 푡 is a graph-field automorphism of 퐿, then 퐿 ⟨푡⟩ satisfies (QD)2. In that case, 푞 = 32푎+1 and +by Proposition 2.5, 퐶퐿(푡) = 2퐺2(푞), which has 2-rank 3. Therefore 푚2(퐿 ⟨푡⟩) = 4. However, by +Theorem B of [10], every maximal subgroup of 퐿 ⟨푡⟩ containing 푡 is either 2-local or has 2-rank +at most 3. This shows that we cannot proceed as before via maximal subgroups. In view of this, +we will proceed by using the long exact sequence of Lemma 3.10. +We have subgroups 푀0 := 퐶퐿(푡) = 2퐺2(푞), 푀1 := 퐺2(3) ⟨푡⟩ ≤ 퐿 ⟨푡⟩ and 푀2 := 2퐺2(3) such +that 푀2 ≤ 푀1 ∩ 푀0. Fix 퐴 a Sylow 2-subgroup of 푀2. By Proposition 5.1(2) and [10, Lm. +2.4], 퐴 is also a Sylow 2-subgroup of 푀0 and it is self-centralising in 퐿, i.e. 퐶퐿(퐴) = 퐴. A +direct computation also shows that 푁푀2(퐴) = 퐴. PSL3(2), which immediately implies 푁퐿(퐴) = +퐴. PSL3(2). +Now, suppose by the way of contradiction that 퐿 ⟨푡⟩ fails (QD)2, that is, the homology group +퐻3(A2(퐿 ⟨푡⟩)) vanishes. Recall that 퐶퐿(푡) = 2퐺2(푞) and there is a unique 퐿-conjugacy class +of involutions 푡′ ∈ 퐿 ⟨푡⟩ − 퐿 by Proposition 2.5(4). Let 푋 = � +퐶퐿 (푡)푥∈퐿/퐶퐿 (푡) A2(퐶퐿(푡푥)). By +Lemma 3.10, we get inclusions +(5.2) +� +퐿/퐶퐿 (푡) +퐻2(A2(퐶퐿(푡))) ↩→ 퐻2 (푋) ↩→ 퐻2(A2(퐿)). +Set +푑 := dim 퐻2(푋), +푑′ := dim +� +퐿/퐶퐿 (푡) +퐻2(A2(퐶퐿(푡))) = |퐿 : 퐶퐿| dim 퐻2(A2(2퐺2(푞))). +Eq. (5.2) shows that 푑′ ≤ 푑. However, we will prove that 푑 < 푑′, arriving then at a contradiction. +On one hand, we have that 푋 is a union of A2-posets. Therefore, below each point, we have +a wedge of spheres of maximal possible dimension. This means that the homology of 푋 can be +obtained from the chain complex that in degree 푖 is freely generated by the spheres below each +point of 푋 of height 푖. In particular, for 푖 = 2, the points of height 2 correspond to the conjugates +of 퐴, the fixed Sylow 2-subgroup of 푀0 = 퐶퐿(푡) and 푀2. Thus, +푑 = dim 퐻2(푋) ≤ |퐿 : 푁퐿(퐴)| · # spheres below 퐴 = 푞6(푞6 − 1)(푞2 − 1) +168 +. +On the other hand, by Proposition 5.1(4), +푑′ ≥ |퐿 : 퐶퐿(푡)| · �χ(A2(2퐺2(푞))) = 푞3(푞3 − 1)(푞 + 1) 1 +21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21). + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +19 +Finally, from these bounds for 푑 and 푑′, it is not hard to conclude that 푑′ > 푑 for all prime +power 푞 ≥ 7, which is our case. This gives a contradiction to Eq. (5.2), and thus shows that +퐻3(A2(퐿 ⟨푡⟩)) ≠ 0, that is, 퐿 ⟨푡⟩ satisfies (QD)2. This finishes the proof of the proposition. +□ +Example 5.3. Let 퐿 = 퐺2(3). We show that A2(퐿) is homotopy equivalent to a wedge of spheres +of dimension 1. In particular, since 푚2(퐿) = 3, 퐿 fails (QD)2. Moreover, by Lemma 3.10, also +the unique nontrivial 2-extension of 퐿 (by a graph-field automorphism) fails (QD)2. +We construct a subposet of A2(퐿) of dimension 1 and homotopy equivalent to A2(퐿). First, +take the subposet 픦(A2(퐿)) = {퐴 ∈ A2(퐿) : 퐴 = Ω1(푍(Ω1(퐶퐿(퐴))))}, which is homotopy +equivalent to A2(퐿) (see [14, Rk. 4.5]). Next, there are two conjugacy classes of elementary +abelian 2-subgroups of order 8, and both are contained in 픦(A2(퐿)). For one of these classes, say +represented by 퐴, the normalizer 푁퐿(퐴) has order 192. Then it can be shown that 픦(A2(퐿))<퐴 +is contractible. Therefore, if we remove the 퐿-conjugates of 퐴 from 픦(A2(퐿)) we get a subposet +픰픦(A2(퐿)) homotopy equivalent to 픦(A2(퐿)). Now, there is a unique conjugacy class of four- +subgroups in this new subposet 픰픦(A2(퐿)), and each such subgroup is contained in a unique +element of order 8 of 픰픦(A2(퐿)). Again, we can remove all the four-subgroups from 픰픦(A2(퐿)) +and obtain a new subposet 푇 homotopy equivalent to A2(퐿). Since 푇 consists only of elements +of order 2 and 8, we conclude that 푇 has dimension 1. Finally, an extra computation shows that +�χ(A2(퐿)) = −11584. Therefore A2(퐿) is homotopy equivalent to a wedge of 11584 spheres of +dimension 1. In particular, 퐿 fails (QD)2. +This also shows that 퐿 = 퐺2(9) extended by a field automorphism of order 2 fails (QD)2: +if 휙 is a field involution, then 퐶퐿(휙) = 퐺2(3), and thus 퐻2(A2(퐶퐿(휙))) = 0 by the previous +computation. Then by Lemma 3.10, we conclude that 퐻3(A2(퐿)) = 0. +5.2. Cases 3퐷4 and 퐹4(푞). +Proposition 5.4. The group 퐿 = 3퐷4(푞) satisfies (E-(QD)) if 푞 ≠ 9 is odd. Also 3퐷4(9) satisfies +(QD)2. +Proof. Recall that 푚2(퐿) = 3 by Table 3. Then a graph automorphism of order 3 of 3퐷4(푞) +centralises a subgroup 퐾 = 퐺2(푞). Also, if 휙 denotes a field automorphism of order 2 of 퐿, then, +after choosing a suitable conjugate, we may assume that 휙 induces a field automorphism on 퐾. +By Proposition 5.2 and its proof, 푚2(퐾) = 3 = 푚2(퐿), 푚2(퐾 ⟨휙⟩) = 4 = 푚2(퐿 ⟨휙⟩), and both +퐾 and 퐾 ⟨휙⟩ satisfy (QD)2 for 푞 ≠ 3, 9 respectively. Also note that 퐺2(9) satisfies (QD)2. By +Lemma 3.3, 퐿 and 퐿 ⟨휙⟩ satisfy (QD)2 if 푞 ≠ 3, 9 respectively. Since these are the only possible +2-extensions of 퐿 by Table 3, this concluded with the proof of our proposition for 푞 ≠ 3. +If 푞 = 3 then a computation of the Euler characteristic of 퐿 in GAP shows that �χ(A2(퐿)) = +882634225472. Since A2(퐿) is connected by Theorem 3.8, we see that 퐻2(A2(퐿)) ≠ 0, that is, 퐿 +satisfies (QD)2. +□ +Proposition 5.5. If 퐿 = 퐹4(푞), with 푞 ≠ 3, 9 odd, then 퐿 satisfies (E-(QD)). Also 퐹4(9) satisfies +(QD)2. +Proof. Suppose that 푞 ≠ 3, 9 is an odd prime power. Then 퐿 contains a subgroup 퐻 := PGL2(푞) × +퐺2(푞) (cf. the main result of [13]). Note that 퐻 satisfies (QD)2 by Propositions 3.1, 4.2 and 5.2. +Since both 퐿 and 퐻 have 2-rank 5 by Table 3, we conclude that 퐿 satisfies (QD)2. +Let 퐵 ∈ O2(퐿), so 퐵 is generated by a field automorphism of order 2. Thus it acts by field +automorphisms in a direct product subgroup isomorphic to 퐻, which we may assume without + +20 +KEVIN I. PITERMAN +loss of generality that it is our 퐻. Then � +퐻 = PGL2(푞)퐵 × 퐺2(푞1/2), which is a subgroup of 퐻퐵, +satisfies (QD)2 by Propositions 3.1, 4.2 and 5.2. Since 푚2( � +퐻) = 6 = 푚2(퐿퐵), we conclude that +퐿퐵 also satisfies (QD)2. +We have shown that every possible 2-extension of 퐿 satisfies (QD)2, so 퐿 satisfies (E-(QD)). +If 푞 = 9, then PGL2(9) × 퐺2(9) satisfies (QD)2 by Propositions 3.1, 4.2 and 5.2. Therefore, +퐹4(9) satisfies (QD)2. +□ +5.3. Cases 퐸6(푞) and 2퐸6(푞). +Proposition 5.6. Let 퐿 = 퐸 휖 +6 (푞) (any version), 휖 ∈ {±1}, and 푞 odd. Then 퐿 satisfies (E-(QD)). +Proof. Let 퐿 = 퐸 휖 +6 (푞) in adjoint version, where 휖 ∈ {±1}. For a 2-extension 퐿퐵 of the adjoint +version 퐿, we see that 푚2(퐿퐵) = 푚2(퐿푢 �퐵), where 퐿푢 is the universal version of 퐸 휖 +6 (푞) and �퐵, +isomorphic to 퐵, is just a lift of the action of 퐵 on 퐿푢 (this is possible since 푍(퐿푢) = (3, 푞 − 휖) is +odd). Thus 퐿퐵 = 퐿푢 �퐵/푍(퐿푢), and by Lemma 3.2, � +퐻∗(A2(퐿퐵)) ⊆ � +퐻∗(A2(퐿푢퐵)). Therefore, if +퐿 satisfies (E-(QD)), then so does the universal version of 퐸 휖 +6 (푞). +We will show that there exists a parabolic subgroup 푃 of 퐿 such that for any 2-extension 퐿퐵, +a suitable conjugate of 푃 is normalised by 퐵 (so we can suppose it is 푃 itself), and 푚2(푃퐵) = +푚2(퐿퐵). +This parabolic subgroup 푃 arises from the 퐴5 subdiagram in 퐸6, so 푃 = 푈 GL휖 +6 (푞)/푍(퐿푢), +where GL휖 +6 (푞)/푍(퐿푢) denotes the Levi complement. Then 푚2(푃) = 6, which realises the 2- +rank of 퐿. +Furthermore, a graph, graph-field or field automorphism of 퐿 (the last two only +for 휖 = 1) stabilises this subdiagram (and hence 푃), inducing a graph (resp. +graph-field or +field) automorphism on GL휖 +6 (푞)/푍(퐿푢). Denote by 푡 such automorphism. Then 푚2(퐿 ⟨푡⟩) ≤ +푚2(퐿) + 1 = 7. We claim that +(5.3) +푚2(푃 ⟨푡⟩) = 푚2(GL휖 +6 (푞) ⟨푡⟩) = 7 = 푚2(퐿 ⟨푡⟩). +Note that 푚2(푃 ⟨푡⟩) = 푚2(GL휖 +6 (푞) ⟨푡⟩), for the lifted action of 푡 on GL휖 +6 (푞). Then it is clear that +Eq. (5.3) holds if 푡 induces a field automorphism (so 휖 = 1), since the stabiliser of 푡 in GL6(푞) is +GL6(푞1/2). Similarly, if 푡 is a graph-field automorphism then 휖 = 1 and 퐶GL6(푞) (푡) = GU6(푞1/2), +which has 2-rank 6. Then, in these two situations, 푚2(푃 ⟨푡⟩) = 7. +Now assume that 푡 is a graph involution. For 휖 = 1, 푡 acts on GL6(푞), so GL6(푞) ⟨푡⟩ contains +a graph automorphism 푔 inducing the map 푥 ↦→ (푥′)−1, where 푥′ denotes the transpose of 푥. +Therefore, 퐶GL6(푞) (푔) = GO6(푞), which has 2-rank 6. This implies that 푚2(GL6(푞) ⟨푡⟩) = 6. If +휖 = −1, 푡 is a graph involution acting on GU6(푞), so up to conjugation 푡 is indeed the map 푥 ↦→ 푥푞. +Therefore, 퐶GU6(푞) (푡) = GO6(푞), and again we get 푚2(GU6(푞) ⟨푡⟩) = 6. In any case, we see that +푚2(푃 ⟨푡⟩) = 7. +Finally, suppose that we have 퐵 = ⟨휙, 훾⟩, where 휙 is a field automorphism of order 2 and +훾 a graph automorphism of order 2 of 퐿 = 퐸6(푞). We can suppose that 퐵 stabilises 푃 (and +thus its unipotent radical), and its Levi complement GL6(푞). Thus, 훾 acts as a graph automor- +phism on the stabiliser of 휙 in GL6(푞), which is isomorphic to GL6(푞1/2). As we saw above, +푚2(GL6(푞1/2) ⟨훾⟩) = 7. Therefore, 푚2(GL6(푞)퐵) = 8. Since 푚2(퐵) = 2 and 푚2(퐸6(푞)) = 6, we +conclude that 푚2(퐸6(푞)퐵) = 8, so the 2-rank is realised in 푃퐵. +To conclude, note that a 2-extension of 퐿 is one of: +(1) 퐿, of 2-rank 6 = 푚2(푃), +(2) 퐿 ⟨훾⟩ of 2-rank 7, with 훾 a graph automorphism of order 2, which also stabilises 푃 and +푚2(푃 ⟨훾⟩) = 7, + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +21 +(3) 퐿 ⟨휙⟩ of 2-rank 7, with 휙 a field automorphism of order 2 (휖 = 1), which also stabilises 푃 +and 푚2(푃 ⟨휙⟩) = 7, +(4) 퐿 ⟨훾휙⟩ of 2-rank 7, with 훾휙 a graph-field automorphism of order 2 (휖 = 1), which also +stabilises 푃 and 푚2(푃 ⟨훾휙⟩) = 7, +(5) 퐿 ⟨훾, 휙⟩ of 2-rank 8, with 휙 a field automorphism of order 2 (휖 = 1) commuting with 훾 a +graph automorphism of order 2, and ⟨훾, 휙⟩ also stabilises 푃 with 푚2(푃 ⟨훾, 휙⟩) = 8. +From this, we conclude that any 2-extension of the adjoint versions of 퐸 휖 +6 (푞) satisfies (QD)2. +By the remark at the beginning of the proof, we conclude that any version of 퐸 휖 +6 (푞) satisfies +(E-(QD)). +□ +5.4. Case 퐸7(푞). +Proposition 5.7. Let 퐿 = 퐸7(푞) (adjoint version), with 푞 odd. Then 퐿 satisfies (E-(QD)). +Proof. Let 퐿 = 퐸7(푞). By Table 3, if 휙 denotes a field automorphism of order 2 of 퐿, the +2-extensions of 퐿 are: +퐿, +Inndiag(퐿), +퐿 ⟨휙⟩ . +Note that Inndiag(퐿) ⟨휙⟩ is not a 2-extension since field and diagonal automorphisms of order 2 +do not commute in view of Lemma 2.6. +Next, we study the 2-ranks of these extensions, so we need to understand the centralisers of +the outer involutions. From Table 3, 푚2(퐿) = 8. We claim that 푚2(Inndiag(퐿)) = 8 = 푚2(퐿). +Indeed, consider 퐾 = 퐸7(푞2) in adjoint version. Then 푚2(퐾) = 8. Let 휙′ be a field automorphism +of order 2 for 퐾. Then, by Proposition 2.5 and Lemma 2.6, +퐾 ≥ 퐶퐾 (휙′) = 퐶Inndiag(퐾) (휙′) = Inndiag(퐸7(푞)) � Inndiag(퐿). +From this we see that 푚2(Inndiag(퐿)) = 8 = 푚2(퐿). In particular, Inndiag(퐿) satisfies (QD)2 if +퐿 does. Moreover, this also proves that if 휙 is a field automorphism of order 2 for 퐿 then +푚2(Inndiag(퐿) ⟨휙⟩) = 9 = 푚2(퐿 ⟨휙⟩). +From these observations, we conclude that, in order to establish (E-(QD)) for 퐸7(푞), it is +enough to show that 퐸7(푞) and 퐸7(푞) ⟨휙⟩ satisfy (QD)2. +To this end, we exhibit a maximal parabolic subgroup of 퐸7(푞) of 2-rank 8. We see that 퐷6 is +a subdiagram of 퐸7, so we have a maximal parabolic subgroup in 퐸7(푞) of the form +푃 = 푈 : (퐷6(푞).(푞 − 1)). +Here 푈 denotes the unipotent radical of 푃, and the subgroup 퐻 = 퐷6(푞) is a quotient of Spin+ +12(푞) +by a central subgroup of order 2. Indeed, 퐻 = HSpin+ +12(푞) and it lies in the centraliser of the +involution that generates the centre of a Sylow 2-subgroup 푇 of 퐿 (see the 푡1 involution of the 퐸7(푞) +entry in Table 4.5.1 of [8]). From this, we show that the Levi complement 퐿푃 = 퐷6(푞).(푞−1) of 푈 +has 2-rank 8. Let 푡 be the involution in the centre of 퐿푃. Then 퐶퐿(푡) = (SL2(푞) ◦2 HSpin+ +12(푞)).2 +by Table 4.5.1 of [8]. Since 푡 ∈ 푍(푇), 푇 ≤ 퐶퐿(푡). Also, SL2(푞) has a unique involution, so the 2- +rank of푇 is realised in a subgroup of the extension 푀 := HSpin+ +12(푞).2. The 2 here at the end comes +from diagonal automorphisms of the half-spin group, as in the Levi complement above. Therefore, +if we identify 푀 as a subgroup of 퐿푃, we conclude that 푚2(퐿푃) = 푚2(푀) = 푚2(퐸7(푞)). +Moreover, after suitable choices of conjugates, a field automorphism 휙 of order 2 must normalise +푃 and act as a field automorphism on our 푀. Since 퐶푀 (휙) contains a subgroup isomorphic to +HSpin+ +12(푞1/2).2, we see that 푃 ⟨휙⟩ has 2-rank 9, which is the 2-rank of the 2-extension 퐸7(푞) ⟨휙⟩. + +22 +KEVIN I. PITERMAN +By Proposition 3.5, 퐿 and 퐿 ⟨휙⟩ satisfy (QD)2. Finally, by the previous discussion, we conclude +that 퐿 satisfies (E-(QD)). +□ +5.5. Case 퐸8(푞). +Proposition 5.8. The simple group 퐸8(푞), 푞 ≠ 3, 9 odd, satisfies (E-(QD)). Also 퐸8(9) satisfies +(QD)2. +Proof. Let 퐿 = 퐸8(푞). By Table 5.1 of [11], 퐿 contains a maximal subgroup +퐻 � (3, 푞 − 1).(PSL3(푞) × 퐸6(푞)).(3, 푞 − 1).2. +Note that 퐹∗(퐻) = (3, 푞 − 1).(PSL3(푞) × 퐸6(푞)), and 퐻+ := 퐻/푍(퐹∗(퐻)) = (PSL3(푞) × +퐸6(푞)).(3, 푞 − 1).2, where (3, 푞 − 1) induces diagonal automorphism on each component of 퐻+, +and the 2 induces a graph involution, also acting on both components. In particular, by taking the +centraliser of a graph involution on the PSL3(푞) component, we see that 퐻0 contains a subgroup +퐾0 isomorphic to +PGL2(푞) × Inndiag(퐸6(푞)) ⟨훾⟩ , +where 훾 is a graph involution of 퐸6(푞) centralising PGL2(푞). Now, recall that 푚2(퐿) = 9 and +푚2(PGL2(푞)) = 2. Since 푚2(퐸6(푞) ⟨훾⟩) = 7 by item (2) of the proof of Proposition 5.6, we see +that +푚2(퐾0) = 푚2(PGL2(푞)) + 푚2(퐸6(푞) ⟨훾⟩) = 2 + 7 = 9 = 푚2(퐿). +Therefore 퐾0 realises the 2-rank of 퐿. +By Table 3, 퐸8(푞) extended by a field automorphism of order 2, say 휙, is the unique nontrivial +2-extension. From the construction of the maximal subgroup 퐻 and 퐾0 (cf. [11]), we can pick a +suitable 퐿-conjugate of 휙 (and we suppose it is the same 휙) such that it normalises 퐻 and, after +passing to the quotient, normalises 퐾0 and induces a field automorphism on both factors of 퐾0. In +particular, we have a subgroup 퐾1 of 퐾0 ⟨휙⟩ of the form +PGL2(푞1/2) × Inndiag(퐸6(푞)) ⟨훾′, 휙⟩ , +where we have chosen 훾′ ∈ Inndiag(퐸6(푞)) ⟨훾⟩ to be a graph automorphism commuting with 휙, +and PGL2(푞1/2) = 퐶PGL2(푞) (휙). Therefore, by item (5) in the proof of Proposition 5.6, +푚2(퐾1) = 2 + 푚2(퐸6(푞) ⟨훾′, 휙⟩) = 2 + 8 = 10. +Since 푚2(퐿 ⟨휙⟩) ≤ 푚2(퐿) + 1 = 10, we conclude that 푚2(퐾1) = 푚2(퐿 ⟨휙⟩). +Finally, note that 퐾0 and 퐾1 satisfy (QD)2 if 푞 ≠ 3, 9 respectively, by Propositions 4.2, 5.6 and +3.1. Hence, by Lemmas 3.2 and 3.3, 퐿 and 퐿 ⟨푡⟩ satisfy (QD)2 if 푞 ≠ 3, 9 respectively. +Therefore every 2-extension of 퐸8(푞) satisfies (QD)2, with the exceptions given in the state- +ment. This concludes the proof of the proposition. +□ +References +[1] M. Aschbacher and S. D. Smith. On Quillen’s conjecture for the 푝-groups complex. Ann. of Math. (2), 137(3):473– +529, 1993. 2, 3, 5, 7, 8 +[2] K. S. Brown. Euler characteristics of groups: the 푝-fractional part. Invent. Math., 29(1):1–5, 1975. 1 +[3] A. M. Cohen, M. W. Liebeck, J. Saxl, and G. M. Seitz. The local maximal subgroups of exceptional groups of +Lie type, finite and algebraic. Proc. London Math. Soc. (3), 64(1):21–48, 1992. 3 +[4] A. M. Cohen and G. M. Seitz. The 푟-rank of the groups of exceptional Lie type. Nederl. Akad. Wetensch. Indag. +Math., 49(3):251–259, 1987. 16 + +MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION +23 +[5] A. Díaz Ramos. On Quillen’s conjecture for 푝-solvable groups. J. Algebra, 513:246–264, 2018. 2, 4 +[6] A. Díaz Ramos and N. Mazza. A geometric approach to Quillen’s conjecture. J. Group Theory, 25(1):91–112, +2022. 4 +[7] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.12.1, 2022. 4 +[8] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups. Number 3. Part I. Chapter +A, volume 40 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998. +Almost simple 퐾-groups. 2, 4, 6, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21 +[9] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups. Number 8. Part III. +Chapters 12–17. The generic case, completed, volume 40 of Mathematical Surveys and Monographs. American +Mathematical Society, Providence, RI, 2018. 6 +[10] P. B. Kleidman. The maximal subgroups of the Chevalley groups 퐺2(푞) with 푞 odd, the Ree groups 2퐺2(푞), and +their automorphism groups. J. Algebra, 117(1):30–71, 1988. 3, 17, 18 +[11] M. W. Liebeck, J. Saxl, and G. M. Seitz. Subgroups of maximal rank in finite exceptional groups of Lie type. +Proc. London Math. Soc. (3), 65(2):297–325, 1992. 3, 22 +[12] M. W. Liebeck and G. M. Seitz. Maximal subgroups of exceptional groups of Lie type, finite and algebraic. Geom. +Dedicata, 35(1-3):353–387, 1990. 3 +[13] M. W. Liebeck and G. M. Seitz. The maximal subgroups of positive dimension in exceptional algebraic groups. +Mem. Amer. Math. Soc., 169(802):vi+227, 2004. 3, 19 +[14] K. I. Piterman. A stronger reformulation of Webb’s conjecture in terms of finite topological spaces. J. Algebra, +527:280–305, 2019. 19 +[15] K. I. Piterman. An approach to Quillen’s conjecture via centralisers of simple groups. Forum Math. Sigma, 9:Paper +No. e48, 23, 2021. 2 +[16] K. I. Piterman, I. Sadofschi Costa, and A. Viruel. Acyclic 2-dimensional complexes and Quillen’s conjecture. +Publ. Mat., 65(1):129–140, 2021. 2, 5 +[17] K. I. Piterman and S. D. Smith. Eliminating components in Quillen’s conjecture. J. Algebra, 607(part A):681–732, +2022. 2, 12 +[18] K. I. Piterman and S. D. Smith. Some results on quillen’s conjecture via equivalent-poset techniques. Submitted, +arXiv:2204.13055, 2022. 2, 3, 4 +[19] K. I. Piterman and V. Welker. Homotopy properties of the complex of frames of a unitary space. Submitted, +arXiv:2208.12626, 2022. 2, 13 +[20] D. Quillen. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2), 94:549–572; ibid. (2) 94 +(1971), 573–602, 1971. 1 +[21] D. Quillen. Homotopy properties of the poset of nontrivial 푝-subgroups of a group. Adv. in Math., 28(2):101–128, +1978. 1, 2, 7, 8, 13 +[22] Y. Segev and P. Webb. Extensions of 퐺-posets and Quillen’s complex. J. Austral. Math. Soc. Ser. A, 57(1):60–75, +1994. 9 +[23] S. D. Smith. Subgroup complexes, volume 179 of Mathematical Surveys and Monographs. American Mathematical +Society, Providence, RI, 2011. 2, 7 +[24] H. N. Ward. On Ree’s series of simple groups. Trans. Amer. Math. Soc., 121:62–89, 1966. 17 +[25] V. Welker. Direct sum decompositions of matroids and exponential structures. J. Combin. Theory Ser. B, +63(2):222–244, 1995. 12, 13 +Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, 35032 Marburg, Germany +Email address: piterman@mathematik.uni-marburg.de + diff --git a/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/load_file.txt b/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa6b86597fcdbb4d699dc1d87cb35879b697fd36 --- /dev/null +++ b/tNE0T4oBgHgl3EQfrwF-/content/tmp_files/load_file.txt @@ -0,0 +1,1191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf,len=1190 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='02570v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='GR] 6 Jan 2023 MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN* Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Given a finite group 퐺 and a prime 푝, let A푝(퐺) be the poset of nontrivial elementary abelian 푝-subgroups of 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The group 퐺 satisfies the Quillen dimension property at 푝 if A푝(퐺) has non-zero homology in the maximal possible degree, which is the 푝-rank of 퐺 minus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For example, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Quillen showed that solvable groups with trivial 푝-core satisfy this property, and later, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Aschbacher and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Smith provided a list of all 푝-extensions of simple groups that may fail this property if 푝 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, a group 퐺 with this property satisfies Quillen’s conjecture: 퐺 has trivial 푝-core and the poset A푝(퐺) is not contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In this article, we focus on the prime 푝 = 2 and prove that the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen’s conjecture at 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Introduction Since the early 70s, there has been a growing interest in the 푝-subgroup posets and their connections with finite group theory, the classification of the finite simple groups, finite geometries, group cohomology and representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The poset S푝(퐺) of nontrivial 푝-subgroups of a group 퐺 was introduced by Kenneth Brown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In that paper, Brown worked with the Euler characteristic χ(퐺) of groups 퐺 satisfying certain finiteness conditions andestablishedconnections between the 푝-fractional part of χ(퐺) and the 푝-subgroup structure of 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' One of the consequences of his results is the commonly known “Homological Sylow theorem”, which states that the Euler characteristic of S푝(퐺) is 1 modulo |퐺| 푝, the order of a Sylow 푝-subgroup of 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Some years later, Daniel Quillen introduced the poset A푝(퐺) of nontrivial elementary abelian 푝-subgroups of a finite group 퐺 and exhibited several applications of the topological properties of these posets [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, the study of elementary abelian 푝-subgroups goes back to Quillen’s earlier work on the Bredon cohomology of 퐺-spaces and his proof of the Atiyah-Swan conjecture, that relates the Krull dimension of a ring to the dimension of A푝(퐺) (see [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In [21], Quillen showed that S푝(퐺) and A푝(퐺) are (퐺-equivariantly) homotopy equivalent, and provided a new proof of Brown’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In fact, when 퐺 is the set of rational points of a semisimple algebraic group over a finite field of characteristic 푝, these posets are homotopy equivalent to the building of 퐺 and, hence, they have the homotopy type of a wedge of spheres of dimension 푙 − 1, where 푙 is the rank of the underlying algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Furthermore, in that case, the homology � 퐻∗(A푝(퐺)) affords the classical Steinberg module for 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 20G41, 20D20, 20D30, 05E18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 푝-subgroups, exceptional groups of Lie type, Quillen’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Supported by a postdoctoral fellowship of the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 1 2 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN Quillen also exhibited other connections between intrinsic algebraic properties of 퐺 and the topology of these posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For instance, he showed that A푝(퐺) is disconnected if and only if 퐺 contains a strongly 푝-embedded subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall that the classification of the groups with this property is indeed one of the many important steps towards the classification of the finite simple groups (see for example Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' On the other hand, Quillen proved that if 퐺 has a fixed point on A푝(퐺) (or, equivalently on S푝(퐺)), then these posets are contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that 퐺 has a fixed point if and only if its 푝-core 푂 푝(퐺) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In view of this and further evidence, Quillen conjectured that the reciprocal to this statement should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' That is, if A푝(퐺) is contractible then there is a fixed point, or, equivalently, 푂 푝(퐺) ≠ 1 (see Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9 of [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In other words, Quillen’s conjecture asserts that A푝(퐺) is contractible if and only if 푂 푝(퐺) ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A significant part of Quillen’s article is devoted to proving the solvable case of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In [21] it is shown that for a 푝-nilpotent group 퐺 with abelian Sylow 푝-subgroups and 푂 푝(퐺) = 1, A푝(퐺) is homotopy equivalent to a nontrivial wedge of spheres of the maximal possible dimension, which is 푚 푝(퐺) − 1, the 푝-rank of 퐺 minus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then, if 퐺 is any solvable group with 푂 푝(퐺) = 1, 퐺 contains a 푝-nilpotent subgroup 푂 푝′(퐺)퐴, with 퐴 ∈ A푝(퐺) of maximal 푝-rank and 푂 푝(푂 푝′(퐺)퐴) = 1, and thus � 퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Later, Michael Aschbacher and Stephen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Smith formalised this property and gave a name to it: an arbitrary group 퐺 with � 퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0 is said to satisfy the Quillen dimension property at 푝, or (QD)푝 for short (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, a solvable group 퐺 with 푂 푝(퐺) = 1 satisfies (QD)푝 and thus Quillen’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Furthermore, it was shown that 푝-solvable groups satisfy this property by using Quillen’s techniques and, in addition, the CFSG (see [5, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' These results also suggest that a stronger statement of the conjecture may hold: if 푂 푝(퐺) = 1 then � 퐻∗(A푝(퐺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Q) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, from now on, by Quillen’s conjecture we will be referring to this stronger version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' It is not hard to see that not every group 퐺 with 푂 푝(퐺) = 1 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For example, we mentioned that finite groups of Lie type in characteristic 푝 satisfy the conjecture, but since the Lie rank is usually strictly smaller than the 푝-rank, they fail (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This has led to the development of new methods to prove Quillen’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' One of the most notorious advances in the conjecture was achieved by Aschbacher-Smith in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' There, they established Quillen’s conjecture for a group 퐺 if 푝 > 5 and in addition, roughly, all the 푝-extensions of finite unitary groups PSU푛(푞), with 푞 odd and 푝 | 푞 + 1, satisfy (QD)푝 (see Main Theorem of [1] for the precise statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here, a 푝-extension of a group 퐿 is a split extension of 퐿 by an elementary abelian 푝-subgroup of Out(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In [1] it is not shown that the group 퐺 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Instead, they proved that if every 푝-extension of a fixed component of 퐺 satisfies (QD)푝, then � 퐻∗(A푝(퐺);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Q) ≠ 0 if 푂 푝(퐺) = 1 (under suitable inductive hypotheses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This result restricts the possibilities of the components of a minimal counterexample to Quillen’s conjecture: every component has a 푝-extension failing (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In view of this result and the classification of the finite simple groups, Aschbacher and Smith described for 푝 ≥ 3, all the possible 푝-extensions of simple groups which may potentially fail (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This is the (QD)-List, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1, of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, it is conjectured in [1] that the unitary groups PSU푛(푞) with 푞 odd and 푝 | 푞 + 1 should not appear in this list, and so the extra hypothesis on the unitary groups in the main result of [1] could be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Nevertheless, this problem remains open (see [19] for recent results in this direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In the last few years, there have been further developments in the Quillen conjecture [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recently, in [18], new tools for the study of the conjecture have been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For example, MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 3 it is shown that the Aschbacher-Smith general approach to the conjecture can be extended to every prime 푝 by reducing reliance on results of [1] stated only for odd primes and invoking the Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [18] shows that the Main Theorem of [1] extends to 푝 ≥ 3, keeping the additional constraint on the unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' On the other hand, for 푝 = 2, one important obstruction for this extension is the lack of a (QD)-List for this prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Roughly, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8 of [18] concludes that a minimal counterexample to Quillen’s conjecture contains a component of Lie type in characteristic 푟 ≠ 3, and every component of 퐺 has a 2-extension failing (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In view of these results on Quillen’s conjecture, in this article, we focus on showing that the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy (QD)2, with a small number of exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This improves the conclusions of [18] on Quillen’s conjecture for 푝 = 2, and allows us to conclude then that exceptional groups of Lie type in odd characteristic different from 3 cannot be components of a minimal counterexample to the conjecture (see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The main result of this article is the following theorem, whose proof is given in different propositions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 be an exceptional finite simple group of Lie type in odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' That is, 퐿 = 3퐷4(푞), 퐹4(푞), 퐺2(푞), 2퐺2(푞)′, 퐸6(푞), 2퐸6(푞), 퐸7(푞) or 퐸8(푞), with 푞 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then every 2-extension of 퐿 satisfies the Quillen dimension property at 푝 = 2, except possibly in the following cases: 3퐷4(9) extended with field automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐹4(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐹4(9) extended with field automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2-extensions of 퐺2(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐺2(9) extended with field automorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2퐺2(3)′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐸8(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐸8(9) extended with field automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, the extensions of 퐺2 (3), 퐺2(9) and 2퐺2(3)′ mentioned above do fail (QD)2 by Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To achieve the conclusions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1, in most cases we exhibit a maximal subgroup 푀 of a 2-extension 퐿퐵 of 퐿 such that 푚2(푀) = 푚2(퐿퐵) and 푀 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since there is an inclusion � 퐻푚2(퐿퐵)−1(A2(푀)) ↩→ � 퐻푚2(퐿퐵)−1(A2(퐿퐵)) in the top-degree homology groups, this establishes (QD)2 for 퐿퐵 (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In some cases, the subgroup 푀 arises from suitable parabolic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' More concretely, when it is possible, we pick 푃 to be a maximal parabolic subgroup of 퐿 which is stabilised by 퐵 and such that 푀 := 푃퐵 realises the 2-rank of 퐿퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then we get a 2-nilpotent configuration 푈퐴, where 푈 is the unipotent radical of 푃, 퐴 is an elementary abelian 2-subgroup realising the 2-rank of 푃퐵, and 푂2(푈퐴) = 퐶퐴(푈) = 1 by one of the corollaries of the Borel-Tits theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Hence, by Quillen’s results on the solvable case, 푈퐴 satisfies (QD)2, and thus also 푀 and 퐿퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' When the choice of such parabolic 푃 is not possible, we pick one of the maximal rank subgroups of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here, the components of the maximal subgroup 푀 are usually smaller exceptional groups, low-dimensional linear group 퐴1(푞) and 퐴2(푞) or unitary groups 퐴 2 (푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, we first prove that the 2-extensions of simple linear and unitary groups in dimensions 2 and 3 satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Although there is a large literature on maximal subgroups of exceptional groups of Lie type, we will only need the results from [3, 10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 and the results of [18] for 푝 = 2, we can conclude: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐺 be a minimal counterexample to Quillen’s conjecture for 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐺 contains a component of Lie type in characteristic 푟 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, every such component fails 4 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN (QD)2 in some 2-extension and belongs to one of the following families: PSL푛(2푎)(푛 ≥ 3), 퐷푛(2푎)(푛 ≥ 4), 퐸6(2푎), PSL± 푛(푞)(푛 ≥ 4), 퐵푛(푞)(푛 ≥ 2), 퐶푛(푞)(푛 ≥ 3), 퐷± 푛(푞)(푛 ≥ 4), where 푞 = 푟푎 and 푟 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The 2-extensions of PSL2(푞), PSL3(푞) and PSU3(푞) satisfy (QD)2 by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6 respectively, with exceptions when 푞 = 3, 5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Nevertheless, the results of [18] eliminate these possibilities from a minimal counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Further results on the Quillen dimension property at 푝 = 2 for the classical groups could be pursued by combining the methods presented in this article with the results of [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In Section 2, we set the notation and conventions that we will need to work with the finite groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We also provide some useful properties to work out the 푝-extension and compute 푝-ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In Section 3 we gather previous results on the Quillen dimension property and related tools that will help us establish this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then in Section 4 we establish (QD)2 for some 2-extensions of linear groups and recall the structure of the centralisers of graph automorphisms, following Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In Section 5 we prove each case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' All groups considered in this article are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We suppress the notation for the homology coefficients, and we assume that they are always taken over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The interested reader may note that our results can be extended to homology with coefficients in other rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, we emphasise that we adopt the language and conventions of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This is particularly important when we name the different types of automorphisms of groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Computer calculations were performed with GAP [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The author thanks Stephen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Smith for many helpful discussions con- cerning the algebraic properties of groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' He also thanks Volkmar Welker for his suggestions on a preliminary version of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Preliminaries We assume that the reader is familiar with the construction of the finite groups of Lie type as fixed points of Steinberg endomorphisms, and the basic properties concerning root systems of reductive algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will follow the language of [8], which also contains the required background on finite groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In this section, we will only recall some notations and names, and state results that will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We denote by C푛, D푛, Sym푛 and Alt푛 the cyclic group of order 푛, the dihedral group of order 푛, the symmetric group on 푛 points and the alternating group on 푛 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐺 is a group, then Aut(퐺), Inn(퐺) and Out(퐺) denote the automorphism group, the group of inner automorphisms and the outer automorphism group of 퐺 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We denote by 푍(퐺) the centre of 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We usually write 퐺 : 퐻, or simply 퐺퐻, for a split extension of 퐺 by 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' When an extension of 퐺 by 퐻 may not split, we denote it by 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By an element 푔 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' a subgroup 퐵) of 퐺 inducing outer automorphisms on 퐿 ≤ 퐺 we mean that 푔 embeds into Aut(퐿) \\ Inn(퐿) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐵 embeds in Aut(퐿) with 퐵 ∩ Inn(퐿) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, 퐻 ◦푚 퐾 denotes a central product of 퐻 and 퐾 by a central cyclic subgroup of order 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' That is, 퐻 ◦푚 퐾 = (퐻 × 퐾)/C푚 where C푚 embeds into both 푍(퐻) and 푍(퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 5 We will usually use the notation 푛 in a group extension to denote a cyclic group of order 푛, and 푛푚 a direct product of 푚 copies of cyclic groups of order 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A number between brackets [푛] in the structure description of an extension means some group of order 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In this article, we are mainly interested in extensions by elementary abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Below we recall the definition of 푝-extension given in the introduction and introduce some useful notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 be a finite group and 푝 a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A 푝-extension of 퐿 is a split extension 퐿퐵 of 퐿 by an elementary abelian 푝-group 퐵 inducing outer automorphisms on 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐿 ≤ 퐺, we denote by O퐺(퐿) the poset of elements 퐵 ∈ A푝(퐺) such that 퐵 ∩ 퐿퐶퐺(퐿) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus O퐺(퐿) is the set of 퐵 ∈ A푝(퐺) inducing outer automorphisms on 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We write O2(퐿) for OAut(퐿) (퐿) at 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We also let ˆO퐺(퐿) = O퐺(퐿) ∪ {1} and ˆO2(퐿) = O2(퐿) ∪ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For a prime number 푝, we say that a group 퐺 satisfies the Quillen dimension property at 푝 if A푝(퐺) has non-zero homology in dimension 푚 푝(퐺) − 1, where 푚 푝(퐺) denotes the 푝-rank of 퐺: (QD)푝 � 퐻푚푝 (퐺)−1(A푝(퐺)) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A remarkable study of the Quillen dimension property for odd primes 푝 was carried out in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This theorem contains a list of the potential 푝-extensions of simple groups that might fail (QD)푝, for 푝 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, this list contains the 푝-extensions of unitary groups PSU푛(푞) with 푞 odd and 푝 | 푞 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' However, Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [1] basically claims that these groups should not belong to this list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In fact, it is shown there that if 푛 < 푞(푞 − 1) then these 푝-extensions satisfy (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Nevertheless, this problem remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The aim of this article is to achieve some progress on a similar list for the prime 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, we will focus on showing that 2-extensions of certain simple groups satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To that end, we introduce the following convenient definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A group 퐿 satisfies (E-(QD)) if every 2-extension of 퐿 satisfies (QD)2: (E-(QD)) For every 퐵 ∈ ˆO2(퐿), 퐿퐵 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In order to establish (QD)푝 for 푝-extensions, it is crucial to be able to compute 푝-ranks of extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following result, extracted from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 in [16], will be a useful tool to compute 푝-ranks of extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 (푝-rank of extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐺 = 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='퐾 be an extension of finite groups, and let 푝 be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푚 푝(퐺) = max 퐴∈S �푚 푝(퐶푁 (퐴)) + 푚 푝(퐴)�, where S = {퐴 ∈ A푝(퐺) ∪ {1} : 퐴 ∩ 푁 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, 푚 푝(퐺) ≤ 푚 푝(푁) + 푚 푝(퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will implicitly use this result at many points of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that, in order to apply this lemma, we should be able to compute centralisers of elementary abelian 2-subgroups, usually inducing outer automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will often proceed as follows: if 퐿퐵 is a 2-extension of 퐿, then take a suitable decomposition 퐵 = 퐵0 ⊕ 퐵1, with |퐵1| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that we can inductively compute the 2-rank of 퐿퐵0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1) 푚2(퐿퐵) = max � 푚2(퐿퐵0), 1 + 푚2(퐶퐿퐵0(푡)) : 푡 ∈ 퐿퐵 \\ 퐿퐵0 is an involution � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, this computation depends only on the conjugacy classes of the involutions 푡, and, in most of the cases that we are interested in, such classes are completely classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 6 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN Now we recall, rather informally, the names of the different types of automorphisms of a simple group of Lie type 퐾 defined over a field of odd characteristic, following Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='13 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We refer to [8] for the full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푡 ∈ Aut(퐾) be an involution and 퐾∗ = Inndiag(퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then we have the following names for 푡: (1) inner-diagonal if 푡 ∈ 퐾∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) inner if 푡 ∈ Inn(퐾);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3) diagonal if 푡 ∈ 퐾∗ \\ Inn(퐾);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4) field automorphism if 푡 ∈ Aut(퐾) \\ 퐾∗ is Aut(퐾)-conjugated to a field automorphism of the ground field and 퐾 is not 2퐴푛(푞), 2퐷푛(푞) or 2퐸6(푞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5) graph if 푡 ∈ Aut(퐾) \\ 퐾∗, roughly, is Aut(퐾)-conjugated to an involution arising as an automorphism of the underlying Dynkin diagram (except for 퐾 = 퐺2(푞)), or else from a field automorphism in cases 2퐴푛(푞), 2퐷푛(푞) and 2퐸6(푞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' and (6) graph-field automorphism if it can be expressed as a product 푔 푓 of a graph involution 푔 and a field automorphism 푓 , or else 퐾 = 퐺2(푞) and 푡 arises from a Aut(퐾)-conjugated of an involution automorphism of the underlying Dynkin diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' It follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8] that the centralisers of field involutions 푡 verify that 푚2(퐶퐾 (푡)) = 푚2(퐾) and 푚2(퐶퐾 ∗(푡)) = 푚2(퐾∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1), we see that 푚2(퐾 ⟨푡⟩) = 푚2(퐾) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Below we reproduce a simplified version of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐾 = 푑Σ(푞) be a group of Lie type in adjoint version in characteristic 푟, and let 푥 be a field or graph-field automorphism of prime order 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Set 퐾푥 = 푂푟′(퐶퐾 (푥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the following hold: (1) If 푥 is a field automorphism then 퐾푥 � 푑Σ(푞1/푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) If 푥 is a graph-field automorphism then 푑 = 1, 푝 = 2 or 3, and 퐾푥 � 푝Σ(푞1/푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3) 퐾푥 is adjoint and 퐶Inndiag(퐾) (푥) � Inndiag(퐾푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4) Fields (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' graph-field) automorphism are all Inndiag(퐾)-conjugated, except for graph- fields for 퐾 = 퐷4(푞) and 푝 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The previous proposition does not determine, a priori, the structure of 퐶퐾 (푥), but just of the centraliser taken over the inner-diagonal automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since we are interested in computing 푚2(퐶퐾 (푥)), it will be crucial for us to decide when a diagonal involution can centralise a field or graph-field automorphism 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We recall below Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8 of [9, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 17], which provides a partial solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐾 � PSL2(푞), PΩ2푛+1(푞), PSp2푛(푞) or 퐸7(푞), where 푞 is apower of an odd prime 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 휙 be a field automorphism of order 2, and let 퐾휙 = 푂푟′(퐶퐾 (휙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then Inndiag(퐾휙) = 퐶Inndiag(퐾) (휙) = 퐶Inn(퐾) (휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, 휙 does not commute with diagonal involutions of Inndiag(퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will mainly work with Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8] to compute the 2-ranks of extensions by diagonal and graph involutions, mostly for the groups of type 퐴± 푚(푞) and the exceptional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In the next paragraph, we briefly and informally describe how to read such table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' See [8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 171-182] for a complete and accurate description of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This table records the 퐾∗-conjugacy classes of inner-diagonal and graph involutions 푡 of a finite group of Lie type 퐾 in adjoint version, and the structure of their centralisers when taken over 퐾∗ = Inndiag(퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The centraliser of an involution 푡 is denoted by 퐶∗ = 퐶퐾 ∗(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The first column of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 denotes the family for which the involutions are listed (퐴푛, 퐵푛, 퐶푛, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=') The second MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 7 column indicates the restrictions for these classes to exist, while the third column is a label for the conjugacy class of that involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For the purposes of this article, we will not need to interpret the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In the fifth column, it is indicated when such classes are of inner type (denoted by 1), diagonal type (denoted by 푑) or graph type (several notations like 푔, 푔′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The notation 1/푑 indicates that it is inner if the condition inside the parentheses at the right holds, and it is diagonal otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From the sixth column to the end, the structure of the centraliser 퐶∗ is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Roughly, 퐶∗ is an extension of a central product of groups of Lie type 퐿∗ = 푂푟′(퐶∗) (column six), whose versions are specified in the column “version” and whose centres can be recovered from the column 푍(퐿∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' An extra part centralising this product can be computed from the column 퐶퐶∗◦(퐿∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here 퐶∗◦ = 퐿∗푇∗ is the connected-centraliser, where 푇∗ is a certain 푟′-subgroup arising from a torus 푇 normalised by 푡 and inducing inner-diagonal automorphisms on 퐿∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From the columns 퐿∗, version, 푍(퐿∗) and 퐶퐶∗◦(퐿∗), one can compute the “inner-part” of 퐶∗◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, from the last two columns we can recover the outer automorphisms of 퐿∗ arising in 퐶∗◦ (in general of diagonal type) and the remaining part of 퐶∗/퐶∗◦, which is often an involution acting on the components of 퐿∗ (as field or graph automorphism, or by switching two components) and on the central part 퐶퐶∗(퐿∗) (which is usually cyclic and the involution acts by inversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To recover the action of the last column, the symbols 푖, ↔, 휙, 훾, 1 mean, respectively, an action by inversion, a swap of two components, a field automorphism of order 2, a graph automorphism of order 2, and an inner action on a component or trivial action on 퐶퐶∗◦ (퐿∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Tools to achieve (QD)푝 In this section, we provide tools and collect results that will help us to establish (QD)2 on certain 2-extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Many of these tools were introduced and exploded by Aschbacher-Smith to determine the (QD)-list in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following proposition is an easy consequence of the Künneth formula for the join of spaces and the fact that A푝(퐻 × 퐾) ≃ A푝(퐻) ∗ A푝(퐾) (see [21, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 푝 is a prime and 퐻, 퐾 satisfy (QD)푝, then 퐻 × 퐾 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following lemma corresponds to Lemmas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='11 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='12 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푁 ≤ 퐺 be such that 푁 ≤ 푂 푝′(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then there is an inclusion � 퐻∗(A푝(퐺/푁)) ⊆ � 퐻∗(A푝(퐺)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, 푚2(퐺) = 푚2(퐺/푁), and if 퐺/푁 satisfies (QD)푝 then so does 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 푁 ≤ 푍(퐺), then indeed A푝(퐺) ≡ A푝(퐺/푁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following observation is an easy consequence of the inclusion between the homology groups of top-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐻 ≤ 퐺 be such that 푚 푝(퐻) = 푚 푝(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐻 satisfies (QD)푝, then so does 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, we recall one of the essential results on the Quillen dimension property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 (Quillen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐺 is a solvable group with 푂 푝(퐺) = 1, then 퐺 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This theorem settles the solvable case of Quillen’s conjecture (see [21, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Later, it was extended to the family of 푝-solvable groups by using the CFSG if 푝 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We refer to Chapter 8 of [23] for further details on Quillen’s conjecture and the Quillen dimension property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 8 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN In view of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 and the Inclusion Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3, it is convenient to look for solvable subgroups of 퐺 with maximal 푝-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Some standard solvable subgroups in a group of Lie type 퐿 arise by taking extensions of unipotent radicals by elementary abelian subgroups of their normalisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' These extensions lie then inside parabolic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following result on parabolic subgroups will help us to achieve (E-(QD)) for arbitrary groups of Lie type (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Step v at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='506 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 be a simple group of Lie type, and 푝 a prime not dividing the characteristic of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that 퐿퐵 is a 푝-extension of 퐿 and that there exists a 퐵-invariant parabolic subgroup 푃 ≤ 퐿 such that 푚 푝(퐿퐵) = 푚 푝(푃퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿퐵 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푅 := 푂푟 (푃), where 푟 is the characteristic of the ground field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then, as a consequence of the Borel-Tits theorem, 퐶Aut(퐿) (푅) ≤ 푅 (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, if 푇 ≤ 푃퐵 realises the 푝-rank of 푃퐵, then 푇 normalises 푅, and 퐶푇 (푅) ≤ 푅 ∩ 푇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This means that 푇 is faithful on 푅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 푂 푝(푅푇) = 1, and 푚 푝(푅푇) = 푚 푝(푃퐵) = 푚 푝(퐿퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푅푇 is a solvable group with trivial 푝-core, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4, 푅푇 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3, 퐿퐵 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 be a simple group of Lie type defined in odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that 푃 is a proper parabolic subgroup of 퐿 containing a Sylow 2-subgroup of 퐿 (that is, |퐿 : 푃| is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿 and the extension of 퐿 by a field automorphism of order 2 satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 and 푃 be as in the hypotheses of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푃 has odd index in 퐿, it contains a Sylow 2-subgroup of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 푚2(푃) = 푚2(퐿) and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5, 퐿 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, let 퐵 ∈ ˆO2(퐿) be cyclic inducing field automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By passing through algebraic groups and root systems, it can be shown that 퐵 normalises some conjugate of 푃, which we may assume is 푃 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, after conjugation, we suppose that 퐵 ≤ 푁Aut(퐿) (푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that a Sylow 2-subgroup of 푃퐵 is a Sylow 2-subgroup of 퐿퐵, so 푚2(푃퐵) = 푚2(퐿퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5, 퐿퐵 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ We close this section with a few more results on low 푝-ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following lemma follows from the 푝-rank 2 case of Quillen’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' See [21, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If A푝(퐺) is connected, 푚 푝(퐺) = 2 and 푂 푝(퐺) = 1, then 퐺 satisfies (QD)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' It will be convenient to recall the classification of groups with a strongly 2-embedded subgroup, that is, those groups with disconnected 2-subgroup poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' See [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1] and [21, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푝 = 2 and 퐺 be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then A2(퐺) is disconnected if and only if 푂2(퐺) = 1 and one of the following holds: (1) 푚2(퐺) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) Ω1(퐺)/푂 푝′(Ω1(퐺))) � PSL2(2푛), PSU3(2푛) or Sz(22푛−1) for some 푛 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, from the isomorphisms among the simple groups, we see that Alt5 � PSL2(5) � PSL2(22), 2퐺2(3)′ � PSL2(23), are included in the list of item (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, sometimes in low dimensions, we will be able to conclude (QD)푝 by computing the sign of the Euler characteristic of A푝(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, we will use the following well-known expression of this invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 9 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The reduced Euler characteristic of A푝(퐺) is: �χ(A푝(퐺)) = � 퐸 ∈A푝 (퐺)/퐺∪{1} (−1)푚푝 (퐸)−1푝( 푚푝 (퐸) 2 )|퐺 : 푁퐺(퐸)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, the next lemma will help us to produce non-zero homology by inductively looking into the homology of the Quillen poset of a certain normal subgroup and centralisers of outer elements acting on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The main reference for this lemma is [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐺 be a finite group and 푝 a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that 퐿 ⊴ 퐺 is a normal subgroup such that O퐺(퐿) consists only of cyclic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then we have a long sequence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' → � 퐻푚+1(A푝(퐺)) → � 퐵∈O퐺 (퐿) � 퐻푚(A푝(퐶퐿(퐵))) 푖∗→ � 퐻푚(A푝(퐿)) 푗∗→ � 퐻푚(A푝(퐺)) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' where 푖∗ and 푗∗ are the natural maps induced by the inclusions A푝(퐶퐿(퐵)) ⊆ A푝(퐿) and A푝(퐿) ⊆ A푝(퐺), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, the following hold: (1) Let 푋 be the union of the subposets A푝(퐶푁 (퐵)) for 퐵 ∈ O퐺(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We have indeed a factorisation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1) � 퐵∈O퐺 (퐿) � 퐻푚(A푝(퐶퐿(퐵))) 푖∗ � 푖′∗ �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ � 퐻푚(A푝(퐿)) � 퐻푚(푋) 푘∗ �r r r r r r r r r r where also 푖′ ∗ and 푘∗ are induced by the inclusions A푝(퐶퐿(퐵)) ⊆ 푋 and 푋 ⊆ A푝(퐿), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) 푚 푝(퐺) ≤ 푚 푝(퐿) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3) If � 퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) = 0 for all 퐵 ∈ O퐺(퐿), then 퐻푚푝 (퐿) (A푝(퐺)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4) We have a bound dim 퐻푚푝 (퐿) (A푝(퐺)) ≥ � 퐵∈O퐺 (퐿) dim � 퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) − dim � 퐻푚푝 (퐿)−1(푋) ≥ � 퐵∈O퐺 (퐿) dim � 퐻푚푝 (퐿)−1(A푝(퐶퐿(퐵))) − dim � 퐻푚푝 (퐿)−1(A푝(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5) If 푚 푝(퐺) = 푚 푝(퐿) + 1 and 퐺 fails (QD)푝, then, for 푚 = 푚 푝(퐿) − 1, we get inclusions � 퐵∈O퐺 (퐿) � 퐻푚(A푝(퐶퐿(퐵))) ↩→ � 퐻푚(푋) ↩→ � 퐻푚(A푝(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The long exact sequence arises from the main result of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1) in item (1) is an immediate consequence of this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Item (2) holds by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Items (3-5) follow by looking into the last terms of the long exact sequence, at 푚 = 푚 푝(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Some linear groups satisfy (QD)2 In this section, we prove that the linear groups PSL2(푞) and PSL3(푞) satisfy (E-(QD)) for every 푞, with a few exceptions for 푞 = 3, 5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' These cases will serve as basic cases for the exceptional groups, where we will occasionally find linear groups as direct factors in some of their maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5], we recall the 2-ranks of the small dimensional linear groups: 10 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 푞 is a power of an odd prime and 푛 = 2, 3, then PSL± 푛(푞) and PGL± 푛(푞) have 2-rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We begin by studying the linear group of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 � PSL2(푞) with 푞 odd and 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then every 2-extension 퐿퐵 of 퐿 satisfies (QD)2, with the following exceptions: (1) 퐿 � PSL2(5), 퐵 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) 퐿 � PSL2(9), 퐵 induces field automorphisms of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, every 2-extension of Inndiag(퐿) � PGL2(푞) satisfies (QD)2, except in case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We consider the possible 2-extensions of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In any case, we know that 퐿 is simple and that Out(퐿) = C2 × C푎, where C2 � Outdiag(퐿) and C푎 is the group of field automorphisms of F푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that 휙 is an order 2-field automorphisms of F푞 (if it exists), and that 푑 ∈ Inndiag(퐿) \\ 퐿 is a diagonal involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the 2-extensions of 퐿 are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This table follows since 2-extension 퐿퐵 퐶퐿(퐵) 푚2(퐿퐵) 퐵 = 1 퐿 2 퐵 = ⟨휙⟩ PGL2(푞1/2) 3 퐵 = ⟨푑⟩ D푞+휖 2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2-extensions of PSL2(푞), 푞 ≥ 5 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here 푞 ≡ 휖 (mod 4), 휖 ∈ {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' every involution of Aut(PSL2(푞)) − PGL2(푞) is a field automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall also that field and diagonal automorphisms of order 2 do not commute by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6 The structure of the centraliser for 푑 follows from the first row of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, observe that 퐿 ⟨푑⟩ = Inndiag(퐿) and 푚2(Inndiag(퐿) ⟨휙⟩) = 3 since 푚2(퐿) = 푚2(Inndiag(퐿)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We prove that each 2-extension of 퐿 satisfies (QD)2 by computing the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' First, 2-extensions 퐿퐵 and Inndiag(퐿) ⟨휙⟩ have connected A2-poset by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8, except for 퐿 = PSL2(5), 퐵 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='7, 퐿 and Inndiag(퐿) satisfy (QD)2, except for 퐿 = PSL2(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that A2(PSL2(5)) = A2(Alt5) = A2(PSL2(4)) is homotopically discrete with 5 points, and the 2-extension PGL2(5) � Sym5 does satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This yields the conclusions of the statement for the case 푞 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next we show (QD)2 for the 2-extensions 퐿 ⟨휙⟩ and Inndiag(퐿) ⟨휙⟩, both of 2-rank 3 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, it is enough to show that 퐿 ⟨휙⟩ satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In order to do this, we compute the dimensions of 퐻1(A2(퐿)) and 퐻1(A2(퐶퐿(휙))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since in this situation 푞 is a square, 푞 ≠ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Second, if 푞 = 25, 퐶퐿(휙) = PGL2(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Hence, in any case, the dimension of these degree 1 homology groups can be computed from the reduced Euler characteristic of the underlying A2-poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here we use the formula given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, for 퐾 = 퐿 or 퐶퐿(휙), dim 퐻1(A2(퐾)) = −�χ(A2(퐾)) = 1 − # of involutions in 퐾 + 2 · # of 4-subgroups of 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1) In Table 2 we describe these numbers: Proof of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The number of involutions and 4-subgroups of PSL2(푞) follows from Dickson’s classification of the subgroups of PSL2(푞) (see also Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 11 Group Number of involutions Number of 4-subgroups PSL2(푞) 푞(푞+휖 ) 2 푞(푞2−1) 24 PGL2(푞) 푞2 푞(푞2−1) 6 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here 푞 ≡ 휖 (mod 4), 휖 ∈ {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The number of involutions of PGL2(푞) follows since there is a unique conjugacy class of diagonal involutions 푑 by Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, the number of elements in such conjugacy class is equal to 푞(푞−휖 ) 2 , which gives 푞2 after adding the number of involutions in PSL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, to compute the number of four-subgroups of PGL2(푞) we proceed as follows: each four-subgroup of PGL2(푞) is either contained in PSL2(푞) or else it contains a unique involution of PSL2(푞) and 2 diagonal involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, for a given diagonal involution 푑, there is a one-to-one correspondence between 4-subgroups containing 푑 and involutions in 퐶퐿(푑) � D푞+휖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This shows that each diagonal involution is contained in (푞 + 휖)/2 4-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since we have 푞(푞−휖 ) 2 diagonal involutions, the total number of 4-subgroups in PGL2(푞) containing diagonal involutions is 푞(푞 − 휖) 2 (푞 + 휖) 2 1 2 = 푞(푞2 − 1) 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus the total number of 4-subgroups in PGL2(푞) is 푞(푞2 − 1) 24 + 푞(푞2 − 1) 8 = 푞(푞2 − 1) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This completes the proof of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ Indeed, by Table 2, we get concrete values for the dimensions of the degree 1 homology groups of A2(PSL2(푞)) and A2(PGL2(푞)): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2) dim 퐻1(A2(PSL2(푞))) = −�χ(A2(PSL2(푞))) = 1 12 (푞 − 휖)(푞2 − (6 − 휖)푞 − 휖12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3) dim 퐻1(A2(PGL2(푞))) = −�χ(A2(PGL2(푞))) = 1 3 (푞 − 3)(푞2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now we need to describe the number of field automorphisms in PSL2(푞) ⟨휙⟩ and in PGL2(푞) ⟨휙⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall that the fieldautomorphisms of PSL2(푞) ⟨휙⟩ are all PGL2(푞)-conjugated, withcentraliser 퐶PGL2(푞) (휙) = 퐶PSL2(푞) (휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, the number of field automorphisms of order 2 in PSL2(푞) ⟨휙⟩ is exactly | PGL2(푞)| |퐶PSL2(푞) (푞)| = 푞(푞2 − 1) 푞1/2(푞 − 1) = 푞1/2(푞 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This gives 푞1/2(푞 + 1) involutions in PSL2(푞) ⟨휙⟩ \\ PSL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = PSL2(푞), 퐵 = ⟨휙⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, the values in Table 2 and formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1), we conclude that: dim 퐻2(A2(퐿퐵)) ≥ 푞1/2(푞 + 1) dim 퐻1(A2(PGL2(푞1/2))) − dim 퐻1(A2(PSL2(푞))) = 푞1/2(푞 + 1) 1 3 (푞1/2 − 3)(푞 − 1) − 1 12 (푞 − 1)(푞2 − 5푞 − 12) = 1 4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that 푞 ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The above number is positive for all 푞 ≥ 13, which is our case since 푞 is an even power of an odd prime and 푞 ≠ 9 by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We conclude that 퐿퐵 = PSL2(푞) ⟨휙⟩ 12 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then also PGL2(푞) ⟨휙⟩ satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, dim 퐻2(A2(PGL2(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSL2(푞) ⟨휙⟩)) ≥ 1 4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4) We have shown that every possible 2-extension of PSL2(푞) and PGL2(푞) satisfies (QD)2, except for the cases described in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ We note that the excepted cases in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 actually fail (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, PSL2(5) fails (QD)2 since it has 2-rank 2 and A2(PSL2(5)) = A2(PSL2(4)) is homotopically discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following example provides the details that show that PSL2(9) ⟨휙⟩ and PGL2(9) ⟨휙⟩ fail (QD)2, where 휙 is a field automorphism of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = PSL2(9) and let 퐴 = Aut(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐴/퐿 � C2 × C2, so every 2-extension of 퐿 is a nontrivial normal subgroup of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This gives 3 possible 2-extensions of 퐿, but not 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 휙 be a field automorphism of 퐿 and 푑 a diagonal automorphism of 퐿, both of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the possible 2-extensions of 퐿 are: (1) 퐿, with 2-rank 2, satisfies (QD)2 with 퐻1(A2(퐿)) of rank 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) 퐿 ⟨휙⟩, with 2-rank 3, fails (QD)2 since 퐶퐿(휙) � Sym4, which has nontrivial 2-core 푂2(퐶퐿(휙)) � C2 × C2 ≠ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3) 퐿 ⟨푑⟩ = PGL2(9), with 2-rank 2, satisfies (QD)2 with 퐻1(A2(퐿) ⟨푑⟩) of rank 160 and 퐶퐿(푑) � D10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that Aut(퐿) has 2-rank 3 and does not satisfy (QD)2, and it is not a 2-extension of 퐿 since diagonal and field automorphisms do not commute in Aut(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also PGL2(9) ⟨휙⟩ fails (QD)2 since 퐶PGL2(9) (휙) = 퐶퐿(휙) has nontrivial 2-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' There is also a remaining almost simple group 푁 with 퐿 < 푁 < Aut(퐿), not contained in the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This is the extension 푁 = PSL2(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 � Alt6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, and it satisfies that A2(푁) = A2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, although this group 푁 is not a 2-extension of 퐿, it is a “non-split 2-extension”, and it does satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, these computations show that A2(퐿) ↩→ A2(Aut(퐿)) induces an inclusion in homology, and hence a non-zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By the main result of [17], PSL2(9) is not a component of a minimal counterexample to Quillen’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Our next aim is to show that 2-extensions of PSL3(푞) satisfy (QD)2, with only a few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will needthe followinglemmawhichrecords the values of the Euler characteristic of the Quillen poset of some linear groups and the unitary groups in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For 퐿 = PSL푛(푞) and 푛 odd, we have �χ(A2(퐿)) = �χ(A2(PGL푛(푞))) = (−1)푛 푛 푛−1 � 푖=1 (푞푖 − 1) 푓푛(푞), where 푓푛(푞) denotes a polynomial as described in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For instance, 푓3(푞) = 푞3 + 3푞2 + 3푞 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, since A2(퐿) is Cohen-Macaulay of dimension 푛 − 2, the above Euler characteristic computes the dimension of 퐻푛−2(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐿 = PSU3(푞), then �χ(A2(퐿)) = �χ(A2(PGU3(푞))) = 1 3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The value of the Euler characteristic for PGL푛(푞) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 and The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 of [25] (note that there is a typo in the formula of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4, and the product over 푖 should be up to 푟 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also, since 푛 is odd, by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5 of [19], A2(PSL푛(푞)) = A2(PGL푛(푞)) = A2(GL푛(푞))>푍 where 푍 is the cyclic subgroup of order 2 of 푍(GL푛(푞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By [21] (see also [25]), A2(PSL푛(푞)) is Cohen-Macaulay of dimension 푛 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The formula for PGU3(푞) follows from Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6 of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ Next, we show that the 2-extensions of PSU3(푞) satisfy (QD)2, except for 푞 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' These cases will be important during our analysis for PSL3(푞), especially when working with 2-extensions by graph-field automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = PSU3(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿 satisfies (E-(QD)) if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, let 휙 be a graph automorphism of order 2 of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then we have dim 퐻2(A2( PGU3(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSU3(푞) ⟨휙⟩)) ≥ 1 3 (푞2 − 1)(푞 + 1) �푞2(푞2 − 푞 + 1) (3, 푞 + 1) (푞 − 3) − (푞3 − 3푞2 + 3푞 − 3) � , which is a positive polynomial for 푞 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, for 푞 = 3, PSU3(3) satisfies (QD)2 but PSU3(3) ⟨휙⟩ fails (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We have that A2(퐿) is connected by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8, and 푚2(퐿) = 2 by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus 퐿 satisfies (QD)2 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5) dim 퐻1(A2(퐿)) = −�χ(A2(퐿)) = 1 3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, the only possible nontrivial 2-extension of 퐿 is by a graph automorphism 휙 of order 2 (which indeed arises from the field automorphism 푥 ↦→ 푥푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿1 = 퐿 ⟨휙⟩ be such extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8], 퐶PGU3(푞) (휙) � Inndiag(Ω3(푞)) = PGL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This implies that 퐶퐿(휙) = PGL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, there is a unique PGU3(푞)-conjugacy class of graph automorphisms, and such elements act by inversion on Outdiag(퐿) = (3, 푞 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus the conjugacy class of 휙 in Out(퐿) has size (3, 푞+1), and this gives rise to exactly (3, 푞+1) extensions 퐿 ⟨휙′⟩ ≤ Aut(퐿) of 퐿 by a conjugate 휙′ of 휙, and these extensions are Aut(퐿)-conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We conclude then that the number of graph automorphisms contained in 퐿1 is 푛푔 := | PGU3(푞)| | PGL2(푞)|(3, 푞 + 1) = 푞2(푞3 + 1) (3, 푞 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, we conclude that dim 퐻2(A2( PGU3(푞) ⟨휙⟩)) ≥ dim 퐻2(A2(PSU3(푞) ⟨휙⟩)) ≥ 푛푔 dim 퐻1(A2(PGL2(푞))) − dim 퐻1(A2(PSU3(푞))) = 푞2(푞3 + 1) (3, 푞 + 1) 1 3 (푞 − 3)(푞2 − 1) − 1 3 (푞6 − 2푞5 − 푞4 + 2푞3 − 3푞2 + 3) = 1 3 (푞2 − 1)(푞 + 1) �푞2(푞2 − 푞 + 1) (3, 푞 + 1) (푞 − 3) − (푞3 − 3푞2 + 3푞 − 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This polynomial is positive for all 푞 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 퐿1 satisfies (QD)2 if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' When 푞 = 3, 퐶퐿(휙) = PGL2(3) has nontrivial 2-core, so 퐻1(A2(퐶퐿(휙))) = 0, and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, 퐻2(A2(퐿1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 14 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN Now we have the necessary background to prove that PSL3(푞) satisfies (E-(QD)), except for a small number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = PSL푛(푞) with 푛, 푞 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The following assertions hold: (1) 퐿 and 퐿 extended by a field involution satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) If 푛 = 3, then every 2-extension of 퐿 satisfies (QD)2, with the following exceptions that fail (QD)2: 퐿 = PSL3(3) extended by a graph automorphism, and 퐿 = PSL3(9) extended by a group generated by a field involution and a graph automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = PSL푛(푞), with 푛 odd, and consider the stabiliser 푃 of a 1-dimensional sub- space of the underlying module 푉 = F푛 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푃 is a parabolic subgroup with structure 푃 � [푞푛−1]퐿푃, where 퐿푃, a Levi complement for 푃, has structure SL푛−1(푞) ◦(푛,푞−1) C푞−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus |퐿푃| = | GL푛−1(푞)|/(푛, 푞 − 1) and the index of 푃 in 퐿 is: |퐿 : 푃| = 푞푛(푛−1)/2 �푛 푖=2(푞푖 − 1) 푞푛−1 · 푞(푛−1)(푛−2)/2 �푛−1 푖=1 (푞푖 − 1) = 푞푛 − 1 푞 − 1 = 푞푛−1 + 푞푛−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' + 푞 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푛 is odd, the index of 푃 in PSL푛(푞) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, 퐿 = PSL푛(푞) and 퐿 extended by a field involution satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This proves item (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Before moving to the case 푛 = 3, we list all the possible 2-extensions of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Denote by 휙, 훾 and 훿 a field automorphism of order 2, a graph automorphism and a graph-field automorphism of 퐿, respectively, such that [휙, 훾] = 1 and 훿 = 휙훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let also 퐿∗ = PGL푛(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the 2-extensions of 퐿 are: (i) 퐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (ii) 퐿 ⟨휙⟩, with 퐶퐿∗(휙) � PGL푛(푞1/2) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (iii) 퐿 ⟨훾⟩, with 퐶퐿(훾) � Inndiag(Ω푛(푞)) by Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (iv) 퐿 ⟨훿⟩, with 퐶퐿∗(훿) � PGU푛(푞1/2) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (v) 퐿 ⟨휙, 훾⟩, with 퐶퐿(휙, 훾) � Inndiag(Ω푛(푞1/2)) by (iii) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now suppose that 푛 = 3, that is 퐿 = PSL3(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We know that the extensions of cases (i) and (ii) above satisfy (QD)2 by the parabolic argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' So it remains to show that the 2-extensions by graph, graph-field and both graph and field automorphisms, satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To that end, we compute the dimensions of the top-degree homology groups, similar to what we did for PSL2(푞) in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' First, recall that we have the following number of involutions of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐵 = ⟨휙, 훾⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 푛 푓 := # field involutions in 퐿 ⟨휙⟩ = # field involutions in 퐿퐵 = | PGL3(푞)| | PGL3(푞1/2)|(3, 푞1/2 + 1) , 푛푔 := # graph involutions in 퐿 ⟨훾⟩ = # graph involutions in 퐿퐵 = | PGL3(푞)| | PGL2(푞)|(3, 푞 − 1) , 푛푔 푓 := # graph-field involutions in 퐿 ⟨훿⟩ = # graph-field involutions in 퐿퐵 = | PGL3(푞)| | PGU3(푞1/2)|(3, 푞1/2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 15 To compute these numbers, we have used the structure of the centraliser in each case, the fact that there is a unique 퐿∗-conjugacy class for each type of involution, and the structure of Out(퐿) = (3, 푞 − 1) : ⟨휙, 훾⟩ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='12 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푡 be a field, graph or graph-field involution of 퐿, and let 퐿1 = 퐿 ⟨푡⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the number 푛1 of involutions in 퐿1 \\ 퐿 is 푛 푓 , 푛푔 or 푛푔 푓 , accordingly to the type of 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note also that 푚2(퐿1) = 푚2(퐿) + 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6) dim 퐻2(A2(퐿1)) ≥ 푛1 · dim 퐻1(A2(퐶퐿(푡))) − dim 퐻1(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We compute 푑(푡) := dim 퐻1(A2(퐶퐿(푡))) in each case, by using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that Ω1(퐶퐿(휙)) = PSL3(푞1/2) by item (ii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also 퐶퐿(훾) = PGL2(푞) by the classical isomorphism Inndiag(Ω3(푞)) � PGL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4, we have: 푑(휙) = dim 퐻1(A2(PSL3(푞1/2))) = 1 3 (푞1/2 − 1)(푞 − 1)(푞3/2 + 3푞 + 3푞1/2 + 3), 푑(훾) = dim 퐻1(A2(PGL2(푞))) = 1 3 (푞 − 3)(푞2 − 1), 푑(훿) = dim 퐻1(A2(PGU3(푞1/2))) = 1 3 (푞3 − 2푞5/2 − 푞2 + 2푞3/2 − 3푞 + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푑 := dim 퐻1(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2), this dimension is 푑 = 1 12 (푞 − 휖)(푞2 − (6 − 휖)푞 − 휖12), with 푞 ≡ 휖 (mod 4) and 휖 ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now it is routine to verify that 푛1푑(푡) > 푑 if 푡 = 훾 or 푡 = 훿, if and only if (푡, 푞) ≠ (훾, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, for 푞 = 3, 퐶퐿(훾) = PGL2(3) � Sym4 has non-trivial 2-core, so 푑(훾) = 0 and in consequence, 퐻2(퐿 ⟨훾⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This shows that 퐿 ⟨훾⟩ fails (QD)2 if 푞 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, a 2-extension of 퐿 by a field, graph or graph-field involution satisfies (QD)2 if and only if 푞 ≠ 3 when 퐿 is extended by a graph involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' It remains to show that 퐿퐵 = 퐿 ⟨휙, 훾⟩ verifies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For this case, we take 퐿 푓 = 퐿 ⟨휙⟩, 퐿2 = 퐿퐵 and consider the long exact sequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10 at 푚 = 2 there (since 푚2(퐿2) = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' That is, we need to show that 퐻3(A2(퐿2)) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that the set of involutions 푡 ∈ 퐿2 \\ 퐿1 is exactly the set of all graph and graph-field automorphisms of the extension 퐿2 = 퐿퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푑푔 := dim 퐻2(A2(PGL2(푞) ⟨휙⟩)), 푑푔 푓 := dim 퐻2(A2(PGU3(푞1/2) ⟨휙⟩)) and 푑 푓 := dim 퐻2(A2(퐿 푓 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='7) dim 퐻3(A2(퐿2)) ≥ 푛푔푑푔 + 푛푔 푓 푑푔 푓 − 푑 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We show that the right-hand side of this equation is positive if 푞 ≠ 9 by providing proper bounds of the dimensions 푑푔, 푑푔 푓 and 푑 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8) 푑푔 = dim 퐻2(A2(PGL2(푞) ⟨휙⟩)) ≥ 1 4 (푞1/2 − 1)(푞 − 1)(푞3/2 − 3푞 − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9) 푑푔 푓 ≥ 1 3 (푞 − 1)(푞1/2 + 1) �푞(푞 − 푞1/2 + 1) (3, 푞1/2 + 1) (푞1/2 − 3) − (푞3/2 − 3푞 + 3푞1/2 − 3) � , which is positive for all 푞 > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 16 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN Finally, we need to bound 푑 푓 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10 at 푚 = 2, we have 푑 푓 = dim 퐻2(A2(퐿 푓 )) = dim 퐻2(A2(PSL3(푞) ⟨휙⟩)) ≤ 푛 푓 dim 퐻1(A2(PSL3(푞1/2))) = 푞3/2(푞 + 1)(푞3/2 + 1) (3, 푞1/2 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now we check with the given bounds that 푛푔푑푔 + 푛푔 푓 푑푔 푓 − 푑 푓 is positive if and only if 푞 > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, if 푞 = 9, similar arguments show 퐻3(A2(퐿퐵)) = 0 since 푑푔 = 0 by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3 and 푑푔 푓 = 0 by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We conclude that every 2-extension of PSL3(푞) satisfies (QD)2, except for PSL3(3) extended by a graph automorphism and for PSL3(9) extended by field and graph automorphisms, which actually fail (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The Quillen dimension property on exceptional groups of Lie type We use the results of the preceding sections to show that, with only finite exceptions, the 2-extensions of the exceptional groups of Lie type satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For that purpose, it will be convenient to recall first which 2-extension can arise in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Table 3 records the 2-ranks of the exceptional groups of Lie type in adjoint version and the structure of the outer automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From this, we can compute the possible 2-extensions in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall that we follow the terminology of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The 2-ranks were extracted from [4] and [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Group 2-rank Outdiag Out/Outdiag 3퐷4(푞) 3 1 3Φ 퐺2(푞) 3 1 ΦΓ, where |ΦΓ : Γ| = 2 if 푞 = 3푎, and Γ = 1 otherwise 2퐺2(푞) 3 1 Φ (odd order) 퐹4(푞) 5 1 Φ 퐸6(푞) 6 (3, 푞 − 1) Φ × Γ, Γ � C2 2퐸6(푞) 6 (3, 푞 + 1) 2Φ 퐸7(푞) 8 2 Φ 퐸8(푞) 9 1 Φ Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Out/Outdiag is cyclic unless specified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Φ = Aut(F푞) � C푎, where 푞 = 푟푎, 푟 is an odd prime, and the usual conventions for the twisted types hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also, Γ is a set of graph automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Cases 퐺2(푞) and 2퐺2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We start by proving that the Ree groups 2퐺2(푞) satisfy (QD)2 if and only if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that, by Table 3 for example, 2퐺2(푞) has no non-trivial 2-extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 be the Ree group 2퐺2(푞), where 푞 is a power of 3 by an odd positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then the following hold: (1) 퐿 has no non-trivial 2-extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) A Sylow 2-subgroup of 퐿 is an elementary abelian group of order 8, so 푚2(퐿) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3) 2-subgroups of equal order of 퐿 are conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4) 퐿 satisfies (QD)2 if and only if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, if 푞 > 3 then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1) dim 퐻2(A2(퐿)) ≥ �χ(A2(퐿)) = 1 21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 17 (5) For 푞 = 3, A2(퐿) = A2(PSL2(8)) is homotopy equivalent to a discrete space of 8 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Items (1-3) are well-known facts about the Ree groups and can be found in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐿 = 2퐺2(3), then 퐿′ = PSL2(8) has index 3 in 퐿, and A2(퐿) � A2(PSL2(8)) is homotopy equivalent to a discrete space with 8 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푚2(퐿) = 3, we conclude that 퐿 fails (QD)2 for 푞 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This proves item (5) and the “only if” part of item (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now suppose that 푞 ≠ 3 and 퐿 = 2퐺2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since A2(퐿) has dimension 2 by item (2), we show that its second homology group is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To that end, it is enough to see that its Euler characteristic is positive since A2(퐿) is connected for 푞 ≠ 3 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, �χ(A2(퐿)) = dim 퐻2(A2(퐿)) − dim 퐻1(A2(퐿)) ≤ dim 퐻2(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We invoke Theorem C of [10] to describe the normalisers of 2-subgroups: the centraliser of an involution is 2×PSL2(푞), the normaliser of a four-subgroup is (22 ×D 푞+1 2 ) : 3, and the normaliser of a Sylow 2-subgroup is 23 : 7 : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From this information, items (2,3) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='9, we can compute the Euler characteristic of A2(퐿): �χ(A2(퐿)) = −1 + |퐿| 2| PSL2(푞)| − 2 |퐿| 6(푞 + 1) + 8 |퐿| 168 = −1 + 푞3(푞3 + 1)(푞 − 1) � 1 푞(푞2 − 1) − 1 3(푞 + 1) + 1 21 � = 1 21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since the polynomial 푞5 − 8푞4 + 15푞3 + 21 is positive for every prime power 푞 ≠ 4, we conclude that 퐻2(A2(퐿)) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In consequence, 퐿 satisfies (QD)2 if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This completes the proof of item (4), and hence of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ For the case 퐺2(푞), we refer the reader to the classification of maximal subgroups of 퐺2(푞) by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Kleidman [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will follow the terminology of that article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐺2(푞), with 푞 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then every 2-extension of 퐿 satisfies (QD)2, except possibly for the 2-extensions of 퐺2(3) and the 2-extension of 퐺2(9) by a field involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐺2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We prove first that 퐺2(푞) and its extension by a field automorphism of order 2 satisfy (QD)2, by exhibiting a maximal subgroup of the same rank that satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Theorem A in [10], 퐺2(푞) contains a subgroup 퐾+ = SL3(푞) : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿+ = 퐹∗(퐾+) � SL3(푞) and 푍 = 푍(퐿+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿0 := 퐿+/푍 = PSL3(푞) and 퐻0 := 퐾+/푍 = 퐿0 ⟨훾⟩, where 훾 induces a graph automorphism on 퐿0 (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and its proof in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, 퐿0 satisfies (QD)2 if 푞 ≠ 3, so 퐻0 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' On the other hand, 푚2(퐿) = 3 by Table 3, and also 푚2(퐿0) = 3 by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 that � 퐻∗(A2(퐻0)) = � 퐻∗(A2(퐾+/푍)) ⊆ � 퐻∗(A2(퐾+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, we get the following inclusions between the top-degree homology groups � 퐻2(A2(퐻0)) ⊆ � 퐻2(A2(퐾+)) ⊆ � 퐻2(A2(퐿)), which show that 퐿 satisfies (QD)2 if 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, a nontrivial 2-extension of 퐿 = 퐺2(푞) can only be given by field automorphisms of order 2 if 푞 is not a power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, by the construction of the subgroup 퐾+ given in [10], field automorphisms of 퐺2(푞) induce field automorphisms on (a suitable conjugate 18 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN of) 퐾+, and hence on the quotient 퐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, for 퐵 ∈ O2(퐿) inducing field automorphisms, we may take 퐾+ fixed by 퐵, and then 퐾+퐵 � SL3(푞) : (2 × 퐵) after a suitable choice of conjugates (recall that Out(SL3(푞)) = (3, 푞 − 1) : (Aut(F푞) × Γ), where Γ = 2 is a group of graph automorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Similar as before, we have a split extension 퐾+퐵/푍 = 퐿0퐵′, where 퐵′ = ⟨훾⟩ × 퐵 ∈ O(퐿0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, 퐿0퐵′ satisfies (QD)2 if 푞 ≠ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Analogously to the previous case, 푚2(퐿0퐵′) = 4 = 푚2(퐿) = 푚2(퐾+퐵), and we get an inclusion in the degree 3 homology groups, showing that 퐾+퐵 and 퐿퐵 satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, an extension of 퐿 by a field automorphism of order 2 satisfies (QD)2 if 푞 ≠ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' It remains to analyse the case 푞 = 3푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8] (see also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='12 of [8]), only field or graph-field automorphisms can arise in Aut(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We have shown above that the extension of 퐿 by a field automorphism of order 2 satisfies (QD)2 if 푞 ≠ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus we need to prove that if 푡 is a graph-field automorphism of 퐿, then 퐿 ⟨푡⟩ satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In that case, 푞 = 32푎+1 and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5, 퐶퐿(푡) = 2퐺2(푞), which has 2-rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore 푚2(퐿 ⟨푡⟩) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' However, by Theorem B of [10], every maximal subgroup of 퐿 ⟨푡⟩ containing 푡 is either 2-local or has 2-rank at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This shows that we cannot proceed as before via maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In view of this, we will proceed by using the long exact sequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We have subgroups 푀0 := 퐶퐿(푡) = 2퐺2(푞), 푀1 := 퐺2(3) ⟨푡⟩ ≤ 퐿 ⟨푡⟩ and 푀2 := 2퐺2(3) such that 푀2 ≤ 푀1 ∩ 푀0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Fix 퐴 a Sylow 2-subgroup of 푀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1(2) and [10, Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4], 퐴 is also a Sylow 2-subgroup of 푀0 and it is self-centralising in 퐿, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 퐶퐿(퐴) = 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' A direct computation also shows that 푁푀2(퐴) = 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PSL3(2), which immediately implies 푁퐿(퐴) = 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PSL3(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now, suppose by the way of contradiction that 퐿 ⟨푡⟩ fails (QD)2, that is, the homology group 퐻3(A2(퐿 ⟨푡⟩)) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall that 퐶퐿(푡) = 2퐺2(푞) and there is a unique 퐿-conjugacy class of involutions 푡′ ∈ 퐿 ⟨푡⟩ − 퐿 by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푋 = � 퐶퐿 (푡)푥∈퐿/퐶퐿 (푡) A2(퐶퐿(푡푥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, we get inclusions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2) � 퐿/퐶퐿 (푡) 퐻2(A2(퐶퐿(푡))) ↩→ 퐻2 (푋) ↩→ 퐻2(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Set 푑 := dim 퐻2(푋), 푑′ := dim � 퐿/퐶퐿 (푡) 퐻2(A2(퐶퐿(푡))) = |퐿 : 퐶퐿| dim 퐻2(A2(2퐺2(푞))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2) shows that 푑′ ≤ 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' However, we will prove that 푑 < 푑′, arriving then at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' On one hand, we have that 푋 is a union of A2-posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, below each point, we have a wedge of spheres of maximal possible dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This means that the homology of 푋 can be obtained from the chain complex that in degree 푖 is freely generated by the spheres below each point of 푋 of height 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, for 푖 = 2, the points of height 2 correspond to the conjugates of 퐴, the fixed Sylow 2-subgroup of 푀0 = 퐶퐿(푡) and 푀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, 푑 = dim 퐻2(푋) ≤ |퐿 : 푁퐿(퐴)| · # spheres below 퐴 = 푞6(푞6 − 1)(푞2 − 1) 168 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' On the other hand, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1(4), 푑′ ≥ |퐿 : 퐶퐿(푡)| · �χ(A2(2퐺2(푞))) = 푞3(푞3 − 1)(푞 + 1) 1 21 (푞2 − 1)(푞5 − 8푞4 + 15푞3 + 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 19 Finally, from these bounds for 푑 and 푑′, it is not hard to conclude that 푑′ > 푑 for all prime power 푞 ≥ 7, which is our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This gives a contradiction to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2), and thus shows that 퐻3(A2(퐿 ⟨푡⟩)) ≠ 0, that is, 퐿 ⟨푡⟩ satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This finishes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐺2(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We show that A2(퐿) is homotopy equivalent to a wedge of spheres of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, since 푚2(퐿) = 3, 퐿 fails (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, also the unique nontrivial 2-extension of 퐿 (by a graph-field automorphism) fails (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We construct a subposet of A2(퐿) of dimension 1 and homotopy equivalent to A2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' First, take the subposet 픦(A2(퐿)) = {퐴 ∈ A2(퐿) : 퐴 = Ω1(푍(Ω1(퐶퐿(퐴))))}, which is homotopy equivalent to A2(퐿) (see [14, Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, there are two conjugacy classes of elementary abelian 2-subgroups of order 8, and both are contained in 픦(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For one of these classes, say represented by 퐴, the normalizer 푁퐿(퐴) has order 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then it can be shown that 픦(A2(퐿))<퐴 is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, if we remove the 퐿-conjugates of 퐴 from 픦(A2(퐿)) we get a subposet 픰픦(A2(퐿)) homotopy equivalent to 픦(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now, there is a unique conjugacy class of four- subgroups in this new subposet 픰픦(A2(퐿)), and each such subgroup is contained in a unique element of order 8 of 픰픦(A2(퐿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Again, we can remove all the four-subgroups from 픰픦(A2(퐿)) and obtain a new subposet 푇 homotopy equivalent to A2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푇 consists only of elements of order 2 and 8, we conclude that 푇 has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, an extra computation shows that �χ(A2(퐿)) = −11584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore A2(퐿) is homotopy equivalent to a wedge of 11584 spheres of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, 퐿 fails (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This also shows that 퐿 = 퐺2(9) extended by a field automorphism of order 2 fails (QD)2: if 휙 is a field involution, then 퐶퐿(휙) = 퐺2(3), and thus 퐻2(A2(퐶퐿(휙))) = 0 by the previous computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='10, we conclude that 퐻3(A2(퐿)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Cases 3퐷4 and 퐹4(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The group 퐿 = 3퐷4(푞) satisfies (E-(QD)) if 푞 ≠ 9 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also 3퐷4(9) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Recall that 푚2(퐿) = 3 by Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then a graph automorphism of order 3 of 3퐷4(푞) centralises a subgroup 퐾 = 퐺2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also, if 휙 denotes a field automorphism of order 2 of 퐿, then, after choosing a suitable conjugate, we may assume that 휙 induces a field automorphism on 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and its proof, 푚2(퐾) = 3 = 푚2(퐿), 푚2(퐾 ⟨휙⟩) = 4 = 푚2(퐿 ⟨휙⟩), and both 퐾 and 퐾 ⟨휙⟩ satisfy (QD)2 for 푞 ≠ 3, 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also note that 퐺2(9) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3, 퐿 and 퐿 ⟨휙⟩ satisfy (QD)2 if 푞 ≠ 3, 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since these are the only possible 2-extensions of 퐿 by Table 3, this concluded with the proof of our proposition for 푞 ≠ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 푞 = 3 then a computation of the Euler characteristic of 퐿 in GAP shows that �χ(A2(퐿)) = 882634225472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since A2(퐿) is connected by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8, we see that 퐻2(A2(퐿)) ≠ 0, that is, 퐿 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 퐿 = 퐹4(푞), with 푞 ≠ 3, 9 odd, then 퐿 satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also 퐹4(9) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Suppose that 푞 ≠ 3, 9 is an odd prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿 contains a subgroup 퐻 := PGL2(푞) × 퐺2(푞) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' the main result of [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that 퐻 satisfies (QD)2 by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since both 퐿 and 퐻 have 2-rank 5 by Table 3, we conclude that 퐿 satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐵 ∈ O2(퐿), so 퐵 is generated by a field automorphism of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus it acts by field automorphisms in a direct product subgroup isomorphic to 퐻, which we may assume without 20 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN loss of generality that it is our 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then � 퐻 = PGL2(푞)퐵 × 퐺2(푞1/2), which is a subgroup of 퐻퐵, satisfies (QD)2 by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푚2( � 퐻) = 6 = 푚2(퐿퐵), we conclude that 퐿퐵 also satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We have shown that every possible 2-extension of 퐿 satisfies (QD)2, so 퐿 satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 푞 = 9, then PGL2(9) × 퐺2(9) satisfies (QD)2 by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 퐹4(9) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Cases 퐸6(푞) and 2퐸6(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐸 휖 6 (푞) (any version), 휖 ∈ {±1}, and 푞 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿 satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐸 휖 6 (푞) in adjoint version, where 휖 ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For a 2-extension 퐿퐵 of the adjoint version 퐿, we see that 푚2(퐿퐵) = 푚2(퐿푢 �퐵), where 퐿푢 is the universal version of 퐸 휖 6 (푞) and �퐵, isomorphic to 퐵, is just a lift of the action of 퐵 on 퐿푢 (this is possible since 푍(퐿푢) = (3, 푞 − 휖) is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus 퐿퐵 = 퐿푢 �퐵/푍(퐿푢), and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, � 퐻∗(A2(퐿퐵)) ⊆ � 퐻∗(A2(퐿푢퐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, if 퐿 satisfies (E-(QD)), then so does the universal version of 퐸 휖 6 (푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We will show that there exists a parabolic subgroup 푃 of 퐿 such that for any 2-extension 퐿퐵, a suitable conjugate of 푃 is normalised by 퐵 (so we can suppose it is 푃 itself), and 푚2(푃퐵) = 푚2(퐿퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This parabolic subgroup 푃 arises from the 퐴5 subdiagram in 퐸6, so 푃 = 푈 GL휖 6 (푞)/푍(퐿푢), where GL휖 6 (푞)/푍(퐿푢) denotes the Levi complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푚2(푃) = 6, which realises the 2- rank of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Furthermore, a graph, graph-field or field automorphism of 퐿 (the last two only for 휖 = 1) stabilises this subdiagram (and hence 푃), inducing a graph (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' graph-field or field) automorphism on GL휖 6 (푞)/푍(퐿푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Denote by 푡 such automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푚2(퐿 ⟨푡⟩) ≤ 푚2(퐿) + 1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We claim that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3) 푚2(푃 ⟨푡⟩) = 푚2(GL휖 6 (푞) ⟨푡⟩) = 7 = 푚2(퐿 ⟨푡⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that 푚2(푃 ⟨푡⟩) = 푚2(GL휖 6 (푞) ⟨푡⟩), for the lifted action of 푡 on GL휖 6 (푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then it is clear that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3) holds if 푡 induces a field automorphism (so 휖 = 1), since the stabiliser of 푡 in GL6(푞) is GL6(푞1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Similarly, if 푡 is a graph-field automorphism then 휖 = 1 and 퐶GL6(푞) (푡) = GU6(푞1/2), which has 2-rank 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then, in these two situations, 푚2(푃 ⟨푡⟩) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now assume that 푡 is a graph involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' For 휖 = 1, 푡 acts on GL6(푞), so GL6(푞) ⟨푡⟩ contains a graph automorphism 푔 inducing the map 푥 ↦→ (푥′)−1, where 푥′ denotes the transpose of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 퐶GL6(푞) (푔) = GO6(푞), which has 2-rank 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This implies that 푚2(GL6(푞) ⟨푡⟩) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' If 휖 = −1, 푡 is a graph involution acting on GU6(푞), so up to conjugation 푡 is indeed the map 푥 ↦→ 푥푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 퐶GU6(푞) (푡) = GO6(푞), and again we get 푚2(GU6(푞) ⟨푡⟩) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In any case, we see that 푚2(푃 ⟨푡⟩) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, suppose that we have 퐵 = ⟨휙, 훾⟩, where 휙 is a field automorphism of order 2 and 훾 a graph automorphism of order 2 of 퐿 = 퐸6(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We can suppose that 퐵 stabilises 푃 (and thus its unipotent radical), and its Levi complement GL6(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Thus, 훾 acts as a graph automor- phism on the stabiliser of 휙 in GL6(푞), which is isomorphic to GL6(푞1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' As we saw above, 푚2(GL6(푞1/2) ⟨훾⟩) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, 푚2(GL6(푞)퐵) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푚2(퐵) = 2 and 푚2(퐸6(푞)) = 6, we conclude that 푚2(퐸6(푞)퐵) = 8, so the 2-rank is realised in 푃퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To conclude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' note that a 2-extension of 퐿 is one of: (1) 퐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' of 2-rank 6 = 푚2(푃),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (2) 퐿 ⟨훾⟩ of 2-rank 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' with 훾 a graph automorphism of order 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' which also stabilises 푃 and 푚2(푃 ⟨훾⟩) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS AND QUILLEN’S DIMENSION 21 (3) 퐿 ⟨휙⟩ of 2-rank 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' with 휙 a field automorphism of order 2 (휖 = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' which also stabilises 푃 and 푚2(푃 ⟨휙⟩) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (4) 퐿 ⟨훾휙⟩ of 2-rank 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' with 훾휙 a graph-field automorphism of order 2 (휖 = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' which also stabilises 푃 and 푚2(푃 ⟨훾휙⟩) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (5) 퐿 ⟨훾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 휙⟩ of 2-rank 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' with 휙 a field automorphism of order 2 (휖 = 1) commuting with 훾 a graph automorphism of order 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' and ⟨훾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 휙⟩ also stabilises 푃 with 푚2(푃 ⟨훾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 휙⟩) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From this, we conclude that any 2-extension of the adjoint versions of 퐸 휖 6 (푞) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By the remark at the beginning of the proof, we conclude that any version of 퐸 휖 6 (푞) satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Case 퐸7(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐸7(푞) (adjoint version), with 푞 odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐿 satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐸7(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Table 3, if 휙 denotes a field automorphism of order 2 of 퐿, the 2-extensions of 퐿 are: 퐿, Inndiag(퐿), 퐿 ⟨휙⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that Inndiag(퐿) ⟨휙⟩ is not a 2-extension since field and diagonal automorphisms of order 2 do not commute in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Next, we study the 2-ranks of these extensions, so we need to understand the centralisers of the outer involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From Table 3, 푚2(퐿) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We claim that 푚2(Inndiag(퐿)) = 8 = 푚2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, consider 퐾 = 퐸7(푞2) in adjoint version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 푚2(퐾) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 휙′ be a field automorphism of order 2 for 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, 퐾 ≥ 퐶퐾 (휙′) = 퐶Inndiag(퐾) (휙′) = Inndiag(퐸7(푞)) � Inndiag(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From this we see that 푚2(Inndiag(퐿)) = 8 = 푚2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, Inndiag(퐿) satisfies (QD)2 if 퐿 does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, this also proves that if 휙 is a field automorphism of order 2 for 퐿 then 푚2(Inndiag(퐿) ⟨휙⟩) = 9 = 푚2(퐿 ⟨휙⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From these observations, we conclude that, in order to establish (E-(QD)) for 퐸7(푞), it is enough to show that 퐸7(푞) and 퐸7(푞) ⟨휙⟩ satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' To this end, we exhibit a maximal parabolic subgroup of 퐸7(푞) of 2-rank 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' We see that 퐷6 is a subdiagram of 퐸7, so we have a maximal parabolic subgroup in 퐸7(푞) of the form 푃 = 푈 : (퐷6(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (푞 − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Here 푈 denotes the unipotent radical of 푃, and the subgroup 퐻 = 퐷6(푞) is a quotient of Spin+ 12(푞) by a central subgroup of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Indeed, 퐻 = HSpin+ 12(푞) and it lies in the centraliser of the involution that generates the centre of a Sylow 2-subgroup 푇 of 퐿 (see the 푡1 involution of the 퐸7(푞) entry in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From this, we show that the Levi complement 퐿푃 = 퐷6(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (푞−1) of 푈 has 2-rank 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 푡 be the involution in the centre of 퐿푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Then 퐶퐿(푡) = (SL2(푞) ◦2 HSpin+ 12(푞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 by Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푡 ∈ 푍(푇), 푇 ≤ 퐶퐿(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also, SL2(푞) has a unique involution, so the 2- rank of푇 is realised in a subgroup of the extension 푀 := HSpin+ 12(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The 2 here at the end comes from diagonal automorphisms of the half-spin group, as in the Levi complement above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, if we identify 푀 as a subgroup of 퐿푃, we conclude that 푚2(퐿푃) = 푚2(푀) = 푚2(퐸7(푞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Moreover, after suitable choices of conjugates, a field automorphism 휙 of order 2 must normalise 푃 and act as a field automorphism on our 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 퐶푀 (휙) contains a subgroup isomorphic to HSpin+ 12(푞1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, we see that 푃 ⟨휙⟩ has 2-rank 9, which is the 2-rank of the 2-extension 퐸7(푞) ⟨휙⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' 22 KEVIN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' PITERMAN By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5, 퐿 and 퐿 ⟨휙⟩ satisfy (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, by the previous discussion, we conclude that 퐿 satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Case 퐸8(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The simple group 퐸8(푞), 푞 ≠ 3, 9 odd, satisfies (E-(QD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Also 퐸8(9) satisfies (QD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Let 퐿 = 퐸8(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1 of [11], 퐿 contains a maximal subgroup 퐻 � (3, 푞 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (PSL3(푞) × 퐸6(푞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3, 푞 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Note that 퐹∗(퐻) = (3, 푞 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (PSL3(푞) × 퐸6(푞)), and 퐻+ := 퐻/푍(퐹∗(퐻)) = (PSL3(푞) × 퐸6(푞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' (3, 푞 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, where (3, 푞 − 1) induces diagonal automorphism on each component of 퐻+, and the 2 induces a graph involution, also acting on both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, by taking the centraliser of a graph involution on the PSL3(푞) component, we see that 퐻0 contains a subgroup 퐾0 isomorphic to PGL2(푞) × Inndiag(퐸6(푞)) ⟨훾⟩ , where 훾 is a graph involution of 퐸6(푞) centralising PGL2(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Now, recall that 푚2(퐿) = 9 and 푚2(PGL2(푞)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푚2(퐸6(푞) ⟨훾⟩) = 7 by item (2) of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, we see that 푚2(퐾0) = 푚2(PGL2(푞)) + 푚2(퐸6(푞) ⟨훾⟩) = 2 + 7 = 9 = 푚2(퐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore 퐾0 realises the 2-rank of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' By Table 3, 퐸8(푞) extended by a field automorphism of order 2, say 휙, is the unique nontrivial 2-extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' From the construction of the maximal subgroup 퐻 and 퐾0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' [11]), we can pick a suitable 퐿-conjugate of 휙 (and we suppose it is the same 휙) such that it normalises 퐻 and, after passing to the quotient, normalises 퐾0 and induces a field automorphism on both factors of 퐾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' In particular, we have a subgroup 퐾1 of 퐾0 ⟨휙⟩ of the form PGL2(푞1/2) × Inndiag(퐸6(푞)) ⟨훾′, 휙⟩ , where we have chosen 훾′ ∈ Inndiag(퐸6(푞)) ⟨훾⟩ to be a graph automorphism commuting with 휙, and PGL2(푞1/2) = 퐶PGL2(푞) (휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore, by item (5) in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6, 푚2(퐾1) = 2 + 푚2(퐸6(푞) ⟨훾′, 휙⟩) = 2 + 8 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Since 푚2(퐿 ⟨휙⟩) ≤ 푚2(퐿) + 1 = 10, we conclude that 푚2(퐾1) = 푚2(퐿 ⟨휙⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Finally, note that 퐾0 and 퐾1 satisfy (QD)2 if 푞 ≠ 3, 9 respectively, by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Hence, by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content='3, 퐿 and 퐿 ⟨푡⟩ satisfy (QD)2 if 푞 ≠ 3, 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Therefore every 2-extension of 퐸8(푞) satisfies (QD)2, with the exceptions given in the state- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' This concludes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' □ References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Aschbacher and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Lyons, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Solomon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' The classification of the finite simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Number 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Part III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} +page_content=' Chapters 12–17.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE0T4oBgHgl3EQfrwF-/content/2301.02570v1.pdf'} diff --git a/utAzT4oBgHgl3EQfBvrw/content/tmp_files/2301.00949v1.pdf.txt b/utAzT4oBgHgl3EQfBvrw/content/tmp_files/2301.00949v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f56426059ce51e01d0cdf2a22b0a4efe0eef0b3 --- /dev/null +++ b/utAzT4oBgHgl3EQfBvrw/content/tmp_files/2301.00949v1.pdf.txt @@ -0,0 +1,219 @@ +arXiv:2301.00949v1 [astro-ph.SR] 3 Jan 2023 +1 +Very low state in PY Per in 2022 +Taichi Kato1 +tkato@kusastro.kyoto-u.ac.jp +1 Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan +Abstract +Using VSNET, VSOLJ, ASAS-SN and ATLAS observations, I found that the Z Cam star PY Per spent a +long, faint low state reaching 19.1 mag at least between 2022 June and November. No dwarf nova outburst +was recorded during this interval. TESS data during this low state showed two maxima in one orbital cycle +and can be interpreted as an ellipsoidal modulation arising from the secondary. These observations suggest +that the mass-transfer almost stopped during this low state and strengthen the identification of PY Per as a +VY Scl star. PY Per had shown an unusual outburst resembling an SU UMa-type superoutburst less than +half a year before (Kato 2022, arXiv:2204.12056) and these phenomena may have been physically related. +In Kato (2022), I reported the detection of an outburst (2021 December–2022 January) resembling an +SU UMa-type superoutburst in the Z Cam star PY Per with an orbital period of 0.15468(5) d (Kato 2022; +Taylor and Thorstensen 1996). Kato (2022) also detected faint states in observations by the American Association +of Variable Stars (AAVSO), and showed that this object is also an VY Scl-type cataclysmic variable (CV) [see +e.g., Warner (1995) for CVs in general and their subtypes]. VY Scl-type faint states are sometimes associated +with Z Cam stars, most notably the 1996–1997 totally unexpected long-lasting fading in RX And (Sion et al. +2001; Kato et al. 2002; Schreiber et al. 2002; Kato 2004). +After the solar conjunction following the unusual outburst mentioned above, I noticed that PY Per was in +low state without any outburst in Variable Star Observers League in Japan (VSOLJ) and VSNET (Kato et al. +2004) observations (vsnet-alert 27029.1). This has been confirmed by using the All-Sky Automated Survey for +Supernovae (ASAS-SN) Sky Patrol (Shappee et al. 2014; Kochanek et al. 2017) data. These observations started +in 2022 June (ASAS-SN) and 2022 July (VSOLJ). Using the data by the Asteroid Terrestrial-impact Last Alert +System (ATLAS: Tonry et al. 2018; Heinze et al. 2018; Smith et al. 2020) Forced Photometry (Shingles et al. +2021)2, I further confirmed that the 2022 fading episode was the deepest (reaching 19.1 mag) and longest (more +than 150 d) ever recorded in this object (figure 1). +For a comparison, a light curve of the preceding seasons is given in figure 2. +In the 2019–2020 season +(before BJD 2458950), the object showed frequent low-amplitude outbursts as was typical for a Z Cam star. The +behavior was similar to this at least between 2016 and 2019. In the 2020–2021 season (BJD 2459030–2459290), +the quiescence became fainter (near 18.0 mag) and longer and brighter outbursts in addition to smaller ones +became more prominent than in the previous season. Although the quiescent brightness (18.0 mag) was almost +as faint as the VY Scl-type low state mentioned in Kato (2022), the object definitely showed dwarf nova-type +outbursts. The light curve in 2020–2021 probably illustrated the behavior when the mass-transfer rate decreased. +Compared to the 2020–2021 season, the mass-transfer rate probably returned normal in the 2021–2022 season +(left part of figure 1). There was, however, an interval lacking outbursts (BJD 2459525–2459568), but not as +faint as the 2020–2021 quiescence, preceding the unusual outburst resembling an SU UMa-type superoutburst. +Accumulation of matter in the disk during this interval may have caused the unusual outburst. Although this +unusual outburst may have been physically related to the very faint low state in 2022, observations were impossible +due to the solar conjunction and how this very faint state started remains a mystery. Note that the 1996–1997 +low state in RX And was preceded by an unusually long standstill. These rare phenomena might have been +physically related and the case would also be suspected in PY Per. +I analyzed Transiting Exoplanet Survey Satellite (TESS) observations obtained in 2022.3 The full light- +curve is available at the Mikulski Archive for Space Telescope (MAST4). The TESS observations started on 2022 +October 28 (BJD 2459882) and ended on 2022 November 26 (BJD 2459910). PY Per started rising in the TESS +data on the final two days. Since most of the TESS data were obtained when PY Per was in deep low state and +since the object has nearby (unrelated) contaminating stars, I did not attempt to extract the flux of PY Per but +used the flux combined with contaminating stars. Using the data before BJD 2459908.6 (object in low state), +I could detect the orbital period and modulations (figure 3). The period was determined to be 0.15453(2) d +1 +2The ATLAS Forced Photometry is available at . +3. +4. + +2 +59400 +59500 +59600 +59700 +59800 +59900 +12 +14 +16 +18 +20 +ATF o +ATF c +ASN g +C +vis +Figure 1: +2021–2022 light curve of PY Per using ATLAS forced photometry (ATF), ASAS-SN (ASN), VSOLJ +and VSNET (C for CCD close to visual and vis for visual) observations. The left part of this figure corresponds +to the fourth panel of Fig. 1 in Kato (2022). The outburst resembling an SU UMa-type superoutburst started +on BJD 2459570. After the solar conjunction, the object was found already in a deep, low state below 18 mag. +The object gradually started to brighten after BJD 2459880 and there was a short outburst on BJD 2459910. A +more ordinary outburst started on BJD 2459927, which slowly rose to a maximum of 13.8 mag. +with the Phase Dispersion Minimization (PDM, Stellingwerf 1978) method after removing long-term trends by +locally-weighted polynomial regression (LOWESS: Cleveland 1979). The errors of periods by the PDM method +were estimated by the methods of Fernie (1989) and Kato et al. (2010). Although the obtained period is in +agreement with that obtained in high state in Kato (2022), the orbital profile is very different. In high state, +there was a single peak in one orbit [figure 3 in Kato (2022)], while the current observations clearly show two +maxima in one orbit. This feature most likely represents an ellipsoidal variation of the secondary and TESS +photometry supports the very weak (or no) contribution from the accretion disk. These observations support +that the mass accretion almost stopped in this very low state in PY Per and strengthen the identification of this +object as a VY Scl star. +Acknowledgements +This work was supported by JSPS KAKENHI Grant Number 21K03616. The author is grateful to the ASAS-SN, +ATLAS and TESS teams for making their data available to the public. I am grateful to VSOLJ and VSNET +observers for reporting observations and to Naoto Kojiguchi for providing a script for downloading TESS data. +The contributors from VSNET and VSOLJ in 2022 were Pavol A. Dubovsky, Masao Funada, Hiroshi Itoh, Eddy +Muyllaert, Yutaka Maeda, Masayuki Moriyama and Gary Poyner. +This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. +The Asteroid Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to search for near earth +asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO + +3 +58700 +58800 +58900 +59000 +59100 +59200 +59300 +12 +14 +16 +18 +20 +ATF o +ATF c +ASN g +C +vis +Figure 2: +Light curve of PY Per in the 2019–2021 season. The symbols are the same as in figure 1. In the +2019–2020 season (before BJD 2458950), the object showed frequent low-amplitude outbursts as was typical for a +Z Cam star. In the 2020–2021 season (BJD 2459030–2459290), the quiescence became fainter (near 18.0 mag) and +longer and brighter outbursts in addition to smaller ones became more prominent than in the previous season. + +4 +0.148 +0.150 +0.152 +0.154 +0.156 +0.158 +0.160 +0.162 +0.98 +0.99 +1.00 +(d) +θ +P=0.15453 +−0.5 +0.0 +0.5 +1.0 +1.5 +−0.04 +−0.02 +0.00 +0.02 +Figure 3: Period analysis of the TESS data during the low state. (Upper): We analyzed 100 samples which +randomly contain 50% of observations, and performed the PDM analysis for these samples. The bootstrap result +is shown as a form of 90% confidence intervals in the resultant PDM θ statistics. (Lower): Orbital variation. Two +peaks in one orbit are clearly visible. Note that the amplitude was smaller than real due to the contaminating +stars. + +5 +search include images and catalogs from the survey area. This work was partially funded by Kepler/K2 grant +J1944/80NSSC19K0112 and HST GO-15889, and STFC grants ST/T000198/1 and ST/S006109/1. The ATLAS +science products have been made possible through the contributions of the University of Hawaii Institute for +Astronomy, the Queen’s University Belfast, the Space Telescope Science Institute, the South African Astronomical +Observatory, and The Millennium Institute of Astrophysics (MAS), Chile. +List of objects in this paper +RX And, Z Cam, PY Per, VY Scl, SU UMa +I provide two forms of the references section (for ADS and as published) so that the references can be easily +incorporated into ADS. +References (for ADS) +Cleveland, W. S. 1979, J. Amer. Statist. Assoc., 74, 829 (https://doi.org/10.2307/2286407) +Fernie, J. D. 1989, PASP, 101, 225 (https://doi.org/10.1086/132426) +Heinze, A. N., et al. 2018, AJ, 156, 241 (arXiv:1804.02132) +Kato, T. 2004, PASJ, 56, S55 (arXiv:astro-ph/0308086) +Kato, T. 2022, VSOLJ Variable Star Bull., 100, (arXiv:2204.12056) +Kato, T., et al. 2010, PASJ, 62, 1525 (arXiv:1009.5444) +Kato, T., Nogami, D., & Masuda, S. 2002, PASJ, 54, 575 (arXiv:astro-ph/0205363) +Kato, T., Uemura, M., Ishioka, R., Nogami, D., Kunjaya, C., Baba, H., & Yamaoka, H. 2004, PASJ, 56, S1 +(arXiv:astro-ph/0310209) +Kochanek, C. S., et al. 2017, PASP, 129, 104502 (arXiv:1706.07060) +Schreiber, M. R., Gänsicke, B. T., & Mattei, J. A. 2002, A&A, 384, L6 (https://doi.org/10.1051/0004- +6361:20020122) +Shappee, B. J., et al. 2014, ApJ, 788, 48 (arXiv:1310.2241) +Shingles, L., et al. 2021, Transient Name Server AstroNote, 7, 1 +Sion, E. M., Szkody, P., Gaensicke, B., Cheng, F. H., La Dous, C., & Hassall, B. 2001, ApJ, 555, 834 +(https://doi.org/10.1086/321529) +Smith, K. W., et al. 2020, PASP, 132, 085002 (arXiv:2003.09052) +Stellingwerf, R. F. 1978, ApJ, 224, 953 (https://doi.org/10.1086/156444) +Taylor, C. J., & Thorstensen, J. R. 1996, PASP, 108, 894 (https://doi.org/10.1086/133810) +Tonry, J. L., et al. 2018, PASP, 130, 064505 (arXiv:1802.00879) +Warner, B. 1995, Cataclysmic Variable Stars (Cambridge: Cambridge University Press) + +6 +References (as published) +Cleveland, W. S. (1979) Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. +74, 829 +Fernie, J. D. (1989) Uncertainties in period determinations. PASP 101, 225 +Heinze, A. N. et al. (2018) A first catalog of variable stars measured by the Asteroid Terrestrial-impact Last +Alert System (ATLAS). AJ 156, 241 +Kato, T. (2004) Detection of short fading episodes in two dwarf novae from VSNET observations. PASJ 56, S55 +Kato, T. (2022) Z Cam star PY Per in SU UMa state? VSOLJ Variable Star Bull. 100, (arXiv:2204.12056) +Kato, T. et al. (2010) Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. II. The Second +Year (2009-2010). PASJ 62, 1525 +Kato, T., Nogami, D., & Masuda, S. (2002) Extended deep minimum and subsequent brightening of RX And in +1996–1997. PASJ 54, 575 +Kato, T., Uemura, M., Ishioka, R., Nogami, D., Kunjaya, C., Baba, H., & Yamaoka, H. (2004) Variable Star +Network: World center for transient object astronomy and variable stars. PASJ 56, S1 +Kochanek, C. S. et al. (2017) The All-Sky Automated Survey for Supernovae (ASAS-SN) light curve server v1.0. +PASP 129, 104502 +Schreiber, M. R., Gänsicke, B. T., & Mattei, J. A. (2002) RX And: An intermediate between Z Cam and VY Scl +stars. A&A 384, L6 +Shappee, B. J. et al. (2014) The man behind the curtain: X-rays drive the UV through NIR variability in the +2013 AGN outburst in NGC 2617. ApJ 788, 48 +Shingles, L. et al. (2021) Release of the ATLAS Forced Photometry server for public use. Transient Name Server +AstroNote 7, 1 +Sion, E. M., Szkody, P., Gaensicke, B., Cheng, F. H., La Dous, C., & Hassall, B. (2001) Hubble Space Telescope +spectroscopy of the dwarf nova RX Andromedae. I. the underlying white dwarf. ApJ 555, 834 +Smith, K. W. et al. (2020) Design and operation of the ATLAS Transient Science Server. PASP 132, 085002 +Stellingwerf, R. F. (1978) Period determination using phase dispersion minimization. ApJ 224, 953 +Taylor, C. J., & Thorstensen, J. R. (1996) Orbital periods of the dwarf novae AR And, AM Cas, and PY Per. +PASP 108, 894 +Tonry, J. L. et al. (2018) ATLAS: A High-cadence All-sky Survey System. PASP 130, 064505 +Warner, B. (1995) Cataclysmic Variable Stars (Cambridge: Cambridge University Press) + diff --git a/utAzT4oBgHgl3EQfBvrw/content/tmp_files/load_file.txt b/utAzT4oBgHgl3EQfBvrw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..330f62074bdbcd77e81b8ff906eec476aed6663c --- /dev/null +++ b/utAzT4oBgHgl3EQfBvrw/content/tmp_files/load_file.txt @@ -0,0 +1,299 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf,len=298 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='00949v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='SR] 3 Jan 2023 1 Very low state in PY Per in 2022 Taichi Kato1 tkato@kusastro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='jp 1 Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan Abstract Using VSNET, VSOLJ, ASAS-SN and ATLAS observations, I found that the Z Cam star PY Per spent a long, faint low state reaching 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='1 mag at least between 2022 June and November.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' No dwarf nova outburst was recorded during this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' TESS data during this low state showed two maxima in one orbital cycle and can be interpreted as an ellipsoidal modulation arising from the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' These observations suggest that the mass-transfer almost stopped during this low state and strengthen the identification of PY Per as a VY Scl star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' PY Per had shown an unusual outburst resembling an SU UMa-type superoutburst less than half a year before (Kato 2022, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='12056) and these phenomena may have been physically related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' In Kato (2022), I reported the detection of an outburst (2021 December–2022 January) resembling an SU UMa-type superoutburst in the Z Cam star PY Per with an orbital period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='15468(5) d (Kato 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Taylor and Thorstensen 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Kato (2022) also detected faint states in observations by the American Association of Variable Stars (AAVSO), and showed that this object is also an VY Scl-type cataclysmic variable (CV) [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=', Warner (1995) for CVs in general and their subtypes].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' VY Scl-type faint states are sometimes associated with Z Cam stars, most notably the 1996–1997 totally unexpected long-lasting fading in RX And (Sion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Kato 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' After the solar conjunction following the unusual outburst mentioned above, I noticed that PY Per was in low state without any outburst in Variable Star Observers League in Japan (VSOLJ) and VSNET (Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2004) observations (vsnet-alert 27029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' This has been confirmed by using the All-Sky Automated Survey for Supernovae (ASAS-SN) Sky Patrol (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2017) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' These observations started in 2022 June (ASAS-SN) and 2022 July (VSOLJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Using the data by the Asteroid Terrestrial-impact Last Alert System (ATLAS: Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Heinze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2020) Forced Photometry (Shingles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' 2021)2, I further confirmed that the 2022 fading episode was the deepest (reaching 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='1 mag) and longest (more than 150 d) ever recorded in this object (figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' For a comparison, a light curve of the preceding seasons is given in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' In the 2019–2020 season (before BJD 2458950), the object showed frequent low-amplitude outbursts as was typical for a Z Cam star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' The behavior was similar to this at least between 2016 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' In the 2020–2021 season (BJD 2459030–2459290), the quiescence became fainter (near 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='0 mag) and longer and brighter outbursts in addition to smaller ones became more prominent than in the previous season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Although the quiescent brightness (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='0 mag) was almost as faint as the VY Scl-type low state mentioned in Kato (2022), the object definitely showed dwarf nova-type outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' The light curve in 2020–2021 probably illustrated the behavior when the mass-transfer rate decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Compared to the 2020–2021 season, the mass-transfer rate probably returned normal in the 2021–2022 season (left part of figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' There was, however, an interval lacking outbursts (BJD 2459525–2459568), but not as faint as the 2020–2021 quiescence, preceding the unusual outburst resembling an SU UMa-type superoutburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Accumulation of matter in the disk during this interval may have caused the unusual outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Although this unusual outburst may have been physically related to the very faint low state in 2022, observations were impossible due to the solar conjunction and how this very faint state started remains a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Note that the 1996–1997 low state in RX And was preceded by an unusually long standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' These rare phenomena might have been physically related and the case would also be suspected in PY Per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' I analyzed Transiting Exoplanet Survey Satellite (TESS) observations obtained in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='3 The full light- curve is available at the Mikulski Archive for Space Telescope (MAST4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' The TESS observations started on 2022 October 28 (BJD 2459882) and ended on 2022 November 26 (BJD 2459910).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' PY Per started rising in the TESS data on the final two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Since most of the TESS data were obtained when PY Per was in deep low state and since the object has nearby (unrelated) contaminating stars, I did not attempt to extract the flux of PY Per but used the flux combined with contaminating stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' Using the data before BJD 2459908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='6 (object in low state), I could detect the orbital period and modulations (figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content=' The period was determined to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf'} +page_content='15453(2) d 1 0, where Br(y) denotes the open ball of radius r centered at y ∈ RN. Then un → 0 +strongly in Lq(RN) for all 2 < q < 2∗, where 2∗ is the limiting exponent in the Sobolev embedding H1(RN) ֒→ +Lp(RN). +This version of the lemma has been used to solve semilinear elliptic equation in the whole space RN, i.e. +−∆u + u = h(u), u ∈ H1(RN). +In [3] and [4] you can find a comprehensive description of the lack of compactness in Sobolev spaces +The Lions Lemma has been generalized in some ways, for example in [5] we can find the formulation of the lemma +for isotropic Orlicz-Sobolev spaces W1 +0 LA(RN), i.e. spaces obtained by the completion of C∞ +0 (RN) with respect to +the norm ∥u∥W1 LA(RN) = ∥|∇u|∥LA(RN) + ∥u∥LA(RN), where +∥u∥LA(RN ) = inf +� +k > 0 : +� +RN A +�|u| +k +� +dt ≤ 1 +� +, +is a Luxemburg norm, A : R → [0, ∞) is an N-function (i.e. is convex, even, coercive and vanishes only at 0) +satisfying ∆2∇2 condition (i.e. there exist K1, K2 > 0, such that K1A(v) ≤ A(2v) ≤ K2A(v) for all v ∈ Rn). +Lemma 1.2 (Theorem. 1.3 in [5]). Assume that a(t)t is increasing in (0, +∞) and that there exist l, m ∈ (1, N) such +that +l ≤ a(|t|)t2 +A(t) +≤ m +for all t ̸= 0, +(1) + +PRIME AI paper +where A(t) = +� |t| +0 a(s)s ds, l ≤ m < l∗ = +lN +N−l. Let {un} ⊂ W1 LA(RN) be a bounded sequence such that there +exists R > 0 satisfying: +lim +n→∞ +� +sup +y∈RN +� +Br(y) +A(|un|) +� += 0. +(L1) +Then, for any N-function B verifying ∆2-condition and satisfying +lim +t→0 +B(t) +A(t) = 0 +and +lim +t→∞ +B(t) +A∗(t) = 0, +where A∗ is a Sobolev conjugate of A, w have +un → 0 in LB(RN). +In [5] authors use lemma 1.2 to prove the existence of solutions to some isotropic quasilinear problems. +It is worth to notice, that in the proof of the lemma above authors essentially use the fact that function A satisfies +∆2∇2 condition, which is guaranteed by condition (1). Isotropic Young function satisfying globally ∆2∇2 condition +is bounded by some power functions with power 1 < p < ∞. If A satisfies ∆2∇2 then W1 LA is a reflexive, separable +Banach space (see e.g. [6]). +There are also papers, where authors consider non-reflexive spaces, e.g. [7]. In this case instead of condition (L1) +authors use the assumption (L2) (see [8]) and assume that the sequence +�� +RN A∗(|un|) dx +� +is bounded. +Lemma 1.3 (Theorem. 1.3 in [7]). Let A, B be a N-functions, A∗ be a Sobolev conjugate of A and +lim +t→0 +B(t) +A(t) = 0 +and +lim +t→0 +B(t) +A∗(t) = 0. +If {un} ⊂ W1 LA(RN) is a sequence such that +�� +RN A(|un|) dx +� +and +�� +RN A∗(|un|) dx +� +are bounded, and for +each ε > 0 we have +meas(|un| > ε) → 0 +as +n → ∞, +(L2) +then +� +RN B(un) → 0 +as +n → ∞. +In [9] author uses the lemma similar to lemma (1.2), but for sequences from anisotropic Orlicz-Sobolev spaces, to find +solutions of the anisotropic quasilinear problem +−div(∇Φ(∇u)) + V (x)N ′(u) = f(u), +where u ∈ W1 LΦ(Rn), +(AQP) +where Φ is an anisotropic n-dimensional N-function (see more in [10]), satisfying ∆2∇2 condition. +In [11] authors prove Lion’s type lemma for reflexive fractional Orlicz-Sobolev spaces, while in [12] authors prove it +for non-reflexive fractional Orlicz-Sobolev spaces. +2 +Main Theorem +In this paper we generalize the Lions-type lemmas 1.1, 1.2, 1.3, we mentioned in the introduction. The only assumption +on functions is that they are Lebesgue measurable finite and vanish only at zero. It is worth to notice, that they can +have growth which is not bounded by polynomials, so it will be possible to use this lemma also in non-reflexive spaces. +In the proof of the following lemma we use some techniques from [11]. +Theorem 2.1. Assume that Φ1, Φ2, Ψ : Rn → [0, ∞) are Lebesgue-measurable functions vanishing only in zero, +satisfying +lim +|v|→0 +Ψ(v) +Φ1(v) = 0, +(Ψ1) +lim +|v|→∞ +Ψ(v) +Φ2(v) = 0, +(Ψ2) +2 + +PRIME AI paper +Let {uk} be a sequence of Lebesgue-measurable functions uk : RN → Rn such that +�� +RN Φ1(uk) +� +, +�� +RN Φ2(uk) +� +are bounded and +lim +k→∞ +� +sup +y∈RN +� +Br(y) +Φ1(uk) +� += 0 +(2) +for some r > 0. Then +lim +k→∞ +� +RN Ψ(uk) = 0 +Proof. We let |A| denote the Lebesgue measure of subset A. Let {uk} be a sequence of Lebesgue-measurable func- +tions such that +�� +RN Φ1(uk) +� +, +�� +RN Φ2(uk) +� +are bounded. +Define +M1 = sup +k +� +RN Φ1(uk) +M2 = sup +k +� +RN Φ2(uk). +Note that M1, M2 < ∞. Fix ε > 0. From (Ψ1), there exists δ > 0, such that +Ψ(v) +Φ1(v) ≤ +ε +3M1 +(3) +for all |v| ≤ δ. +Similarly from (Ψ2), there exists T > 0, such that +Ψ(v) +Φ2(v) ≤ +ε +3M2 +(4) +for all |v| ≥ T . Let us denote: +Ak = +� +x ∈ RN : |uk(x)| ≤ δ +� +, +Bk = +� +x ∈ RN : δ < |uk(x)| < T +� +, +Ck = +� +x ∈ RN : |uk(x)| ≥ T +� +. +Then +� +RN Ψ(uk) = +� +Ak +Ψ(uk) + +� +Bk +Ψ(uk) + +� +Ck +Ψ(uk). +By (3) we obtain +� +Ak +Ψ(uk) ≤ +ε +3M1 +� +RN Φ1(uk) ≤ ε +3 +and by (4) +� +Ck +Ψ(uk) ≤ +ε +3M2 +� +RN Φ2(uk) ≤ ε +3. +We need to show that +� +Bk +Ψ(uk) ≤ ε +3. +We will first show that |Bk| → 0 as k → ∞. +Assume, by contradiction, that (up to subsequence) +|Bk| → L > 0. +Then, for some subsequence {uk}, there exist y0 ∈ RN and σ > 0 such that +|Bk ∩ Br(y0)| ≥ σ > 0. +(5) +Let +CΨ = +max +δ≤|v|≤T Ψ(v), +cΦ = +min +δ≤|v|≤T Φ1(v), +CΦ = +max +δ≤|v|≤T Φ1(v). +We observe that +� +Br(y0) +Φ1(uk) ≥ +� +Br(y0)∩Bk +Φ1(uk) ≥ cΦ|Bk ∩ Br(y0)|. +Hence and by assumption (2) we have that +|Bk ∩ Br(y0)| → 0 +as k → ∞ +3 + +PRIME AI paper +which contradicts with (5). +Since |Bk| → 0 as k → ∞, we have that there exists k0 such that for all k ≥ k0 +|Bk| < cΦ (3CΦCΨ)−1 ε. +Then +|Bk| ≤ (cΦ)−1 +� +Bk +Φ(uk) ≤ CΦ (cΦ)−1 |Bk| +and +� +Bk +Ψ(uk) ≤ CΨ (cΦ)−1 +� +Bk +Φ(uk) ≤ CΨCΦ (cΦ)−1 |Bk| < ε +3. +Remark 2.2. Note that what matters in this theorem (just as in the concentration-compactness lemma of Lions in [8]) +is the behavior of the integral, not the space. +References +[1] Paul L. Lions. The concentration-compactness principle in the calculus of variations. The locally compact case. +I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21(2):109–145, 1984. +[2] David G. Costa. An invitation to variational methods in differential equations. Birkhäuser Boston, Inc., Boston, +MA, 2007. +[3] Mathieu Lewin. Describing lack of compactness in Sobolev spaces. Lecture - Taken from unpublished lecture +notes "Variational Methods in Quantum Mechanics" written for a course delivered at the University of Cergy- +Pontoise in 2010., January 2010. +[4] Michael Struwe. Variational methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. +A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of +Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial +differential equations and Hamiltonian systems. +[5] Claudianor O. Alves, Giovany M. Figueiredo, and Jefferson A. Santos. Strauss and Lions type results for a class +of Orlicz-Sobolev spaces and applications. Topol. Methods Nonlinear Anal., 44(2):435–456, 2014. +[6] Robert A. Adams and John J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Ams- +terdam). Elsevier/Academic Press, Amsterdam, second edition, 2003. +[7] Claudianor O. Alves and Marcos L. M. Carvalho. A Lions type result for a large class of Orlicz-Sobolev space +and applications. Mosc. Math. J., 22(3):401–426, 2022. +[8] Elliott H. Lieb. On the lowest eigenvalue of the laplacian for the intersection of two domains. Inventiones +mathematicae, 74:441–448, 1983. +[9] Karol Wro´nski. Quasilinear elliptic problem in anisotrpic orlicz-sobolev space on unbounded domain, 2022. +[10] G. Barletta and A. Cianchi. Dirichlet problems for fully anisotropic elliptic equations. Proc. Royal Soc. Ed., +147(1):25–60, 2017. +[11] Edcarlos D. Silva, M. L. Carvalho, J. C. de Albuquerque, and Sabri Bahrouni. Compact embedding theorems +and a Lions’ type lemma for fractional Orlicz-Sobolev spaces. J. Differential Equations, 300:487–512, 2021. +[12] Sabri Bahrouni, Hichem Ounaies, and Olfa Elfalah. Problems involving the fractional g-laplacian with lack of +compactness, 2022. +4 + diff --git a/vdE3T4oBgHgl3EQfOAme/content/tmp_files/load_file.txt b/vdE3T4oBgHgl3EQfOAme/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb7e5018a28d84780eb5c0a59f97443ba8ac8f6c --- /dev/null +++ b/vdE3T4oBgHgl3EQfOAme/content/tmp_files/load_file.txt @@ -0,0 +1,167 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf,len=166 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='04389v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='AP] 11 Jan 2023 GENERALIZED VERSION OF THE LIONS-TYPE LEMMA Magdalena Chmara Department of Technical Physics and Applied Mathematics, Gda´nsk University of Technology, Narutowicza 11/12, 80-233 Gda´nsk, Poland magdalena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='chmara@pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='pl ABSTRACT In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue- measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' This lemma generalizes some results for a class of Orlicz-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' What matters here is the behavior of the integral, not the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Keywords Lions-type result · concentration-compactness · unbounded domains 1 Introduction In 1984 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Lions published his celebrated article [1], in which he introduced a concentration-compactness method for solving minimization problems on unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' One of the main tools provided by [1] is lemma I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' A variety of formulations of this lemma has been widely used to deal with the lack of compactness on unbounded domain for different types of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 102] we can find the following version of the Lion’s Lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Suppose {un} ∈ H1(RN) is a bounded sequence satisfying lim n→∞ � sup y∈RN � Br(y) |un|p � = 0 for some p ∈ [2, 2∗] and r > 0, where Br(y) denotes the open ball of radius r centered at y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Then un → 0 strongly in Lq(RN) for all 2 < q < 2∗, where 2∗ is the limiting exponent in the Sobolev embedding H1(RN) ֒→ Lp(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' This version of the lemma has been used to solve semilinear elliptic equation in the whole space RN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' −∆u + u = h(u), u ∈ H1(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In [3] and [4] you can find a comprehensive description of the lack of compactness in Sobolev spaces The Lions Lemma has been generalized in some ways, for example in [5] we can find the formulation of the lemma for isotropic Orlicz-Sobolev spaces W1 0 LA(RN), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' spaces obtained by the completion of C∞ 0 (RN) with respect to the norm ∥u∥W1 LA(RN) = ∥|∇u|∥LA(RN) + ∥u∥LA(RN), where ∥u∥LA(RN ) = inf � k > 0 : � RN A �|u| k � dt ≤ 1 � , is a Luxemburg norm, A : R → [0, ∞) is an N-function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' is convex, even, coercive and vanishes only at 0) satisfying ∆2∇2 condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' there exist K1, K2 > 0, such that K1A(v) ≤ A(2v) ≤ K2A(v) for all v ∈ Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='2 (Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='3 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Assume that a(t)t is increasing in (0, +∞) and that there exist l, m ∈ (1, N) such that l ≤ a(|t|)t2 A(t) ≤ m for all t ̸= 0, (1) PRIME AI paper where A(t) = � |t| 0 a(s)s ds, l ≤ m < l∗ = lN N−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Let {un} ⊂ W1 LA(RN) be a bounded sequence such that there exists R > 0 satisfying: lim n→∞ � sup y∈RN � Br(y) A(|un|) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' (L1) Then, for any N-function B verifying ∆2-condition and satisfying lim t→0 B(t) A(t) = 0 and lim t→∞ B(t) A∗(t) = 0, where A∗ is a Sobolev conjugate of A, w have un → 0 in LB(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In [5] authors use lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='2 to prove the existence of solutions to some isotropic quasilinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' It is worth to notice, that in the proof of the lemma above authors essentially use the fact that function A satisfies ∆2∇2 condition, which is guaranteed by condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Isotropic Young function satisfying globally ∆2∇2 condition is bounded by some power functions with power 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' If A satisfies ∆2∇2 then W1 LA is a reflexive, separable Banach space (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' There are also papers, where authors consider non-reflexive spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In this case instead of condition (L1) authors use the assumption (L2) (see [8]) and assume that the sequence �� RN A∗(|un|) dx � is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='3 (Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='3 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Let A, B be a N-functions, A∗ be a Sobolev conjugate of A and lim t→0 B(t) A(t) = 0 and lim t→0 B(t) A∗(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' If {un} ⊂ W1 LA(RN) is a sequence such that �� RN A(|un|) dx � and �� RN A∗(|un|) dx � are bounded, and for each ε > 0 we have meas(|un| > ε) → 0 as n → ∞, (L2) then � RN B(un) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In [9] author uses the lemma similar to lemma (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='2), but for sequences from anisotropic Orlicz-Sobolev spaces, to find solutions of the anisotropic quasilinear problem −div(∇Φ(∇u)) + V (x)N ′(u) = f(u), where u ∈ W1 LΦ(Rn), (AQP) where Φ is an anisotropic n-dimensional N-function (see more in [10]), satisfying ∆2∇2 condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In [11] authors prove Lion’s type lemma for reflexive fractional Orlicz-Sobolev spaces, while in [12] authors prove it for non-reflexive fractional Orlicz-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 2 Main Theorem In this paper we generalize the Lions-type lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='3, we mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' The only assumption on functions is that they are Lebesgue measurable finite and vanish only at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' It is worth to notice, that they can have growth which is not bounded by polynomials, so it will be possible to use this lemma also in non-reflexive spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' In the proof of the following lemma we use some techniques from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Assume that Φ1, Φ2, Ψ : Rn → [0, ∞) are Lebesgue-measurable functions vanishing only in zero, satisfying lim |v|→0 Ψ(v) Φ1(v) = 0, (Ψ1) lim |v|→∞ Ψ(v) Φ2(v) = 0, (Ψ2) 2 PRIME AI paper Let {uk} be a sequence of Lebesgue-measurable functions uk : RN → Rn such that �� RN Φ1(uk) � , �� RN Φ2(uk) � are bounded and lim k→∞ � sup y∈RN � Br(y) Φ1(uk) � = 0 (2) for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Then lim k→∞ � RN Ψ(uk) = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' We let |A| denote the Lebesgue measure of subset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Let {uk} be a sequence of Lebesgue-measurable func- tions such that �� RN Φ1(uk) � , �� RN Φ2(uk) � are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Define M1 = sup k � RN Φ1(uk) M2 = sup k � RN Φ2(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Note that M1, M2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' From (Ψ1), there exists δ > 0, such that Ψ(v) Φ1(v) ≤ ε 3M1 (3) for all |v| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Similarly from (Ψ2), there exists T > 0, such that Ψ(v) Φ2(v) ≤ ε 3M2 (4) for all |v| ≥ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Let us denote: Ak = � x ∈ RN : |uk(x)| ≤ δ � , Bk = � x ∈ RN : δ < |uk(x)| < T � , Ck = � x ∈ RN : |uk(x)| ≥ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Then � RN Ψ(uk) = � Ak Ψ(uk) + � Bk Ψ(uk) + � Ck Ψ(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' By (3) we obtain � Ak Ψ(uk) ≤ ε 3M1 � RN Φ1(uk) ≤ ε 3 and by (4) � Ck Ψ(uk) ≤ ε 3M2 � RN Φ2(uk) ≤ ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' We need to show that � Bk Ψ(uk) ≤ ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' We will first show that |Bk| → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Assume, by contradiction, that (up to subsequence) |Bk| → L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Then, for some subsequence {uk}, there exist y0 ∈ RN and σ > 0 such that |Bk ∩ Br(y0)| ≥ σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' (5) Let CΨ = max δ≤|v|≤T Ψ(v), cΦ = min δ≤|v|≤T Φ1(v), CΦ = max δ≤|v|≤T Φ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' We observe that � Br(y0) Φ1(uk) ≥ � Br(y0)∩Bk Φ1(uk) ≥ cΦ|Bk ∩ Br(y0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Hence and by assumption (2) we have that |Bk ∩ Br(y0)| → 0 as k → ∞ 3 PRIME AI paper which contradicts with (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Since |Bk| → 0 as k → ∞, we have that there exists k0 such that for all k ≥ k0 |Bk| < cΦ (3CΦCΨ)−1 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Then |Bk| ≤ (cΦ)−1 � Bk Φ(uk) ≤ CΦ (cΦ)−1 |Bk| and � Bk Ψ(uk) ≤ CΨ (cΦ)−1 � Bk Φ(uk) ≤ CΨCΦ (cΦ)−1 |Bk| < ε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Note that what matters in this theorem (just as in the concentration-compactness lemma of Lions in [8]) is the behavior of the integral, not the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' References [1] Paul L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' The concentration-compactness principle in the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' The locally compact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Poincaré Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Non Linéaire, 21(2):109–145, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [2] David G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Costa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' An invitation to variational methods in differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Birkhäuser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=', Boston, MA, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [3] Mathieu Lewin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Describing lack of compactness in Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Lecture - Taken from unpublished lecture notes "Variational Methods in Quantum Mechanics" written for a course delivered at the University of Cergy- Pontoise in 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=', January 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [4] Michael Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Variational methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Folge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 3rd Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Springer-Verlag, Berlin, fourth edition, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Applications to nonlinear partial differential equations and Hamiltonian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [5] Claudianor O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Alves, Giovany M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Figueiredo, and Jefferson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Methods Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=', 44(2):435–456, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [6] Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Adams and John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Fournier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Sobolev spaces, volume 140 of Pure and Applied Mathematics (Ams- terdam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Elsevier/Academic Press, Amsterdam, second edition, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [7] Claudianor O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Alves and Marcos L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Carvalho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' A Lions type result for a large class of Orlicz-Sobolev space and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Mosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Math.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Royal Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=', 147(1):25–60, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [11] Edcarlos D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Silva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Carvalho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' de Albuquerque, and Sabri Bahrouni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Compact embedding theorems and a Lions’ type lemma for fractional Orlicz-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Differential Equations, 300:487–512, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' [12] Sabri Bahrouni, Hichem Ounaies, and Olfa Elfalah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' Problems involving the fractional g-laplacian with lack of compactness, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE3T4oBgHgl3EQfOAme/content/2301.04389v1.pdf'} diff --git a/wdAzT4oBgHgl3EQf7f5k/vector_store/index.pkl b/wdAzT4oBgHgl3EQf7f5k/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..11321a4f39d902c832986ddb9916c26cdbcc9788 --- /dev/null +++ b/wdAzT4oBgHgl3EQf7f5k/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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J. Bambic1★, H. R. Russell2, C. S. Reynolds3,4,5, A. C. Fabian3, B. R. McNamara6,7 and P. E. J. Nulsen8,9 +1 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA +2 School of Physics, Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, UK +3 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK +4 Department of Astronomy, University of Maryland, College Park, MD 20742-2421, USA +5 Joint Space Science Institute (JSI), College Park, MD 20742-2421, USA +6 Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada +7 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada +8 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA +9 ICRAR, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia +31 January 2023 +ABSTRACT +We present the deepest Chandra observation to date of the galaxy M84 in the Virgo Cluster, with over 840 kiloseconds of data +provided by legacy observations and a recent 730 kilosecond campaign. The increased signal-to-noise allows us to study the +origins of the accretion flow feeding the supermassive black hole in the center of M84 from the kiloparsec scales of the X-ray +halo to the Bondi radius, 𝑅B. Temperature, metallicity, and deprojected density profiles are obtained in four sectors about M84’s +AGN, extending into the Bondi radius. Rather than being dictated by the potential of the black hole, the accretion flow is strongly +influenced by the AGN’s bipolar radio jets. Along the jet axis, the density profile is consistent with 𝑛𝑒 ∝ 𝑟−1; however, the +profiles flatten perpendicular to the jet. Radio jets produce a significant asymmetry in the flow, violating a key assumption of +Bondi accretion. Temperature in the inner kiloparsec is approximately constant, with only a slight increase from 0.6 to 0.7 keV +approaching 𝑅B, and there is no evidence for a temperature rise imposed by the black hole. The Bondi accretion rate �𝑀B exceeds +the rate inferred from AGN luminosity and jet power by over four orders of magnitude. In sectors perpendicular to the jet, �𝑀B +measurements agree; however, the accretion rate is > 4𝜎 lower in the North sector along the jet, likely due to cavities in the X-ray +gas. Our measurements provide unique insight into the fueling of AGN responsible for radio mode feedback in galaxy clusters. +Key words: X-rays: galaxies: clusters — galaxies: clusters: M84 — intergalactic medium +1 INTRODUCTION +Accretion onto active galactic nuclei (AGN) at the centers of mas- +sive elliptical galaxies fuels AGN feedback in clusters of galax- +ies. The gravitational potential energy released from plasma flowing +onto these supermassive black holes (SMBHs) powers jets of rela- +tivistic particles which sculpt the surrounding intracluster medium +(ICM). In cool-core clusters where the central cooling time of the hot +(107 − 108 K) ICM is ≲ Gyr, thermalization of the jet kinetic energy +provides the heating necessary to balance radiative cooling, maintain- +ing cluster atmospheres in their observed quasi-thermal equilibrium +and averting a “cooling catastrophe” (Fabian 1994; McNamara & +Nulsen 2007; Fabian 2012). +Deep (≳100 kiloseconds), spatially-resolved Chandra X-ray Space +Telescope observations of nearby galaxy clusters such as Perseus and +Virgo have revealed the signatures of this feedback process: cavities +or bubbles carved out of the ICM by jets (McNamara et al. 2000; +Churazov et al. 2001); weak shocks, ripples, and waves emanating +from newly formed bubbles (Sanders & Fabian 2007, 2008; Forman +et al. 2007); bright filaments formed by gas cooling around these +★ E-mail: cbambic@princeton.edu +cavities (Fabian et al. 2003); and turbulent fluctuations stirred by +the buoyant rise of bubbles through clusters (Churazov et al. 2004; +Zhuravleva et al. 2014; Hitomi Collaboration et al. 2016; Simionescu +et al. 2019). Yet, while the X-ray morphology of clusters has pro- +vided insight into how AGN shape their environments on 10s − 100s +of kiloparsecs (kpc) scales, understanding the connection between +AGN and the sub-kiloparsec scale accretion flows which power them +remains a critical uncertainty in the paradigm of AGN feedback. +Even the basic energetics of large-scale black hole “feeding” is an +open problem (see Abramowicz & Fragile 2013 for a review). In the +standard paradigm, material within the SMBH’s sphere of influence, +the Bondi radius (𝑅B = 2𝐺𝑀BH/𝑐2𝑠 where 𝑀BH is the black hole +mass and 𝑐𝑠 is the speed of sound well beyond 𝑅𝐵), is destined to +either reach the hole or race away in an outflow. Ionized gas pierces +the sphere of influence at a rate +�𝑀B en route to the hole, where +this plasma is consumed at a rate �𝑀. A fraction 𝜂 of the rest mass +power �𝑀𝑐2 is released by the accretion flow in the form of radiation +and outflows—winds or jets—such that the total power (radiative + +outflow) of the AGN is 𝐿 = 𝜂 �𝑀𝑐2. +At the largest scales, a gas inflow with accretion rate �𝑀B is formed +by gas cooling and gravitational infall under the influence of the com- +bined galactic and SMBH potential (Quataert & Narayan 2000). The +© 0000 The Authors +arXiv:2301.11937v1 [astro-ph.HE] 27 Jan 2023 + +2 +C.J. Bambic et al. +accretion rate �𝑀 is then influenced by the structure of this inflow: +the angular momentum (Proga & Begelman 2003) and effective tur- +bulent viscosity (Narayan & Fabian 2011) of the gas, and the relative +contributions of hot X-ray emitting plasma (Matteo et al. 2003) vs. +cold atomic and molecular gas (Pizzolato & Soker 2005) which may +“rain down” through the Bondi radius (Gaspari et al. 2012; Yang & +Reynolds 2016). Magnetic fields certainly complicate this picture, +with the magnetic flux frozen into the flow (Lubow et al. 1994) +competing with dynamo-generated fields (Brandenburg et al. 1995; +Brandenburg & Subramanian 2005; Blackman 2012; Liska et al. +2020) to power relativistic jets (Blandford & Znajek 1977; Komis- +sarov 2001; Tchekhovskoy et al. 2010, 2011) and winds (Blandford +& Payne 1982; Proga 2000), and thereby influence the value of 𝜂. +The complexities of this inflow determine the state of the resulting +accretion disk around the black hole and the relative contribution of +radiation to the flow’s structure. For the jetted systems of interest in +cluster AGN feedback, the accretion flow is likely radiatively ineffi- +cient, forming a virialized, geometrically-thick advection-dominated +accretion flow (ADAF; Ichimaru 1977; Rees et al. 1982; Narayan & +Yi 1994, 1995; Quataert & Narayan 1999), a convection-dominated +accretion flow (CDAF; Quataert & Gruzinov 2000), or when the +net magnetic flux reaching the hole is large, a magnetically arrested +disk (MAD; Bisnovatyi-Kogan & Ruzmaikin 1974; Narayan et al. +2003; Igumenshchev 2008; McKinney et al. 2012; Avara et al. 2016; +Marshall et al. 2018; Ripperda et al. 2022). +While �𝑀B is crucial in determining accretion flow structure, mea- +suring this parameter is a major challenge. Because the true +�𝑀B +cannot be measured directly, large-scale black hole feeding is often +interpreted through a steady, spherically symmetric model of accre- +tion, the Bondi (1952) solution. Within this framework, �𝑀B for a +given SMBH mass is specified entirely by the gas density and tem- +perature at 𝑅B, quantities which in principle can be measured with +deep X-ray observations. +This choice is one of convenience—there are no strong theoreti- +cal reasons to expect the assumptions of the Bondi solution to hold +in real systems. However, some evidence points to the importance +of �𝑀B in setting feedback power. Allen et al. (2006), using a small +sample of X-ray observations of nearby elliptical galaxies, found +an apparent correlation between Bondi accretion rate and AGN jet +power, as measured from the enthalpy of jet-blown cavities. This +method for inferring jet power is subject to significant uncertainties, +e.g. projection effects and the assumption of subsonic inflation. In- +deed, a follow-up study by Russell et al. (2013) using a larger sample +of elliptical galaxies found a less significant correlation. +A direct correlation between Bondi accretion rate and AGN feed- +back power has interesting consequences. The correlation may imply +a universality in the radiatively inefficient accretion flows (RIAFs) +which power AGN in early type galaxies, with �𝑀B serving as the +crucial parameter for regulating power from radiative (𝐿Rad) and +jet (𝐿Jet) feedback on ∼Gyr timescales. In addition, the correlation +could be leveraged in sub-grid models for galaxy formation, where +feedback power from unresolved AGN must be tuned based on re- +solvable properties, such as �𝑀B (Pillepich et al. 2018). Establishing +this correlation necessitates deep X-ray observations which resolve +the density and temperature at 𝑅B. +In this paper, we harness the deepest X-ray observations to date +of the galaxy M84 (NGC 4374) to measure the Bondi accretion rate +of hot phase (≳ 0.5 keV) gas onto a jetted AGN in an early type +galaxy. These measurements are based on a new Chandra campaign +which yielded approximately 730 kiloseconds (ks) on M84. Com- +bined with legacy data published in Finoguenov & Jones (2001, +2002) and Finoguenov et al. (2008), the observations presented com- +prise over 840 ks of X-ray data. +M84 is one of only 5 known systems where the Bondi radius +can be resolved by Chandra, despite the observatory’s remarkable +sub-arcsecond angular resolution. The other 4 systems are Sgr A∗ +(Baganoff et al. 2003), NGC 3115 (Wong et al. 2014), NGC 1600 +(Runge & Walker 2021), and M87 (Russell et al. 2015, hereafter +HRR15). Even within this small class, M84 stands out. Unlike Sgr A∗ +and NGC 1600, M84 has an X-ray detected AGN. In contrast to +NGC 3115, a Fanaroff & Riley (1974) Type I radio jet is clearly +observed in M84. However, unlike that in M87 which hosts a notably +powerful jet, M84’s AGN is not particularly luminous (more than +an order of magnitude dimmer than M87’s AGN) and our extended +campaign caught the SMBH in a relatively quiescent state. Thus, +M84 does not require the same sophisticated treatment of pile-up +as was performed in M87 (HRR15). These factors make M84 an +especially useful object for exploring the interplay of feeding and +feedback in elliptical galaxies. +This paper is organized as follows. We describe our data analysis +in §2 including data reduction, spectral models for the AGN and +galactic gas, and simulations of the detector point spread function +(PSF) used for forward modelling spectral contamination from the +AGN. In §3, we present results: profiles of gas density, temperature, +and metallicity approaching and just within the Bondi radius, and the +measured Bondi accretion rates �𝑀B and efficiencies 𝜂. We discuss +the implications of our measurements in §4, and conclude in §5. +2 CHANDRA DATA ANALYSIS +M84 is a nearby (luminosity distance 𝐷𝐿 = 16.83 Mpc; redshift 𝑧 = +0.00327) giant elliptical galaxy (type E1) and satellite member of the +Virgo Cluster of galaxies. The galaxy has been the subject of three +separate Chandra ACIS-S campaigns which together yield ≈840 ks +of data. While earlier works by Finoguenov & Jones (2001) and +Finoguenov et al. (2008) addressed the detailed structure of M84 and +how the X-ray halo is shaped by feedback, our ultra-deep campaign +is concerned primarily with black hole feeding and gas structure +approaching and just within the Bondi radius of the SMBH. +2.1 Data Reduction +This work is a follow-up to a similar analysis of M87 by HRR15. +Thus, we follow the same data reduction procedure. +Our data reduction was performed using CIAO version 4.11 and +the Calibration Database (CalDB) 4.8.5, updated November 7, 2019 +(Fruscione et al. 2006). This update followed a major revision to the +soft energy response brought about by contaminant build-up over +Chandra’s prolific 23 year lifetime (thus far). Our long campaign +was affected by this contamination, and as we show, the majority of +M84’s galactic gas, especially that approaching the Bondi radius, is +cooler than 1 keV and emitting X-rays within the range of degraded +performance. Given the low temperature of the extended emission +in M84, the calibration of the contaminant build up on Chandra’s +optical block filters is particularly important. We therefore verified +that temperature, metallicity and normalization values measured with +the new observations are consistent with the archival observations, +which were taken only a few years after Chandra’s launch and less +affected. Using the chandra_repro routine, we reprocess our data +to produce second-level event files, removing bad pixels based on +the analysis reference data library (ardlib), detecting point sources +using wavdetect, and creating light curves to filter out bad time +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +3 +Figure 1. Left: Merged 0.5−2 keV image, totaling 798.66 ks of cleaned exposure time with Obs. ID 5908 excluded (see §2.9). Right: 5 GHz radio contours (white) +as measured by the Very Large Array (VLA) overlaid on X-ray image. Contours correspond to 12 logarithmically-spaced levels in flux from 2 × 10−3 − 0.1 mJy, +and colors denote X-ray counts. The H-shaped morphology is carved out of the halo by radio jets, forming bright rims about the radio lobes. +intervals. To produce merged images, we assume an exposure cor- +rection for each Obs. ID’s exposure map. +2.2 X-ray Morphology +Figure 1 displays a merged 0.5−2 keV image based on all three cam- +paigns. Similar images can be found in Finoguenov & Jones (2001) +and Finoguenov et al. (2008) from the first two sets of observations. +Using only limited Chandra data, Finoguenov & Jones (2001) +were able to identify the salient features of the galaxy’s X-ray emis- +sion. Instead of a featureless X-ray halo, M84 hosts depressions in +emissivity North and South of the central AGN, coincident with radio +lobes produced by Fanaroff & Riley (1974) Type I jet activity (Laing +& Bridle 1987). These cavities create an H-shaped structure in the +halo gas, which extends ≈ 150′′ (12.2 kpc) from the Northernmost +edge of the emission to the faint rim in the Southwest of the image. +The crossbar of the H spans ≈ 46′′ (3.7 kpc) and is approximately +aligned with optical dust lanes (Hansen et al. 1985), although the +dust lanes are on a much larger scale, cutting across the X-ray image. +As argued by Finoguenov et al. (2008), these cavities may actually +be comprised of at least two bubbles each, with bright rims (viewed +in projection) demarcating the bubble boundaries. Indeed, our deep +observation is able to clearly detect a tenuous bubble rim extending +toward the Southwest in the image. The Northern bubble is com- +pressed, likely by the ram pressure of ICM gas as the galaxy moves +through the cluster. +While the bubbles are located just to the North and South of +the crossbar, the jet is aligned with the West filament; the galaxy has +drifted over time. Subsequently, ram pressure has swept the Northern +bubble back and “bent” the radio jet—a signature of radio galaxies +moving through clusters (Miley et al. 1972; Owen & Rudnick 1976; +Begelman et al. 1979; Morsony et al. 2013; McBride & McCourt +2014). Intriguingly, the Southern bubble has not been swept in the +same direction. There may be a large-scale shear flow across M84, +or the jet may have reoriented itself over the course of the episodes +recorded in the radio lobes, possibly through precession. +2.3 Spectral Fitting +Unless stated otherwise, we fit spectra simultaneously with the XSPEC +spectral fitting package (Arnaud 1996) using all Obs. IDs listed in +Table 1. Spectra are extracted using CIAO’s specextract function +and grouped such that at least one count is present in all energy bins +over the range of 0.5−7 keV. Fits are performed using the modified C- +statistic (Cash 1979) with elemental abundances taken from Anders +& Grevesse (1989) for comparison with past results. All spectral +models are fit with galactic absorption included via a photoelectric +absorption (phabs) model, with a constant galactic column density +of 𝑁H,gal += 2.9 × 1020 cm2 as measured by the HI4Pi Survey +(HI4PI Collaboration et al. 2016). +2.4 Virgo Cluster Spectral Model +M84 is embedded in the Virgo Cluster, so we must peer through a +“screen” of hard X-ray emission ≳1 keV. Because the Virgo Cluster +occupies the entire field of view, we follow HRR15 and choose to use +blank sky backgrounds for all spectral fits. The appropriate blank- +sky background dataset was processed identically to the event file, +reprojected to the same sky position, and normalized so that the count +rate matched that of the event file for the 9.5 − 12 keV energy band. +We model the bremsstrahlung emission from the Virgo ICM with +a single temperature APEC plasma emission model. The parameters +for this model are determined by fitting a spectrum extracted from a +large region, a 4.6′ × 4.9′ box around M84, excluding point sources +(as detected by wavdetect) and the galaxy, whose X-ray emission +is confined within a 1.9′ × 2.6′ box. +This Virgo spectrum is fit with temperature (𝑇), metallicity +(𝑍), and normalization free. The fit yields reasonable values, +MNRAS 000, 000–000 (0000) + +1.63.kpc +20'arcsec +2 +5 +12 +24 +49 +98 +197 +3931.63 kpc +20 arcsec +2 +4 +9 +18 +37 +74 +150 +2994 +C.J. Bambic et al. +Obs. ID +Date +Exposure +𝑁H +Γ +Flux (2-10 keV) +C-Stat/DOF +(ks) +(1022 cm−2) +(10−13 erg cm−2 s−1) +803 +19/05/2000 +28.47 +0.23+0.09 +−0.07 +1.79+0.20 +−0.19 +0.99+0.14 +−0.12 +142.6/ 184 +5908 +01/05/2005 +46.08 +0.16+0.03 +−0.03 +2.03+0.10 +−0.10 +1.62+0.12 +−0.11 +233.7/ 277 +6131 +07/11/2005 +40.93 +0.81+0.39 +−0.31 +1.68+0.36 +−0.34 +0.63+0.10 +−0.08 +146.3/ 165 +20539 +05/04/2019 +39.54 +0.16+0.29 +−0.16 +1.66+0.36 +−0.31 +0.50+0.09 +−0.07 +121.8/ 141 +20540 +26/02/2019 +30.17 +0.08+0.28 +−0.08 +1.78+0.38 +−0.25 +0.49+0.09 +−0.08 +109.5/ 127 +20541 +10/04/2019 +11.29 +0.37+0.66 +−0.37 +2.11+0.97 +−0.80 +0.46+0.26 +−0.15 +42.1/ 55 +20542 +18/03/2019 +34.61 +0.005+0.28 +−0.005 +1.46+0.36 +−0.18 +0.48+0.08 +−0.08 +120.9/ 122 +20543 +27/04/2019 +54.34 +1.57+0.63 +−0.52 +2.95+0.53 +−0.48 +0.32+0.05 +−0.04 +113.5/ 135 +21845 +28/03/2019 +27.70 +0.50+0.25 +−0.32 +2.05+0.34 +−0.42 +0.47+0.09 +−0.08 +118.8/ 113 +21867 +13/03/2019 +23.63 +0.36+0.53 +−0.35 +2.31+0.62 +−0.52 +0.34+0.08 +−0.07 +80.0/ 104 +22126 +28/02/2019 +35.10 +0.22+0.20 +−0.16 +1.80+0.29 +−0.27 +0.67+0.11 +−0.09 +132.3/ 153 +22127 +02/03/2019 +22.77 +0.27+0.26 +−0.21 +1.75+0.32 +−0.31 +0.85+0.14 +−0.12 +93.2/ 136 +22128 +03/03/2019 +23.75 +0.37+0.32 +−0.25 +1.78+0.40 +−0.36 +0.70+0.14 +−0.11 +94.6/ 124 +22142 +14/03/2019 +20.77 +0.74+0.64 +−0.47 +2.53+0.74 +−0.66 +0.35+0.10 +−0.08 +58.5/ 86 +22143 +16/03/2019 +22.75 +0.59+0.42 +−0.34 +2.05+0.45 +−0.42 +0.70+0.13 +−0.11 +111.0/ 123 +22144 +15/03/2019 +31.75 +0.09+0.19 +−0.09 +2.03+0.31 +−0.26 +0.46+0.08 +−0.07 +128.0/ 142 +22153 +23/03/2019 +21.08 +0.89+0.51 +−0.41 +2.58+0.56 +−0.51 +0.45+0.10 +−0.08 +74.2/ 93 +22163 +29/03/2019 +35.59 +0.54+0.33 +−0.27 +1.86+0.34 +−0.32 +0.65+0.10 +−0.08 +125.3/ 153 +22164 +31/03/2019 +32.63 +0.59+0.42 +−0.39 +1.85+0.40 +−0.41 +0.60+0.10 +−0.08 +134.7/ 142 +22166 +06/04/2019 +38.56 +0.41+0.31 +−0.24 +2.31+0.39 +−0.35 +0.34+0.06 +−0.05 +117.2/ 132 +22174 +11/04/2019 +49.41 +0.84+0.36 +−0.30 +2.20+0.38 +−0.35 +0.52+0.08 +−0.07 +129.4/ 160 +22175 +12/04/2019 +27.20 +0.49+0.41 +−0.34 +1.83+0.43 +−0.40 +0.54+0.11 +−0.09 +93.0/ 114 +22176 +13/04/2019 +51.39 +0.57+0.23 +−0.20 +2.21+0.26 +−0.25 +0.64+0.07 +−0.06 +165.3/ 195 +22177 +14/04/2019 +36.58 +0.86+0.37 +−0.32 +2.59+0.40 +−0.38 +0.49+0.07 +−0.06 +100.9/ 147 +22195 +28/04/2019 +38.07 +0.66+0.61 +−0.49 +1.98+0.54 +−0.50 +0.41+0.08 +−0.06 +133.0/ 122 +22196 +07/05/2019 +20.58 +0.97+0.56 +−0.44 +2.72+0.63 +−0.57 +0.34+0.08 +−0.07 +67.1/ 88 +New Campaign +02/2019-05/2019 +729.26 +0.49+0.07 +−0.07 +2.05+0.08 +−0.08 +0.50+0.02 +−0.02 +2645.8/ 2973 +All Data +05/2000-05/2019 +844.74 +0.44+0.06 +−0.06 +2.00+0.08 +−0.08 +0.52+0.02 +−0.02 +3055.9/ 3328 +Table 1. Summary of observations. Fits to AGN are obtained using the “M84 Model” as presented in the text, with a VAPEC model for the galactic emission and +an APEC model for the Virgo Cluster “screen.” The photon index Γ remains close to 2 as expected for Comptonized emission from an ADAF. We include local +absorption with column density 𝑁H to account for intervening dust lanes or a dusty torus around the AGN. +𝑇 = 2.32 ± 0.06 keV and 𝑍 = 0.429 ± 0.04 𝑍⊙, consistent with +𝑇 ≈ 2.3 keV obtained by Urban et al. (2011) and Ehlert et al. (2013) +using the much higher spectral resolution of XMM-Newton. +2.5 M84 Galactic Gas Spectral Model +The earliest Chandra measurements of M84 by Finoguenov & Jones +(2001) showed an overabundance of metals relative to solar. This +overabundance could be contributed both by iron-peak elements +(Fe and Ni) originating from Type Ia supernovae, or 𝛼 elements +(C, N, O, Al, Si, etc.) produced by Type II supernovae. Since XMM- +Newton lacks the spatial resolution of Chandra, abundance measure- +ments performed by XMM probe larger length scales than we are +studying; we must constrain 𝛼 element metallicities ourselves. +We fit the spectrum of the full 4.6′ × 4.9′ box with the galaxy +included using a VAPEC model for M84’s galactic gas emission, +an APEC component for the Virgo ICM, and an extra power law +component for unresolved point sources (see §2.6). Since the helium +(He) abundance of the VAPEC component cannot be constrained in +the X-ray band, we set the He abundance to solar. VAPEC iron-peak +element abundances 𝑍Fe are tethered together in the fits, as are all +remaining 𝛼 element metallicities 𝑍𝛼. The APEC temperature and +metallicity are fixed based on §2.4, but the normalization is left free. +2.6 Unresolved Point Sources +M84 is known to host a substantial number of X-ray binary (XRB) +point sources (Finoguenov & Jones 2002). While many of these +XRBs can be masked out, unresolved XRBs and AB/CV stars rep- +resent a source of hard emission which can affect temperature and +abundance measurements. We follow the common practice of mod- +eling these unknown populations using a simple power law model +with fixed photon index ΓXRB = 1.6 (Goulding et al. 2016). The nor- +malization of this power law is left free in the fits to the VAPEC+APEC +model, which are designed to provide adequate statistics for con- +straining the 𝛼 element metallicity, 𝑍𝛼. +Unfortunately, the contaminant build-up which has degraded +Chandra’s soft energy response prevents us from constraining the +𝛼 element metallicity from the new extended campaign, even with +ample source counts available. Thus, the only Obs. IDs used for de- +termining 𝑍𝛼 come from legacy observations: Obs. IDs 803, 5908, +and 6131. We find a reasonable constraint on 𝑍𝛼, approximately 0.45 +times solar abundance. This value changes only slightly to 0.51 solar +when the power law component is neglected. For all remaining spec- +tral fits in this paper, we fix 𝑍𝛼 to 0.45 and fit only the temperature, +normalization, and the iron-peak metallicities in the VAPEC model. +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +5 +0 +500 +1000 1500 2000 +13.6 +13.4 +13.2 +13.0 +12.8 +log(Flux) +(erg cm−2 s−1) +0 +500 +1000 1500 2000 +1.5 +2.0 +2.5 +3.0 +Photon Index +Γ +0 +500 +1000 1500 2000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Source NH +(1022 cm2) +6850 +6860 +6870 +6880 +6890 +6900 +6910 +6920 +6930 +6850 +6860 +6870 +6880 +6890 +6900 +6910 +6920 +6930 +6850 +6860 +6870 +6880 +6890 +6900 +6910 +6920 +6930 + +Days since First Observation +Figure 2. Time variability of AGN over all three observational campaigns used in this work. The observation corresponding to Obs. ID 5908 (red) has a much +higher flux than the others, violating an assumption that the AGN has a constant luminosity. We remove this observation from our analysis of Bondi radius scales. +2.7 Accounting for AGN Contamination +While the AGN is a point source, Chandra’s point spread function +(PSF) distributes AGN photons across a number of pixels, including +those containing photons produced by gas at Bondi radius scales. +Therefore, we must account for AGN contamination when fitting +spectra extracted from these sub-kpc scales. +The AGN and galactic gas are spectrally distinct; however, they are +not spectrally separable with Chandra’s spectral resolution. We find +that when fitting the AGN and galactic gas together, we are unable to +adequately constrain the parameters for both models. In response to +this limitation, we follow the example of HRR15 and forward-model +the AGN contamination by simulating Chandra’s energy-dependent +PSF’s effect on the measured AGN spectrum. We first fit the spectrum +of the AGN based on fixing the parameters of the galactic gas spectral +model (§2.5). Using the parameters obtained from this fit, we produce +an AGN spectrum free of contributions from the galactic gas or the +Virgo screen. This spectrum is fed into the Chandra Ray Tracer tool +(ChaRT; Carter et al. 2003) which simulates the detector response. +The MARX software package (Davis et al. 2012) is used to produce a +second-level event file of the simulated AGN. +2.8 AGN Spectral Model +Previous estimates based on the most recent SMBH mass measure- +ments by Walsh et al. (2010) place the angular scale of the SMBH +Bondi radius at approximately 1′′ (Russell et al. 2013); however, +measurements in the literature differ by as much as a factor of ≈4 +(Bower et al. 1998; Maciejewski & Binney 2001). Thus, we choose +to define the region of AGN contamination as a circle with a radius of +1′′ centered on the peak of the AGN surface brightness distribution. +When we refer to the “AGN spectrum” in this paper, we are referring +to the spectrum extracted from this region which includes the AGN, +galactic gas emission, Virgo Cluster emission, and unresolved XRBs +and AB/CV stars. +The Comptonized emission from the AGN is modeled as a red- +shifted power law (Russell et al. 2010; Siemiginowska et al. 2010). +Due to the presence of the intervening dust lanes as well as a +possible “dusty torus” surrounding the AGN, we allow for lo- +cal photoelectric absorption through a zphabs model. Thus, the +spectrum of the 1′′ AGN region is fit using the “M84 model,” +phabs(zphabs(zpowerlw)+VAPEC+APEC+powerlaw). While this +model seems complicated, the only free parameters are the local +column density and the photon index Γ and normalization of the +zpowerlw component. +Appendix A2 of HRR15 presents the method for obtaining the +parameters for the remaining components. Spectra are extracted +from a 2′′ − 4′′ annulus circumscribing the AGN and fit with a +phabs(VAPEC+APEC+powerlaw) model for the galactic gas, Virgo +ICM, and unresolved point source emission. The APEC component is +fixed based on fits in §2.5 and the 𝑇 and 𝑍 from §2.4 (note that all +normalizations are scaled to the appropriate region areas). We then +fit the annulus spectrum with a free VAPEC temperature, iron-peak +metallicity, and normalization as well as a free powerlaw normal- +ization with fixed photon index ΓXRB = 1.6 (see §2.5 and §2.6). +The VAPEC normalization used in the AGN fits is determined by +boosting the normalization from the annulus fit centered at 3′′ to the +1′′ AGN circle based on a power law extrapolation of the 1′′ − 40′′ +surface brightness profile. For the powerlaw component, we assume +the normalization is constant from 1′′ − 3′′. +2.9 AGN Variability +We perform the fit described in §2.8 for all Obs. IDs individually in +addition to a simultaneous fit. The measured parameters for the AGN +model are shown in Table 1 and plotted in Figure 2. +Note that the AGN, unlike the galactic gas emission, is highly +variable, with nearly an order of magnitude variation in flux over the +three campaigns. Serendipitously, the AGN was relatively quiescent +MNRAS 000, 000–000 (0000) + +6 +C.J. Bambic et al. +(a) +10−8 +10−7 +10−6 +(b) +1 +2 +5 +10 +Radius (arcsec) +10−7 +10−6 +(c) +0.04 +0.1 +0.2 +0.4 +0.8 +Radius (kpc) +Surface Brightness (counts/s/cm2/arcsec2) +Figure 3. Left: Sectors (10′′ in radius) overlaid on merged 0.5 − 2 keV image of M84 central region. The East (magenta) and West (green) regions are aligned +perpedicular with the AGN jet which is approximately aligned with the Western filament (Figure 1). Right: Background-subtracted 0.5 − 7 keV surface brightness +profiles in sectors. The upper panel shows profiles of the AGN (triangles) as well as the broadband data, while the lower panel shows the AGN-subtracted surface +brightness profile, used to compute the deprojected density profiles. +during 2019 (observations following the break in Figure 2), implying +that the vast majority of data is subject to minimal AGN contamina- +tion. Unfortunately, one observation, Obs. ID 5908, caught a state of +outburst. Because we are modeling the AGN based on the statistically +powerful simultaneous fit to all usable observations, we necessarily +make the assumption that the AGN flux is constant with time. Obs. +ID 5908 (shown in red in Figure 2) breaks this assumption and is +thus omitted from the remaining analysis of Bondi radius scales. +2.10 Simulating the AGN Spectrum +The fit to the AGN spectrum is consistent with a photon index of +Γ ≈ 2, in accord with other similar ADAF spectra (Gu & Cao 2009; +Younes et al. 2011; Yang et al. 2015; Younes et al. 2019). With +this result, we simulate how the AGN spectrum free of contributions +from the galactic gas, Virgo, and XRB/AB/CVs would appear to +Chandra’s ACIS-S detector. +Appendix A3 of HRR15 describes this process in detail. We boost +the normalization of the spectrum input to ChaRT and use the aspect +solution file for Obs. ID 20543 (the longest exposure observation). +Pile-up is negligible in M84 and not included in our modeling. +The MARX tool is used to produce a second level event file and +exposure map for the simulation, and we reproject the simulation +to the coordinate system of the observation. We have accounted for +galactic emission in the AGN spectrum by measuring the galactic +background from 2′′ − 4′′ and modeling the surface brightness (SB) +profile to account for the increase in background into 1′′. However, +our choice of model introduces a systematic error which may cause us +to inadvertently over or under-subtract the AGN by a small amount— +enough to impact the measured temperature. +Subtraction of the AGN can be tested by comparing the hard band +(4 − 7 keV) SB profile of the simulation with that of the data (see +Appendix A). Bright, lumpy, soft emission from galactic gas, which +may vary on scales of the PSF, is entirely subdominant in the 4 − 7 +keV band. Instead, AGN, Virgo ICM, and unresolved point source +emission dominates. If the AGN is under-subtracted, a hard band ex- +cess will manifest itself at the scales of the PSF. However, if the AGN +is over-subtracted, there should be a drop in hard emission at small +scales. Because the Virgo screen is uniform over these small scales, +any discontinuities in the AGN-subtracted SB profile is evidence of +spatial variation in the unresolved point source flux. +Our AGN simulation leaves a modest hard band excess of 7% at +PSF scales. We compensate for this excess by boosting the AGN +simulation 7% such that the AGN-subtracted 4 − 7 keV band SB +profile flattens from 1′′ − 2′′ (see Figure 9). Continuity of the AGN- +subtracted SB at these small scales indicates that the unresolved point +source emission is relatively constant with radius, and our assumption +that the XRB/AB/CV normalization is constant with radius obtains. +We extract spectra for both the observations and AGN simulation +in 1′′ radial bins extending from 1′′ − 10′′. Simulated spectra are +fit using an absorbed, redshifted power law model, and the fit pa- +rameters are fixed for the AGN components in the combined “M84 +model” used in §2.8. We compute the error bars on temperature and +metallicity for the two points in the innermost 2′′ of each sector by +boosting or diminishing the AGN normalization by 5%, marginaliz- +ing over uncertainties in the AGN flux. The AGN normalization is +set to 0 beyond the innermost three annuli in each sector since the +AGN’s PSF is insignificant beyond 3′′. +2.11 Profile Deprojection +To obtain density profiles, we follow HRR15 and first compute +background-subtracted surface brightness (SB) profiles of the inner +1′′ −10′′ (∼ 0.8 kpc) in sectors, referred to as North, East, West, and +South respectively. The North and South sectors are aligned with the +radio jet, while the East and West sectors are anti-aligned. We sub- +tract off the Virgo and X-ray background from the SB profiles based +on a measurement of SB taken far away from the galaxy and free of +point sources. The sectors and SB profiles are shown in Figure 3. +MNRAS 000, 000–000 (0000) + +N +410 pc +5 arcsecFeeding and Feedback at the Bondi Radius of M84 +7 +Sector +𝑅B +𝑇 (𝑅B) +𝑍 (𝑅B) +𝑛𝑒 (𝑅B) +Index +�𝑀B +𝜂 +𝐿B/𝐿Edd +pc +keV +Z⊙ +cm−3 +𝛼 +10−3 M⊙yr−1 +×10−6 +×10−4 +North +49.0+6.6 +−5.3 +0.71+0.04 +−0.05 +0.14+0.04 +−0.03 +0.11 ± 0.09 +−1.15 ± 0.19 +1.57+1.03 +−1.01 +17.62+31.64 +−7.03 +0.84+0.56 +−0.54 +East +48.4+5.7 +−5.4 +0.72+0.05 +−0.04 +0.09+0.03 +−0.02 +0.45 ± 0.04 +−0.80 ± 0.12 +6.35+1.58 +−1.35 +4.37+1.28 +−0.93 +3.39+0.93 +−0.78 +West +42.8+4.6 +−5.9 +0.82+0.09 +−0.04 +0.12+0.02 +−0.02 +0.47 ± 0.03 +−0.93 ± 0.09 +5.52+1.21 +−1.31 +5.03+1.68 +−0.98 +2.94+0.72 +−0.75 +South +58.7+8.2 +−6.9 +0.60+0.05 +−0.05 +0.29+0.09 +−0.07 +0.26 ± 0.02 +−1.21 ± 0.33 +≤ 4.88 +≥ 5.68 +≤ 2.60 +All +48.1+5.3 +−4.7 +0.73+0.02 +−0.02 +0.15+0.02 +−0.02 +0.27 ± 0.04 +−0.87 ± 0.09 +3.74+1.05 +−0.89 +7.42+2.46 +−1.72 +1.99+0.61 +−0.50 +Table 2. Summary of measurements at 1′′, density profile index 𝛼, accretion rates �𝑀B, efficiencies 𝜂, and Eddington ratios 𝐿B/𝐿Edd for Bondi accretion in +each sector. Because of small-scale cavities evident in the AGN-subtracted SB profiles in the South sector (Figure 3), we are only able to obtain limits on the +density 𝑛𝑒 and quantities derived from density in this region. +SB is a projection of a 3D distribution of X-ray emission onto a 2D +plane. By assuming spherical symmetry, we can deproject each sec- +tor’s SB profile, “peeling back” shells of X-ray emission to determine +a volumetric emissivity at each profile radius. Spherical symmetry is +a poor assumption for M84’s highly-structured H-shape; however, +the assumption may obtain more readily around the quasi-spherical +halo in the inner 10′′. +We apply the deprojection method of Kriss et al. (1983) to the +AGN-subtracted SB profiles, panel (c) in Figure 3. This method only +strictly applies when the SB is monotonically increasing inward. Cav- +ities, evident in the significant drop in the AGN-subtracted surface +brightness profiles for the North and South sectors at a radius of 1′′, +violate this assumption and prevent us from obtaining anything more +than an upper limit on 𝑛𝑒 in the South sector. +Even though the AGN simulation is always sub-dominant to the +observed SB (panel (b) in Figure 3), the simulation is brightest in +the South where the observed profiles show a depression in SB. We +note that in the jet-aligned sectors, the average SB of the innermost +radial bin is ∼20% less than that in the off jet-axis sectors. However, +the AGN simulation tends to favor more photons in the jet-aligned +sectors, with ∼26% more photons in the North and South compared to +the East and West. Chandra’s PSF at sub-arcsecond scales is subject +to a hook feature which is captured in the ChaRT-MARX simulation. +Because the simulation PSF is asymmetric, a simple re-alignement +of the simulation is insufficient to eliminate the cavities from the +AGN-subtracted SB. +Our emissivity profiles, temperatures, and metallicities are all mea- +sured in the same radial bins/ sectors. Thus, we are able to use the +temperatures and metallicities to determine the number density of +X-ray emitting electrons 𝑛𝑒 from the emissivity profiles. In this way, +we obtain profiles of 𝑛𝑒 in each sector separately. +2.12 Contour Binning +The large signal-to-noise afforded by our deep observations allows us +to produce maps tracing the large-scale temperature and metallicity +structure in M84. For this task, we use the contour binning method +presented in Sanders (2006) and made possible through the contbin +software package (Sanders 2016). +Contour binning groups adjacent pixels of similar surface bright- +ness to achieve a requested signal-to-noise ratio. The method groups +gas expected to be spectrally similar, allowing us to extract spectra +with high signal-to-noise. Thus, contour binning produces accurate +temperature maps of spatially-resolved extended sources with non- +smooth surface brightness distributions. +We use a signal-to-noise ratio of 32 and set the smooth signal- +to-noise parameter to 20. Because of M84’s H-shaped emissivity +distribution, contours tend to be elongated along the filaments, con- +necting regions which are too spatially separated to be causally con- +nected. We thus constrain the shape of the contours using contbin’s +constrainval parameter, which we set to 1.2. The spectra extracted +from the regions defined by contbin are fit using the spectral model +defined in §2.5; however, we do not include an XRB/AV/CV com- +ponent in our fits as this component tends to be negligible on the +kiloparsec (kpc) scales relevant for the maps. +3 RESULTS +In this section, we present profiles of temperature, metallicity, and +deprojected density measured in four separate sectors—two aligned +with the jet axis and two anti-aligned (Figure 3a). We use these mea- +surements to calculate the accretion rates �𝑀B and efficiencies 𝜂 for +Bondi accretion in each of the sectors, where 𝐿Jet + 𝐿X = 𝜂 �𝑀𝑐2 +and 𝐿X is the X-ray luminosity of the AGN. Our main results are +summarized in Table 2. These measurements allow us to explore the +large-scale structure of the accretion flow and compare timescales +which dictate the flow’s dynamics, namely the cooling time 𝑡cool +and inflow time 𝑡inflow. We conclude this section with maps of tem- +perature, metallicity, and pseudo-pressure to connect the small-scale +physics of accretion with the kpc-scale structure of the X-ray halo. +3.1 Density Profile +The top panel of Figure 4 displays the density profile for each sector. +Density increases monotonically toward smaller radii in all sectors +from 7′′ to 2′′. The scaling of density with radius provides a direct +comparison between our data and the theoretical prediction from the +adiabatic Bondi solution. We can model the observed density profile +as a simple power law, +𝑛𝑒(𝑟) = 𝑛𝑒,0 +� 𝑟 +𝑅B +� 𝛼 +, +(1) +where 𝑛𝑒,0 is the number density at the Bondi radius, 𝑟 = 𝑅B. Using +a Markov chain Monte Carlo (MCMC; Foreman-Mackey et al. 2013) +method, we fit this power law model to the density data for each +sector. We omit the innermost data points at 1′′ in the North and +South sector fits as these points are strongly affected by the presence +of Bondi radius-scale cavities. Values of 𝛼 are shown in Table 2. +3.2 Temperature Profiles +We see evidence for shock heated gas in the temperature pro- +file of the North sector (middle panel of Figure 4). A jump from +𝑇 = 0.75 ± 0.03 keV at 4′′ to 𝑇 = 0.94+0.02 +−0.04 keV at 3′′ in the North +sector points to the influence of the radio jet. However, this feature +MNRAS 000, 000–000 (0000) + +8 +C.J. Bambic et al. +appears to be exceptional rather than commonplace. We find that the +temperature profiles are relatively flat with radius, increasing only +gradually inward from 0.6 keV at 800 pc to 0.7 − 0.8 keV at 100 pc. +Temperatures at the innermost points, with radial error bars +crossing through the Bondi radius, show substantial scatter from +0.6 − 0.8 keV. We see that temperature in the South begins to de- +crease inward at the radii where M84’s quasi-spherical central halo +begins; gas may be cooling more efficiently in these denser regions. +However, we note that the South sector, especially the innermost +point, is certainly affected by cavities. +Note that points obtained in the South sector lack the constraining +power of the other sectors due to a clear point source throughout +much of the region. In all but the South sector, there are at least 700 +source counts per radial bin; however, from 3′′ − 6′′ in the South, +the number of counts drops below 200. Thus, while the temperature +of the innermost point in the South sector may be reliable (although +certainly a cavity is present), the decreasing trend in temperature +may not be physical. Instead, based on the other sectors, one may +reasonably conclude that temperature is relatively constant over the +inner kpc, with a gradual increase toward Bondi radius scales. +3.3 Bondi Accretion Rate +The adiabatic Bondi accretion rate is given by, +�𝑀B = 0.012 +� 𝑇 +keV +�−3/2 � 𝑛𝑒 +cm−3 +� � 𝑀BH +109 M⊙ +�2 +M⊙yr−1, +(2) +where we have measured 𝑛𝑒 and 𝑇 at the Bondi radius (Rafferty et al. +2006), and we use the most recent measurement of the SMBH mass +from Walsh et al. (2010), 𝑀BH = 8.5+0.9 +−0.8 × 108𝑀⊙. +Central values for �𝑀B are calculated through Equation 2 and the +data in Table 2. Our method for computing errors is described in +Appendix B. We choose to use a Monte Carlo method, drawing sam- +ples from distributions of 𝑛𝑒, 𝑇, and 𝑀BH and applying Equation 2. +The 1𝜎 errors on �𝑀B are then the 16th and 84th percentiles of the +resulting distribution. +Cavities formed by the radio jet and uncertain subtraction of the +AGN PSF have a significant impact on measurements of the Bondi +accretion rate. Averaging together the East and West sectors to yield +�𝑀B = (5.94 ± 0.94) × 10−3 𝑀⊙yr−1 perpendicular to the jet, we +find that the accretion rate in the North sector parallel to the jet, +�𝑀B = 1.57+1.03 +−1.01 × 10−3 𝑀⊙yr−1, is discrepant at the level of 4.6𝜎. +Given the dearth of photons in the South sector, the upper limit ob- +tained from this sector is likely a vast over-estimate, with the true +inferred �𝑀B lying even below that in the North sector. These discrep- +ancies point not only to the difficulty of measuring �𝑀B, but also the +importance of carefully accounting for cavities, which will systemat- +ically suppress the measured �𝑀B. Traditional methods which assume +spherical symmetry to compute a deprojected 𝑛𝑒 in a full annulus +around the AGN rather than in sectors are possibly under-estimating +the true Bondi accretion rate. +We close this section by noting that recent measurements by the +Event Horizon Telescope (EHT; Event Horizon Telescope Collabora- +tion et al. 2019a) may indicate that the gas dynamical measurement +of M84’s SMBH mass is an underestimate of the true value. The +EHT employs an emission modeling technique for assessing SMBH +masses which, in the case of Sgr A∗ (Event Horizon Telescope Col- +laboration et al. 2022) yields a value completely consistent with stel- +lar dynamical measurements (Ghez et al. 2008; Gillessen et al. 2009). +However, when applied to M87∗, the EHT measurement (Event Hori- +zon Telescope Collaboration et al. 2019b) is discrepant with previous +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +ne (cm−3) +North +East +West +South +0.6 +0.8 +1.0 +T (keV) +0.5 +1 +2 +5 +10 +Radius (arcsec) +0.0 +0.2 +0.4 +0.6 +Z (Z⊙) +0.04 +0.1 +0.2 +0.4 +0.8 +Radius (kpc) +Figure 4. Deprojected electron number density 𝑛𝑒, gas temperature 𝑇 , and +iron-peak element metallicity 𝑍 as a function of radius for all four sectors. +Dashed lines indicate the range of our measured Bondi radii. The presence of +cavities in the South sector precludes a measurement of 𝑛𝑒 for the innermost +point; however, because the emissivity profile drops sharply at this radius, we +use the data point at 2′′ as an upper limit. +gas dynamical measurements (Walsh et al. 2013). Thus, the Walsh +et al. (2010) measurements of M84’s SMBH and subsequently our +measurements of the Bondi radius and Bondi accretion rate may also +be underestimates of the true values. +3.4 The Inefficiency of Bondi Accretion +Using the central values and distributions of �𝑀B, we can compute the +efficiency of Bondi accretion. We define this efficiency factor through +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +9 +𝐿Jet + 𝐿X = 𝜂 �𝑀𝑐2, where our combined fit to the AGN gives an X- +ray luminosity 𝐿X = 1.6+0.15 +−0.14 × 1039 erg/s. The jet power 𝐿Jet is +obtained by measuring the enthalpy of M84’s cavities assuming they +are in pressure equilibrium with their surroundings, and dividing by +the characteristic timescale of the bubbles, either the sound crossing +time or buoyancy timescale. Using this method, Russell et al. (2013) +found the jet power to be 𝐿Jet = 1.1+0.9 +−0.4 × 1042 erg/s. +For determining the errors in 𝜂, we assume dimidiated Gaussians +for 𝐿X and 𝐿Jet, but instead of assuming distributions for �𝑀B, we use +the distributions computed in §3.3. The results are shown in Table 2. +While typically ∼10% of the �𝑀𝑐2 power is released by gravita- +tional infall through an accretion disk, Bondi accretion onto M84’s +SMBH is far less efficient, with 𝜂 ∼ 10−6 in the East and West sectors +unaffected by cavities. These low efficiencies imply that M84 hosts +a radiatively inefficient accretion flow (RIAF). A similar conclusion +follows from the Eddington ratios for Bondi accretion. Using the +definition 𝐿B = �𝑀B𝑐2, and the Eddington luminosity for the SMBH +(Russell et al. 2013), +𝐿Edd = 1.26 × 1047 +� 𝑀BH +109𝑀⊙ +� +erg/s, +(3) +we can determine the Eddington ratio for Bondi accretion, 𝐿B/𝐿Edd. +In all sectors, M84’s AGN displays Eddington ratios around a +few ×10−4. When the Eddington ratio is based on the true accretion +rate onto the hole �𝑀, the value is much lower, 𝜂𝐿B/𝐿Edd ∼ 10−10. +Accretion proceeds not through a thin, radiatively efficient disk +(Shakura & Sunyaev 1973), but rather via a hot RIAF (Yuan & +Narayan 2014). +3.5 Timescales for Accretion +We can better elucidate the structure of the flow by measuring profiles +of the relevant timescales for accretion, namely the cooling time 𝑡cool, +free-fall time 𝑡ff, and Bondi inflow time 𝑡inflow. The cooling time is +the timescale for a gas with thermal energy density 3 +2𝑛𝑒𝑇 to radiate its +energy away, 𝑡cool = (3/2)𝑛𝑒𝑇/𝑛2𝑒Λ. Here, Λ represents the cooling +function for the X-ray gas and consists of bremsstrahlung continuum +as well as significant line cooling in 𝑇 ≲ 1 keV gas. +The free-fall time is the dynamical timescale for gas to free fall +from a radius 𝑟 under the gravitational influence of M84’s SMBH and +dark matter halo. In line with the Bondi (1952) solution, we assume +that M84’s dark matter distribution is described by a spherically- +symmetric Hernquist (1990) profile. We add an additional point mass +potential with mass 𝑀BH to the dark matter potential, which is used to +determine the gravitational acceleration as a function of radius, 𝑔(𝑟). +The free-fall time can then be computed using the simple expression, +𝑡ff = +√︁ +2𝑟/𝑔. +Traditionally, the ratio 𝑡cool/𝑡ff has been used as a probe of thermal +instability in galaxy clusters. At the small ≲ 1 kpc scales where the +Bondi flow originates, the free fall time is far too short to be relevant +for the structure of the flow (see Discussion). Rather, the dynamical +timescale for the accretion flow is the Bondi inflow time, 𝑡inflow. For +steady-state Bondi accreton, +�𝑀B = 4𝜋𝑟2𝜌(𝑟)𝑢𝑟 (𝑟) = constant, +(4) +where we have introduced the mass density 𝜌(𝑟) and the radial +inflow velocity 𝑢𝑟 (𝑟). The mass density is related to our mea- +sured number density by assuming quasi-neutrality and introducing +the mean-molecular weight 𝜇, which we take to be 0.6 such that, +𝜌(𝑟) = 1.15 𝑚 𝑝𝑛𝑒(𝑟), and 𝑚 𝑝 is the proton mass. We can approxi- +0.1 +1.0 +tcool (Gyr) +North +East +West +South +0.01 +0.1 +1.0 +10.0 +tinflow (Gyr) +0.5 +1 +2 +5 +10 +Radius (arcsec) +0.01 +0.1 +1.0 +10.0 +tcool/tinflow +0.04 +0.1 +0.2 +0.4 +0.8 +Radius (kpc) +Figure 5. Profiles of cooling time and inflow time in the inner 10′′ (0.8 kpc). +The cooling time 𝑡cool gradually decreases inward from 200 Myr at 0.5 kpc +to 60 Myr near the Bondi radius (dashed lines). Inflow times show a similar +trend; however, the inflow time is ∼10 Myr at the Bondi radius in all sectors. +The ratio of timescales shows a clear transition above 𝑡cool/𝑡inflow = 1 at 2′′ +as the flow approaches the Bondi radius. +mate the inflow time as +𝑡inflow = +𝑟 +𝑢𝑟 (𝑟) = 4𝜋𝑟3𝜌(𝑟) +�𝑀B +. +(5) +Note that this expression is particularly simple as the number density +𝑛𝑒 drops out when substituting in the expression for �𝑀B (Equation 2), +𝑡inflow ≈ 30 +� 𝑟 +kpc +�3 � 𝑇 +keV +�3/2 � 𝑀BH +109 𝑀⊙ +�2 +Gyr. +(6) +Figure 5 shows profiles of the timescales. The free-fall timescale is +short (𝑡ff < 1.2 Myr) within the inner kpc and does not dictate the gas +flow. Rather, cooling is the dominant process from 200−800 pc. We +emphasize that this flow structure is different from a “cooling flow” +(Fabian 1994) which involves catastrophic levels of star formation +and a deluge of cold gas from the cluster ICM onto the galaxy. Our +observations probe much smaller scales than would be relevant for +MNRAS 000, 000–000 (0000) + +10 +C.J. Bambic et al. +Figure 6. Left: Temperature map of M84, where brighter colors indicate higher temperature, measured in keV. Right: Metallicity map, where darker colors +indicate higher metallicity, measured relative to solar metallicity 𝑍⊙. The same 2 × 10−3 − 0.1 mJy VLA radio contours from Figure 1 are overlaid to emphasize +the structure of the H-shaped filaments. +cooling flows, which are suppressed by AGN feedback (Fabian 2012). +At these small scales, gas slowly condenses and begins to flow inward +as the atmosphere cools and loses thermal pressure support. +As material approaches the Bondi radius, the dominant (shorter) +timescale becomes the Bondi inflow time 𝑡inflow. Thermal pressure +support loses relevance and the flow begins to experience the in- +fluence of the SMBH. In this way, detection of the transition from +𝑡cool/𝑡inflow < 1 to 𝑡cool/𝑡inflow > 1 may indicate that we are probing +the beginning of the large-scale accretion flow onto the SMBH. +3.6 Temperature and Metallicity Maps +The temperature and metallicity profiles presented in Figure 4 pro- +vide insight into the structure of the beginnings of the accretion +flow feeding the AGN in M84. In this section, we “zoom out” from +these small, sub-kpc scales to explore the temperature and metallicity +structure of the galactic gas. +Figure 6 shows the temperature and metallicity maps produced us- +ing the methods described in §2.12. The maps are fit using the com- +bined VAPEC+APEC model described in §2.5 whenever the surface +brightness of the region is larger than 10−7 counts/s/cm2/arcsec2. +Otherwise, we use an APEC model since low surface brightness re- +gions are likely dominated by emission from the Virgo ICM rather +than from M84’s galactic gas. This choice does not have a significant +effect on the structure of the maps; the use of an APEC+APEC model +for all regions yields similar maps. +The temperature map shows that the galactic gas is remarkably +isothermal, with the vast majority of gas occupying a narrow range +of temperatures from 0.5−0.8 keV, similar to what is seen in the +temperature profiles of the inner kpc around the AGN. There ap- +pears to be a large-scale, although weak, temperature gradient, with +colder material located West of the radio jet and warmer material +sandwiched between the radio lobes. Gas is generally hotter within +the radio lobes; however this trend may be due to the fact that the +cavities are dominated by dim Virgo ICM emission. In the Northern +radio lobe, we see rich temperature structure, with a colder filament +bridging through the bubble. This feature may be a projection effect. +Instead, a cold filament could be wrapping around in front of or +behind the bubble in three dimensional space. +Colder temperatures trace out filaments; however, the H-shape is +far more apparent in the metallicity map in Figure 6. In this map, +higher metallicities, with values around 0.3−0.5 𝑍⊙ define the H, +and metallicity appears relatively symmetric about the radio jet. We +see evidence for a drop in metallicity approaching the interior of the +galaxy, and this trend is clearly apparent in the metallicity profiles +in Figure 4. Radio cavities appear to clear out the X-ray halo of +metal-enriched material; however, rather than pulling high metallicity +material into the wake of the bubbles, the jet appears to simply push +metal enriched material aside into the H-shaped filaments. +While jets shape the filaments, feedback is gentle and does not lead +to a substantial over-pressurization of the filaments. We can study this +process via the pseudo-pressure map (Figure 7). Pseudo-pressure is +calculated by multiplying the gas temperature in each region with the +square root of the normalization per unit area. For bremsstrahlung +emission, which has a weak 𝑇1/2 temperature dependence, the nor- +malization is proportional to the emission measure, +∫ +𝑛2𝑒𝑑ℓ, where +𝑑ℓ is the differential path length through the cluster. +Pseudo-pressure increases only by a factor 1.5 between the Virgo +ICM regions located beyond about 55′′ East of the SMBH and the +outer halo of M84. The filaments, which begin 30′′ along the same +direction, show another factor of 1.5 increase. Such an increase can- +not be explained by the temperature, which instead decreases inward +along the same East-pointing ray. The increase is also too steep to +be attributable simply to an increase in the path length (and thus +emission measure) if we assume M84’s galactic gas is distributed +quasi-spherically. Instead, gas density is enhanced in the filaments, +likely mediated by cooling in the dense, metal-rich gas. +MNRAS 000, 000–000 (0000) + +N +1.63 kpc +20 arcsec +0.48 +0.56 +0.64 +0.72 +0.8 +0.88 +0.96 +1.11.63 kpc +20 arcsec +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45Feeding and Feedback at the Bondi Radius of M84 +11 +Figure 7. Pseudo-pressure map, formed by multiplying the temperature map +in Figure 6 with a map of the square root of area-corrected normalization. +The scale shown is logarithmic and spans 2 orders of magnitude. Filaments +are not substantially over-pressurized relative to the surrounding cluster. +The crossbar of the H can be interpreted as a disk of dense gas +around the AGN, viewed in projection. Cooling in the dense disk +leads to a collapse into a thin, over-pressurized structure. In this +way, rather than being squeezed by the radio lobes, the crossbar +may simply be condensing through the cooling and gravitational +collapse of a large-scale (35′′ or ∼3 kpc) centrifugally-supported +disk. Similarly, features of the pressure map which seem to extend +into the radio lobes, such as a “fish-tail” like structure visible in +the Southeast in Figure 7, can be attributed to filaments wrapping +around the radio lobes, viewed in projection. If such filaments are +uplifted with the bubbles, cooling may be encouraged, leading to the +formation of the dense, metal rich structures clinging to the bubbles. +4 DISCUSSION +The density, temperature, and metallicity profiles in Figure 4 provide +a direct comparison to the spherically-symmetric Bondi (1952) so- +lution. Similarly, our measurements of the profile index 𝛼 and Bondi +accretion rate �𝑀B (Table 2) quantify the degree of asymmetry in the +flow for sectors aligned with the jet axis and those anti-aligned. +In this section, we discuss the origins of the observed deviations +from the Bondi solution, namely the influence of the jet on the +flow. We include a brief discussion of multiphase structure and ther- +mal instability in M84, presenting the entropy profiles based on our +temperature and density measurements, and discuss the lack of an +observed temperature rise at the Bondi radius. Finally, we close with +an analysis of the “hot blob” of material noted in §3.2. +4.1 Asymmetry Imposed by the Jet +Within the 1𝜎 error bars, all density profiles are just slightly flatter +than 𝑛𝑒 ∝ 𝑟−1, which is consistent with findings by HRR15 in M87. +This profile however is completely inconsistent with the density pro- +file predicted by the Bondi (1952) solution for an adiabatic gas, which +instead would predict a density profile of 𝑑 ln 𝜌/𝑑 ln 𝑟 = −0.373 at +𝑟 = 𝑅B—a 5.5𝜎 discrepancy from the “All” sectors value in Table 2. +This discrepancy points to the fact that the flow may be strongly in- +fluenced by the galactic gravitational potential rather than the SMBH +point mass alone (Quataert & Narayan 2000). +Both jet-aligned sectors show steeper radial profiles compared +with the shallow profiles perpendicular to the jet. Because the points +most affected by the presence of cavities were removed when fitting +for 𝛼, the cavities do not account for this steepening in the density +profile. Instead, jet-inflated bubbles may entrain dense material from +the core of the galaxy in their wakes, buoyantly lifting this gas to +larger radii. The result is a dearth of material at the Bondi radius +along the jet and density enhancement at larger radii (Churazov et al. +2001; Fabian et al. 2003). Alternatively, because the highest densities +in the North sector (𝑛𝑒 = 0.26 ± 0.009 cm−3 at 3′′) are coincident +with the “hot blob” of shocked gas, the density enhancement may be +due to compression in the shocked region itself. +4.2 Mechanical Feedback on the Accretion Flow +The Bondi accretion rates �𝑀B aligned and anti-aligned with the jet +axis are discrepant at a level > 4𝜎. This discrepancy may be at- +tributed to small-scale cavities formed by radio jets blasting through +and clearing out halo gas. However, whether or not this discrepancy +indicates that jets modify the true accretion rate of material through +the Bondi radius remains an open question. Certainly, cavities com- +plicate measurements of �𝑀B. Spherical symmetry does not apply +in the presence of a jet and deprojection is no longer well-posed if +surface brightness decreases inward. +Still, the radio jet may have a negligible impact on the accretion +flow itself. Relativistic jets with small opening angles can impart +substantial energy to the accretion flow via shock heating (see §4.6) +which impedes gas cooling and introduces further asymmetry to the +flow. However, these jets impact a relatively small fraction of the +accreting gas. While a large number of simulations have been able to +explore self-regulation of AGN in cluster environments (Cattaneo & +Teyssier 2007; Sijacki et al. 2007; Dubois et al. 2010; Gaspari et al. +2012; Li & Bryan 2014; Prasad et al. 2015; Yang & Reynolds 2016; +Bourne & Sijacki 2017), these simulations lack the dynamic range +to study black hole feeding at scales below 𝑅B. +Recently, Ressler et al. (2018) and Ressler et al. (2020) demon- +strated a calculation of black hole feeding for the RIAF in Sgr A∗ +which evolved the origins of the flow fed by stellar winds down to the +black hole horizon. A similar procedure has been undertaken by Guo +et al. (2022) for the AGN in M87, with a heating prescription standing +in for jetted AGN feedback. In all of these works, angular momentum +plays a crucial role. Thermal instability, turbulence, stellar winds, and +cloud-cloud or cloud-filament interactions set the angular momen- +tum distribution of accreting gas. High angular momentum gas which +is unable to shed angular momentum through collisions of turbulent +transport (Narayan & Fabian 2011), is flung away as it encounters +the centrifugal barrier of the SMBH. Yet, low angular momentum +gas has the possibility of settling into the observed accretion flow. +These works predict a suppression of the Bondi accretion rate with +the scaling �𝑀 ∼ (𝑟/𝑅B)1/2 �𝑀B. M84’s 𝑀BH = 8.5+0.9 +−0.8 × 108 𝑀⊙ +black hole has an innermost stable circular orbit (ISCO) with a radius +𝑅ISCO ≈ 7.6×1014 cm (assuming no black hole spin). For the Bondi +radius based on all sectors, 𝑅B(All) = 48.1+5.3 +−4.7 pc, the predicted +accretion rate at the ISCO using the scaling inferred from simulations +is �𝑀 = (𝑅ISCO/𝑅B)1/2 �𝑀B ≈ 8.5×10−6𝑀⊙yr−1. If the flow liberates +MNRAS 000, 000–000 (0000) + +1.63 kpc +20 arcsec +0.00011 +0.00017 +0.00040 +0.00134 +0.0050612 +C.J. Bambic et al. +𝜂 ∼ 10% of the �𝑀𝑐2 energy which reaches the hole, the inferred +power is 𝐿ISCO ≈ 5 × 1040 erg/s. This power is well short of the +Gyr-averaged jet power 𝐿Jet = 1.1+0.9 +−0.4 × 1042 erg/s. Thus, if the 𝑟1/2 +scaling obtains in M84, there must be additional sources of accreting +gas beyond the hot phase material inferred from X-ray observations +alone. +Understanding the interaction between jets and the accretion flows +powering them remains an open problem. Self-consistently evolving +the sub-parsec scales responsible for launching jets with the ∼50 pc +scales of the Bondi radius requires resolving gas thermodynamics, +inflows, and outflows over 5 orders of magnitude in scale. We ex- +pect that a combination of increased computational power and deep +observations of molecular gas, enabled by observatories like the At- +acama Large Millimeter Array (ALMA), will serve to better elucidate +how jets and bubbles affect the distribution of mass and angular +momentum in the gas fueling RIAFs in massive elliptical galaxies. +4.3 Cold vs. Hot Mode Accretion +Large-scale accretion at scales comparable to the Bondi radius can +be broadly divided into two classes, similar to those invoked in the +galaxy formation community (Kereš et al. 2005): cold mode and hot +mode accretion. Bondi accretion of ∼keV X-ray gas represents the +hot mode of accretion. As we have shown by our measurements of +density and temperature at the Bondi radius, Bondi accretion alone +is more than sufficient to power the central AGN in M84. However, +if multiphase gas, particularly components much colder than what +we are studying in the X-rays, is present, the cold mode of accretion +may be equally if not more important. +In cold mode accretion, thermally unstable (Field 1965) gas cools, +condenses, and precipitates out of the hot medium, forming dense +structures such as “clouds” (or “blobs”) and filaments. As long as +these cold structures possess a minimal amount of angular momen- +tum, or can shed angular momentum via cloud-cloud, cloud-filament, +etc. collisions, they can chaotically “rain down” onto the central +SMBH, providing a gas supply even in excess of that provided by +Bondi accretion alone (Pizzolato & Soker 2005; Gaspari et al. 2012). +While this picture of accretion is straightforward in principle, in +practice, a number of challenges remain. In cluster environments, +buoyancy acts to negate thermal instability, at least at the level of +linear theory (Defouw 1970; Cowie et al. 1980; Nulsen 1986; Bal- +bus & Soker 1989). Idealized nonlinear simulations with heating and +cooling globally balanced, as carried out by McCourt et al. (2012) +in plane parallel geometry, Sharma et al. (2012) in spherical coordi- +nates, and with jetted feedback as in the simulations by Gaspari et al. +(2012), argue that the existence of multiphase gas depends sensi- +tively on the minimum of the cooling to free-fall time ratio, 𝑡cool/𝑡ff. +Subsequently, simulations by Li & Bryan (2014) and Meece et al. +(2015), as well as observational efforts by Voit & Donahue (2015) +and Voit et al. (2015) have further solidified the importance of the +ratio of these timescales in the literature. +Voit et al. (2017) adds motivation for the minimum 𝑡cool/𝑡ff ratio +in clusters, arguing that the ratio sets a critical slope for the entropy +profiles in clusters. For entropy profiles steeper than this threshold, +multiphase gas cannot precipitate from the hot phase since it is subject +to buoyant oscillations and thus strong buoyancy damping. Yet, when +the slope is flattened by an injection of high entropy material into +the center of the cluster via feedback, thermal instability can proceed +and cold mode accretion is once again relevant. In this way, the +minimum 𝑡cool/𝑡ff ratio alone is not the only crucial parameter in +clusters. Rather, this ratio must be compared to the entropy gradient +to predict the presence of multiphase gas. +0.5 +1 +2 +5 +10 +Radius (arcsec) +0.1 +1 +Tn−2/3 +e +� +keV cm2� +North +East +West +South +0.04 +0.1 +0.2 +0.4 +0.8 +Radius (kpc) +Figure 8. Entropy profiles of all 4 sectors. There is a clear inward decrease +in entropy within the inner kpc of M84, which should indicate a dearth of +multiphase gas based on the 𝑡cool/𝑡ff ratio at these small radii. +We note that while there is some observational support for these +models, the idea of a critical 𝑡cool/𝑡ff ratio setting the conditions +for the formation of multiphase gas is by no means settled physics. +Buoyantly rising bubbles may stimulate cooling and multiphase gas +formation via adiabatic uplift (McNamara et al. 2016), as may be +indicated by our temperature, metallicity, and psuedo-pressure maps +(Figures 6-7). In addition, observational biases may over-emphasize +the importance of 𝑡cool/𝑡ff (Hogan et al. 2017; Pulido et al. 2018). +Our work is focused on AGN fueling at the Bondi radius. Thus, rather +than wade headlong into rather subtle questions of thermal instability +in galaxy clusters, we present a simple test for multiphase gas based +on the measured entropy profiles in M84’s X-ray halo. +Figure 8 presents the radial entropy profiles in each of the four +sectors. Following Voit et al. (2017), we define the dimensionless +entropy gradient as 𝛼𝐾 ≡ 𝒓 · ∇ ln (𝐾), where 𝐾 ≡ 𝑇𝑛−2/3 +𝑒 +is the gas +entropy in units of keV cm2. Equation 22 of Voit et al. (2017) provides +a condition on the entropy gradient for nonlinear condensation, +𝛼𝐾 < 𝛼𝐾 ,crit ≡ 3(2 − 𝜆)2 +40 +� 𝑡ff +𝑡cool +�2 +, +(7) +where 𝜆 ≡ 𝑑 ln Λ/𝑑 ln𝑇 parameterizes the cooling function Λ, and +for the relevant cooling mechanisms in clusters lies in the range +−1 ≲ 𝜆 ≲ 0.5. Thus, entropy profiles steeper than the critical value +𝛼𝐾 ,crit should result in gradually damped buoyant oscillations of +cooling gas, rather than the condensation necessary to fuel cold +mode accretion. Because of the short free fall times so close to +the Bondi radius (𝑡ff ∼ 0.2 Myr) and comparatively long cooling +times (𝑡cool ∼ 0.1 Gyr; Figure 5) at these small scales, the ratio +𝑡ff/𝑡cool ∼ 2 × 10−3 implies that the critical entropy gradient is +essentially flat. Figure 8 indicates that in all sectors, the entropy +gradient is far too steep to admit condensation and the formation of +multiphase gas. If multiphase gas is in fact present, this material must +have been sent toward the Bondi radius from much larger scales. +While we searched for multi-temperature gas in the central kpc as +an indication of gas cooling out of the ionized phase of the X-ray emit- +ting plasma, we were unable to find evidence of a second temperature +component in the X-ray band. Adding in a second VAPEC component +provided no constraint on a second temperature. The likely reason is +that Chandra can only distinguish temperatures separated by ∼ 0.5 +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +13 +keV in energy space. Because M84’s gas is cold (0.6 − 0.7 keV), a +colder component would appear at 0.2−0.3 keV, below the detector’s +sensitive energy range. A hotter component may be detectable; how- +ever, if the AGN was over-subtracted rather than under-subtracted, +this potentially weak signal may be lost. Thus, consistent with the +conclusion from the entropy profiles, we find no evidence for the +presence of multiphase X-ray emitting gas. +We note that the strict criterion presented in Equation 7 may not +be applicable to Bondi radius scales. The central assumption under- +pinning the importance of 𝑡ff/𝑡cool in cluster environments is that +heating and cooling is globally balanced. While such a balance may +apply globally within the Virgo Cluster, locally, at the small sub-kpc +scales probed in our analysis, heating cannot offset cooling. AGN +jet energy is thermalized on 10s of kpc length scales comparable +to or larger than the bubbles, via weak shocks and sound waves +(Fabian et al. 2003; Sanders & Fabian 2007; Graham et al. 2008; +Fabian et al. 2017; Bambic & Reynolds 2019), turbulence driven by +g-modes (Churazov et al. 2002, 2004; Zhuravleva et al. 2014; Zhang +et al. 2018), mixing of high entropy bubble material with cluster +gas (Hillel & Soker 2016, 2018), cosmic rays (Guo & Oh 2008; +Pfrommer 2013; Ruszkowski et al. 2017; Ehlert et al. 2018; Yang +et al. 2019; Kempski & Quataert 2020), etc. Near the Bondi radius of +M84 where the cooling time is ∼0.1 Gyr, these heating mechanisms +operate inefficiently. +For this reason, the traditional comparison of the free-fall and +cooling timescales should give way to a comparison of the cooling +and Bondi inflow timescales (see §3.5). Figure 5 indicates a transition +from a “cooling-dominated” flow to an “inflow-dominated” flow at +scales comparable to 𝑅B. In this way, we see that the accretion flow +around the Bondi radius should not be regarded as a static equilibrium +defined by the interplay of AGN heating and radiative cooling, but a +dynamic inflow of material under the influence of the SMBH. +4.4 Metallicity Structure +When the iron-peak elementmetallicity isfree tovary inthe fit, we see +a clear metallicity gradient. Metallicity decreases inwards in all sec- +tors, with the exception being the cavity-affected point in the South. +This gradient (referred to in the literature as a central abundance +drop) is common in cluster environments and has been observed in +more than 8 objects (Panagoulia et al. 2013, 2015; Lakhchaura et al. +2019; Liu et al. 2019). Though dust from old stars should be increas- +ing the central metallicity, if this dust is locked into filaments, it can +be lifted buoyantly to larger radii by jet-inflated bubbles. Thus, the +same processes which shape the density gradients along the jet axis +may be responsible for the metallicity gradient. +While buoyancy may explain some of the metallicity gradient +along the jet axis, the challenge remains to explain the central abun- +dance drops in sectors perpendicular to the jet axis. Without a mod- +ulation of the SNe Ia rate with radius in the galaxy, this gradient is +difficult to account for. Rather than arising from a physical process, +the drop in metallicity may point to an unresolved second temperature +component and thus, gas cooling en route to a multiphase structure. +While we found no evidence for such multi-temperature structure +(§4.3), future, deeper observations free of the constraints on soft en- +ergy response which afflict Chandra are necessary to tease out the +existence of this cooler material. +We close by noting that absorption, specifically “intrinsic” absorp- +tion due to interlaced cold and hot phase gas may be obstructing our +view of the gas cooling which is responsible for fueling M84’s AGN. +Such a “hidden” cooling flow (Fabian 1994; Fabian et al. 2022) may +be present within M84. Indeed, there is evidence from XMM-Newton +observations that an intrinsic absorption model may describe M84’s +galactic gas (Fabian et al. 2022 in prep.). However, high spectral res- +olution, far beyond what can be achieved by Chandra, is required to +tease out the parameters of this model. Thus, XMM-Newton obser- +vations, which probe much larger scales than Chandra (comparable +with the extent of M84’s H-shaped filaments) are unable to constrain +an accretion rate for a “hidden” cooling flow at Bondi radius scales. +4.5 No Temperature Rise at the Bondi Radius +We find no evidence for a temperature rise approaching the Bondi +radius, a phenomenon that has been proposed as evidence for the +transition from the galactic potential to that of the SMBH. This +conclusion may be a consequence of the changing metallicity, which +decreases by nearly a factor of 4 over the inner few hundred pc in +all but the South sector (although the cavity and limited numbers of +counts may be playing a role). +When we fix the metallicity to the radially averaged value +(∼0.3 𝑍⊙), we see signs of a temperature jump, with the inner- +most points in the East and West sectors reaching 1.5 and 1.2 keV +respectively. While temperature does increase with fixed iron-peak +element metallicity, so also does the reduced C-statistic, indicating +that a temperature rise may not truly be present. Instead, there may +have been an under-subtraction of the AGN which provides an ex- +cess of hard photons, enough to over-estimate the temperature when +metallicity is not a free parameter. +The lack of an observed temperature rise may not be surprising. +Observations of the temperature profiles in M87 by HRR15 find a +similar absence. In some respects this is to be expected: the analytical +Bondi solution at radii comparable to 𝑅B shows a relatively flat tem- +perature profile, with the majority of the adiabatic heating occurring +at small scales, well below the Bondi radius. +However, the notable absence of the temperature rise may be a +result of biases inherent to observations of multiphase gas. Indeed, +Guo et al. (2022) have recently performed simulations of Bondi +accretion from tens of kpc scales down to accretion flow scales +well below the Bondi radius which include gas at a wide range of +temperatures. They find that their simulations tend to predict a flat +emission-weighted temperature profile in the 0.5 − 7 keV band, even +at scales an order of magnitude below 𝑅B, where adiabatic heating +of hot phase gas becomes significant. Our X-ray observations may +be biased by the energy band accessible to Chandra, and even future +missions which probe scales below 𝑅B may similarly never detect a +temperature rise. +4.6 Shock Heating by the Jet or Nonthermal Emission? +While there is no evidence for a temperature rise at the Bondi radius, +we do see a clear (3.8𝜎) temperature increase from 4′′ to 3′′ in the +North sector which we refer to as a “hot blob.” This temperature +increase at 2′′ − 3′′ from the AGN is at the same angular separation +from the AGN as a knot of radio emission detected by VLA in the 5 +and 8.5 GHz bands, and ALMA in the 97 and 236 GHz bands (see +Knot B in Figures 1-3 of Meyer et al. (2018)). This “blob” or “knot” +of X-ray emission, first detected by Harris et al. (2002), stands out +clearly in X-rays, even with the limited exposure time (≈29 ks) of +Chandra’s first observation of M84. +The 0.5 − 7 keV X-ray spectrum of the region containing the “hot +blob” is well described (reduced C-statistic of 1.08 and 1.06 for +the North Sector at 2′′ and 3′′ respectively) by a VAPEC model in +our analysis, indicating thermal X-rays. However, the radio emission +MNRAS 000, 000–000 (0000) + +14 +C.J. Bambic et al. +is far more complicated. Early results favored a synchrotron origin +for the emission (Harris et al. 2002); however, Meyer et al. (2018) +argue that the radio and X-ray spectra of the knots cannot both be +explained by standard models for jet emission. In their analysis, the +X-ray emission is modeled as both a power law representing the jet +and an APEC component representing the thermal gas. +Motivated by these works, we re-fit the spectra from the North +sector at 2′′ and 3′′ using our M84 model (§2.8), with the addition of +a red-shifted power law (zpowerlw) component meant to represent +the X-ray jet detected by Meyer et al. (2018). When fitting the 2′′ +and 3′′ North spectra with the extra zpowerlw “jet” component, the +resulting temperature and metallicity of the VAPEC component are +unconstrained (in the case of the 3′′ point, temperature is constrained +but metallicity is not). We thus proceeded to leave the temperature +and zpowerlw normalization free in the fit, but fix the VAPEC metal- +licity to three different values of metallicity: 0.1 (consistent with +our measurements), 0.2, and 0.3 solar (consistent with that used by +Meyer et al. (2018)). For each metallicity, we scan zpowerlw photon +indices from Γ = 1 − 3, which we fix in the fit. +This procedure yields improved C-statistics over the 1.08 and 1.06 +found initially, in some cases comparable to or better than the fit to +the full annulus spectrum including all sectors at 2′′ and 3′′. Rea- +sonable photon indices near Γ ≈ 2 yield good fits. As expected, the +corresponding VAPEC temperature is lower when the jet component +is included; however, rather intriguingly, the fit temperature is lower +than all other sectors save that in the West at the same radii. The +temperature never exceeds 0.7 keV for all photon indices and metal- +licities tested. For the metallicity of 0.1 solar consistent with what +was found in our profiles (Figure 4), the fit finds VAPEC temperatures +below 0.6 keV for the point at 2′′. When using the metallicitity of 0.3 +solar assumed in Meyer et al. (2018), the fits settle around 𝑇 = 0.65 +keV, which is just below the temperature at a radius of 2′′ in the East +Sector (𝑇 = 0.67+0.04 +−0.03 keV). Note that all of these temperatures are +well below the 3 keV thermal model for the X-ray emission proposed +as an alternative to synchrotron emission in Harris et al. (2002). +The takeaway message from this analysis is clear: a model with +nonthermal emission from an X-ray jet and colder (𝑇 ≈ 0.65 keV) +galactic gas describes the “hot blob” as well as a purely thermal +emission model of 𝑇 ≳ 0.9 keV gas shock heated by the jet. Because +of Chandra’s limited spectral resolution, we cannot discriminate +between these models. However, given the complexity of M84’s jet +emission in the radio band and the co-spatial temperature increase +observed in our X-ray observations, a deeper study of M84’s jet +which can harness the full power of our 840 ks data set is merited. +4.7 Comparison to Other Measurements +Bondi accretion is incredibly inefficient. The Bondi accretion rate +�𝑀B is measured to be a few × 10−3𝑀⊙yr−1 in each of the sectors +in this work. Thus, the accretion flow in M84 need only liberate +𝜂 ∼ 10−6 of the �𝑀B𝑐2 fuel provided by Bondi accretion to power the +galaxy’s relativistic jets and X-ray AGN. +We are not the first group to arrive at this conclusion in M84. +The first measurement of the Bondi accretion rate can be at- +tributed to Allen et al. (2006), who found an accretion rate of +�𝑀B = 8.5+8.4 +−4.1×10−3𝑀⊙yr−1 by measuring the temperature and den- +sity of the full annulus around the AGN, i.e. including all sectors at the +innermost “Bondi radius” point. While this work was in preparation, +another measurement of �𝑀B was performed by Plšek et al. (2022) +which leveraged the data from our new campaign, presented here +and publicly available. They found �𝑀B = 2.4+1.9 +−1.5 × 10−3𝑀⊙yr−1, +again using all sectors. If we compare these values with our “All” +sector measurement of �𝑀B = 3.74+1.05 +−0.89 ×10−3𝑀⊙yr−1, then there is +strong agreement among all published values of the Bondi accretion +rate in M84, with notably tighter error bars in the more recent values +enabled by an extra ≳ 750 ks provided by the new campaign. +5 CONCLUSION +We have presented the deepest Chandra X-ray observations to date +of M84, a jetted elliptical galaxy in the Virgo Cluster. These obser- +vations, which comprise over 840 ks of Chandra data, have enabled +a detailed study of the temperature, density, and metallicity structure +of the galaxy, from kiloparsec scales to ≈ 50 pc scales just inside +the Bondi radius of the galaxy’s SMBH. New images of M84 have +been presented, emphasizing the intricate structure of the soft X-ray +filaments, formed into an H morphology by the action of powerful +(𝐿Jet = 1.1+0.9 +−0.4 × 1042 erg/s; Russell et al. 2013) radio jets. +Density and temperature measurements obtained through spectra +extracted from the innermost 0.5′′ − 1.5′′ bin allowed us to compute +Bondi accretion rates for each of 4 sectors around the central AGN. +“All” sectors are fit together to allow comparison to previous works. +The main conclusions of our analysis are as follows: +(i) Radial profiles of deprojected electron number density 𝑛𝑒 are +consistent with 𝑛𝑒 ∝ 𝑟−1, but slightly flatter (Figure 4 and Table 2). +This profile is in tension at the level of 5.5𝜎 with the expectation of +Bondi accretion, which predicts 𝑑 ln 𝜌/𝑑 ln 𝑟 = −0.373 at 𝑟 = 𝑅𝐵 +and an 𝑟−3/2 scaling at 𝑟 ≪ 𝑅𝐵. +(ii) The radial profile indices 𝛼 are statistically consistent; how- +ever, we see that the profiles are steeper along the jet axis than per- +pendicular to the jet (Table 2). This violation of spherical symmetry +is counter to the assumptions of the Bondi solution. +(iii) There is a discrepancy in the Bondi accretion rate depending +upon which sector is used to measure �𝑀𝐵 (Table 2). This discrepancy +between jet-aligned and mis-aligned sectors is at the level of 4.6𝜎, +which is statistically significant. While the discrepancy may point to +the influence of the jet on the large-scale accretion flow, the disparity +likely arises due to the presence of cavities (Figures 3 and 4) or +uncertainties in modeling the AGN emission at Bondi radius scales +(see Appendix A). +(iv) Temperatures do not vary widely throughout the galaxy (Fig- +ure 6) and only increase gradually from 0.6 keV to 0.7 keV over the +inner kpc approaching the Bondi radius. The exception is that in the +North sector, we see evidence for a temperature increase at points +2′′ − 3′′ from the AGN (Figure 4). We refer to this feature as a “hot +blob” of gas. The physical origin of this “hot blob” remains an open +question. Shock heating by the radio jet or nonthermal emission from +knots in an unresolved X-ray jet (Meyer et al. 2018) are both plausible +explanations; however, Chandra lacks the spectral resolution to rule +out either of these two models. +(v) We detect no temperature rise at the Bondi radius, consistent +with findings by HRR15 in M87. +(vi) By comparing the Bondi inflow time 𝑡inflow to the cooling +time as a function of radius, we observe evidence for a transition +from a “cooling-dominated” flow to an “inflow-dominated” flow at +scales of 1′′ − 2′′, providing support to the conclusion that we have +resolved M84’s Bondi radius. +ACKNOWLEDGEMENTS +The data presented in this work was obtained through Chandra Pro- +posal #19800344: “Fueling and self-regulation of AGN feedback at +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +15 +the Bondi radius of M84” and is publically available on the Chandra +archive. We are grateful to the Chandra X-ray Center for support, +not only with the initial observations and pointing of the telescope, +but also with use of the ChaRT tool. This manuscript was improved +thanks to the careful reading and suggestions of an anonymous ref- +eree. CJB thanks Andy Goulding and Jeremy Sanders for technical +support throughout this project. This work benefited from stimulating +discussions with Eliot Quataert and Minghao Guo at Princeton and +Eileen Meyer at the Texas Symposium on Relativistic Astrophysics. +CJB thanks the graduate students of the Institute of Astronomy, Cam- +bridge and Princeton University for their many insights as well. This +work is possible through the financial support of the Churchill Foun- +dation of the United States and CJB continues to be supported by a +National Science Foundation (NSF) Graduate Research Fellowship. +Early stages of this work were performed at the Multiscale Phe- +nomena in Plasma Astrophysics program at KITP in Santa Barbara, +CA, research supported in part by the NSF under Grant No. NSF +PHY-1748958. HRR acknowledges support from an STFC Ernest +Rutherford Fellowship and an Anne McLaren Fellowship provided +by the University of Nottingham. CSR thanks the STFC for sup- +port under the Consolidated Grant ST/S000623/1, as well as the +European Research Council (ERC) for support under the European +Union’s Horizon 2020 research and innovation programme (grant +834203). +DATA AVAILABILITY +The Chandra data described in this work are available in the Chandra +data archive (https://cxc.harvard.edu/cda/). 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This procedure provided a normalization +for the VAPEC background component at 3′′ from the AGN. Then, +we fit the surface brightness (SB) distribution with the assumption +of spherical symmetry using a simple power law in radius, assuming +that the power law extrapolation from 3′′ to 1′′ accurately described +the SB at 1′′ from the central AGN. The ratio of the SB determined +at 1′′ and 3′′ from this power law was taken as a “boost” factor +multiplied on to the previously fit-for VAPEC normalization. In the +case of M84, we found this boost factor to be 3.98. +Then, the parameters of the input energy spectrum passed to ChaRT +were determined by fitting the full spectrum (source + background) +extracted from the 1′′ circle with a fixed background VAPEC compo- +nent and only the zpowerlw parameters (meant to model the AGN +source) left free. The normalization of this fixed VAPEC compo- +nent is simply the normalization of the 2′′ − 4′′ annulus, multi- +plied by the boost factor (3.98) and corrected by the ratio of the +annulus to 1′′ circle areas. Once the parameters for the zpowerlw +source component were determined, a clean spectrum including only +phabs(zphabs(zpowerlw)) parameters was produced and passed +to the ChaRT tool as the input spectrum. This source spectrum is rep- +resentative of the AGN without contributions from the background. +MNRAS 000, 000–000 (0000) + +Feeding and Feedback at the Bondi Radius of M84 +17 +0.5 +1 +2 +5 +10 +20 +30 40 +Radius (arcsec) +10 +11 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +Surface Brightness +(cts s +1 cm +2arcsec +2) +Hard Band Simulation (4-7 keV) +Hard Band Data (4-7 keV) +Broad Band Data +0.5 +1 +2 +5 +10 +20 +30 40 +Radius (arcsec) +10 +11 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +Surface Brightness +(cts s +1 cm +2arcsec +2) +5% Boost in Simulation +5% Drop in Simulation +Simulation Subtracted +Figure 9. Top: Hard band (4 − 7 keV) surface brightness profiles extracted +from merged event file and ChaRT+MARX AGN simulation, with 7% boost +applied to simulated profiles. Bottom: Subtraction of hard band simulation +profile from hard band data profile. The hard band, where only AGN emis- +sion and background from the Virgo screen and unresolved point sources +remains, is more than an order of magnitude subdominant to the total SB. +By applying a 7% boost to the simulation, we achieve a flattening of the +AGN-subtracted profile within the inner 2′′, indicating that the AGN has +been properly subtracted from the data. All that remains at PSF scales are +negligible contributions from the spatially uniform Virgo screen and unre- +solved point sources. The point source emission appears to be subdominant +given the smoothness of the AGN-subtracted hard band SB from 1′′ − 3′′. +Because the normalization of the input spectrum was determined +by assuming a model for the galactic gas spectrum, the flux of the +simulation may not be an accurate representation of the true AGN. +To test the accuracy of the AGN modeling, we choose an energy +band where the AGN completely dominates and which is free of the +small-scale variations (on scales comparable to the PSF) imposed by +bright, lumpy, soft emission from galactic gas. In this case, following +HRR15, we choose the hard 4 − 7 keV band. At these energies, the +only contributions to the hard band SB should be from the AGN +source, Virgo ICM background (which should be spatially uniform), +and unresolved point source background. If point source emission is +substantial, the hard band SB profile should display discontinuities +and rapid spatial variations on scales of the PSF. +The top panel of Figure 9 shows a comparison of the hard band +profiles for the data (black) and simulation (red). By forward mod- +eling the AGN, we are working to subtract the AGN contribution at +Bondi radius scales from the spectra extracted in each sector. The +blue points in the lower panel of Figure 9 show a subtraction of the +simulation’s hard band SB profile from that of the data. If we were to +overestimate the AGN flux and thus over-subtract the AGN from the +data, we should expect a drop in hard band SB at 1′′, i.e. the scales +of the PSF. Alternatively, if we under-subtract the AGN, we should +expect a hard band excess in the bottom panel of Figure 9. +We find that our AGN simulation based on modeling the galactic +gas background leaves a 7% excess in the hard (4 − 7 keV) band. +Without compensating for this excess, we would under-subtract the +AGN and possibly bias our temperature measurements with excess +hard AGN photons. Thus, we boost the overall AGN simulation nor- +malization by 7%, which means that the difference profile in the +bottom panel of Figure 9 flattens at the scales of the PSF. When +computing errors on temperature, we do not simply report the statis- +tical uncertainties determined by XSPEC. Rather, we boost the AGN +normalization by 5% (on top of the 7% compensation) and decrease +the normalization by 5% (shown as the red and black points in the +bottom panel of Figure 9 respectively), to marginalize over uncer- +tainties in the AGN modeling. As a result, errors in temperature and +metallicity are larger at the Bondi radius. Finally, because the hard +band SB is relatively continuous at PSF scales, we conclude that +unresolved point sources are properly accounted for. +APPENDIX B: ASCERTAINING ERRORS ON �𝑀B AND 𝜂 +For computing errors on �𝑀B, we use a Monte Carlo method, draw- +ing 107 samples from distributions of 𝑛𝑒, 𝑇, and 𝑀BH and applying +Equation 2. Because of asymmetric error bars in 𝑇 and 𝑀BH, we +model the distributions of these variables as “dimidiated Gaussians” +(Barlow 2003), two Gaussians centered on the same mean with dif- +ferent standard deviations above and below the mean based on the 1𝜎 +upper and lower error bars. For equal positive and negative error bars, +the dimidiated Gaussian is equivalent to a normal distribution. We +model the underlying distribution of 𝑛𝑒 as a log-normal with mean +and standard deviation based on the central value and 1𝜎 error bar +respectively. This choice ensures strict positivity of 𝑛𝑒 but only has +significance for the point in the North—a Gaussian yields a similar +error bar for all other points. We take the 1𝜎 errors on �𝑀B to be the +16th and 84th percentile of the resulting distribution. +MNRAS 000, 000–000 (0000) + diff --git a/wtFKT4oBgHgl3EQf5S7u/content/tmp_files/load_file.txt b/wtFKT4oBgHgl3EQf5S7u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed01209a98eed638b821a6b0eec41d723ac63a7e --- /dev/null +++ b/wtFKT4oBgHgl3EQf5S7u/content/tmp_files/load_file.txt @@ -0,0 +1,2240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf,len=2239 +page_content='MNRAS 000, 000–000 (0000) Preprint 31 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 AGN Feeding and Feedback in M84: From Kiloparsec Scales to the Bondi Radius C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} 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+page_content=' Australia 31 January 2023 ABSTRACT We present the deepest Chandra observation to date of the galaxy M84 in the Virgo Cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' with over 840 kiloseconds of data provided by legacy observations and a recent 730 kilosecond campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The increased signal-to-noise allows us to study the origins of the accretion flow feeding the supermassive black hole in the center of M84 from the kiloparsec scales of the X-ray halo to the Bondi radius, 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Temperature, metallicity, and deprojected density profiles are obtained in four sectors about M84’s AGN, extending into the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather than being dictated by the potential of the black hole, the accretion flow is strongly influenced by the AGN’s bipolar radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Along the jet axis, the density profile is consistent with 𝑛𝑒 ∝ 𝑟−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the profiles flatten perpendicular to the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Radio jets produce a significant asymmetry in the flow, violating a key assumption of Bondi accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Temperature in the inner kiloparsec is approximately constant, with only a slight increase from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 keV approaching 𝑅B, and there is no evidence for a temperature rise imposed by the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The Bondi accretion rate �𝑀B exceeds the rate inferred from AGN luminosity and jet power by over four orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In sectors perpendicular to the jet, �𝑀B measurements agree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the accretion rate is > 4𝜎 lower in the North sector along the jet, likely due to cavities in the X-ray gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our measurements provide unique insight into the fueling of AGN responsible for radio mode feedback in galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Key words: X-rays: galaxies: clusters — galaxies: clusters: M84 — intergalactic medium 1 INTRODUCTION Accretion onto active galactic nuclei (AGN) at the centers of mas- sive elliptical galaxies fuels AGN feedback in clusters of galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The gravitational potential energy released from plasma flowing onto these supermassive black holes (SMBHs) powers jets of rela- tivistic particles which sculpt the surrounding intracluster medium (ICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In cool-core clusters where the central cooling time of the hot (107 − 108 K) ICM is ≲ Gyr, thermalization of the jet kinetic energy provides the heating necessary to balance radiative cooling, maintain- ing cluster atmospheres in their observed quasi-thermal equilibrium and averting a “cooling catastrophe” (Fabian 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' McNamara & Nulsen 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Deep (≳100 kiloseconds), spatially-resolved Chandra X-ray Space Telescope observations of nearby galaxy clusters such as Perseus and Virgo have revealed the signatures of this feedback process: cavities or bubbles carved out of the ICM by jets (McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' weak shocks, ripples, and waves emanating from newly formed bubbles (Sanders & Fabian 2007, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Forman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' bright filaments formed by gas cooling around these ★ E-mail: cbambic@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='edu cavities (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' and turbulent fluctuations stirred by the buoyant rise of bubbles through clusters (Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Zhuravleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Hitomi Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Simionescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yet, while the X-ray morphology of clusters has pro- vided insight into how AGN shape their environments on 10s − 100s of kiloparsecs (kpc) scales, understanding the connection between AGN and the sub-kiloparsec scale accretion flows which power them remains a critical uncertainty in the paradigm of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Even the basic energetics of large-scale black hole “feeding” is an open problem (see Abramowicz & Fragile 2013 for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In the standard paradigm, material within the SMBH’s sphere of influence, the Bondi radius (𝑅B = 2𝐺𝑀BH/𝑐2𝑠 where 𝑀BH is the black hole mass and 𝑐𝑠 is the speed of sound well beyond 𝑅𝐵), is destined to either reach the hole or race away in an outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Ionized gas pierces the sphere of influence at a rate �𝑀B en route to the hole, where this plasma is consumed at a rate �𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A fraction 𝜂 of the rest mass power �𝑀𝑐2 is released by the accretion flow in the form of radiation and outflows—winds or jets—such that the total power (radiative + outflow) of the AGN is 𝐿 = 𝜂 �𝑀𝑐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' At the largest scales, a gas inflow with accretion rate �𝑀B is formed by gas cooling and gravitational infall under the influence of the com- bined galactic and SMBH potential (Quataert & Narayan 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The © 0000 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='11937v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='HE] 27 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' accretion rate �𝑀 is then influenced by the structure of this inflow: the angular momentum (Proga & Begelman 2003) and effective tur- bulent viscosity (Narayan & Fabian 2011) of the gas, and the relative contributions of hot X-ray emitting plasma (Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' cold atomic and molecular gas (Pizzolato & Soker 2005) which may “rain down” through the Bondi radius (Gaspari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yang & Reynolds 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Magnetic fields certainly complicate this picture, with the magnetic flux frozen into the flow (Lubow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1994) competing with dynamo-generated fields (Brandenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Brandenburg & Subramanian 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Blackman 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Liska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2020) to power relativistic jets (Blandford & Znajek 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Komis- sarov 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Tchekhovskoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2010, 2011) and winds (Blandford & Payne 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Proga 2000), and thereby influence the value of 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The complexities of this inflow determine the state of the resulting accretion disk around the black hole and the relative contribution of radiation to the flow’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For the jetted systems of interest in cluster AGN feedback, the accretion flow is likely radiatively ineffi- cient, forming a virialized, geometrically-thick advection-dominated accretion flow (ADAF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Ichimaru 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Narayan & Yi 1994, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Quataert & Narayan 1999), a convection-dominated accretion flow (CDAF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Quataert & Gruzinov 2000), or when the net magnetic flux reaching the hole is large, a magnetically arrested disk (MAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bisnovatyi-Kogan & Ruzmaikin 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Igumenshchev 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' McKinney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Avara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While �𝑀B is crucial in determining accretion flow structure, mea- suring this parameter is a major challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the true �𝑀B cannot be measured directly, large-scale black hole feeding is often interpreted through a steady, spherically symmetric model of accre- tion, the Bondi (1952) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Within this framework, �𝑀B for a given SMBH mass is specified entirely by the gas density and tem- perature at 𝑅B, quantities which in principle can be measured with deep X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This choice is one of convenience—there are no strong theoreti- cal reasons to expect the assumptions of the Bondi solution to hold in real systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, some evidence points to the importance of �𝑀B in setting feedback power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2006), using a small sample of X-ray observations of nearby elliptical galaxies, found an apparent correlation between Bondi accretion rate and AGN jet power, as measured from the enthalpy of jet-blown cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This method for inferring jet power is subject to significant uncertainties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' projection effects and the assumption of subsonic inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In- deed, a follow-up study by Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2013) using a larger sample of elliptical galaxies found a less significant correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A direct correlation between Bondi accretion rate and AGN feed- back power has interesting consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The correlation may imply a universality in the radiatively inefficient accretion flows (RIAFs) which power AGN in early type galaxies, with �𝑀B serving as the crucial parameter for regulating power from radiative (𝐿Rad) and jet (𝐿Jet) feedback on ∼Gyr timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In addition, the correlation could be leveraged in sub-grid models for galaxy formation, where feedback power from unresolved AGN must be tuned based on re- solvable properties, such as �𝑀B (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Establishing this correlation necessitates deep X-ray observations which resolve the density and temperature at 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this paper, we harness the deepest X-ray observations to date of the galaxy M84 (NGC 4374) to measure the Bondi accretion rate of hot phase (≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 keV) gas onto a jetted AGN in an early type galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These measurements are based on a new Chandra campaign which yielded approximately 730 kiloseconds (ks) on M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Com- bined with legacy data published in Finoguenov & Jones (2001, 2002) and Finoguenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2008), the observations presented com- prise over 840 ks of X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' M84 is one of only 5 known systems where the Bondi radius can be resolved by Chandra, despite the observatory’s remarkable sub-arcsecond angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The other 4 systems are Sgr A∗ (Baganoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003), NGC 3115 (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2014), NGC 1600 (Runge & Walker 2021), and M87 (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2015, hereafter HRR15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Even within this small class, M84 stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Unlike Sgr A∗ and NGC 1600, M84 has an X-ray detected AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In contrast to NGC 3115, a Fanaroff & Riley (1974) Type I radio jet is clearly observed in M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, unlike that in M87 which hosts a notably powerful jet, M84’s AGN is not particularly luminous (more than an order of magnitude dimmer than M87’s AGN) and our extended campaign caught the SMBH in a relatively quiescent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, M84 does not require the same sophisticated treatment of pile-up as was performed in M87 (HRR15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These factors make M84 an especially useful object for exploring the interplay of feeding and feedback in elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We describe our data analysis in §2 including data reduction, spectral models for the AGN and galactic gas, and simulations of the detector point spread function (PSF) used for forward modelling spectral contamination from the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In §3, we present results: profiles of gas density, temperature, and metallicity approaching and just within the Bondi radius, and the measured Bondi accretion rates �𝑀B and efficiencies 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We discuss the implications of our measurements in §4, and conclude in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2 CHANDRA DATA ANALYSIS M84 is a nearby (luminosity distance 𝐷𝐿 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='83 Mpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' redshift 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00327) giant elliptical galaxy (type E1) and satellite member of the Virgo Cluster of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The galaxy has been the subject of three separate Chandra ACIS-S campaigns which together yield ≈840 ks of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While earlier works by Finoguenov & Jones (2001) and Finoguenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2008) addressed the detailed structure of M84 and how the X-ray halo is shaped by feedback, our ultra-deep campaign is concerned primarily with black hole feeding and gas structure approaching and just within the Bondi radius of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 Data Reduction This work is a follow-up to a similar analysis of M87 by HRR15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, we follow the same data reduction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our data reduction was performed using CIAO version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='11 and the Calibration Database (CalDB) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5, updated November 7, 2019 (Fruscione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This update followed a major revision to the soft energy response brought about by contaminant build-up over Chandra’s prolific 23 year lifetime (thus far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our long campaign was affected by this contamination, and as we show, the majority of M84’s galactic gas, especially that approaching the Bondi radius, is cooler than 1 keV and emitting X-rays within the range of degraded performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Given the low temperature of the extended emission in M84, the calibration of the contaminant build up on Chandra’s optical block filters is particularly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We therefore verified that temperature, metallicity and normalization values measured with the new observations are consistent with the archival observations, which were taken only a few years after Chandra’s launch and less affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using the chandra_repro routine, we reprocess our data to produce second-level event files, removing bad pixels based on the analysis reference data library (ardlib), detecting point sources using wavdetect, and creating light curves to filter out bad time MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Left: Merged 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5−2 keV image, totaling 798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='66 ks of cleaned exposure time with Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID 5908 excluded (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Right: 5 GHz radio contours (white) as measured by the Very Large Array (VLA) overlaid on X-ray image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Contours correspond to 12 logarithmically-spaced levels in flux from 2 × 10−3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 mJy, and colors denote X-ray counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The H-shaped morphology is carved out of the halo by radio jets, forming bright rims about the radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' To produce merged images, we assume an exposure cor- rection for each Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID’s exposure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 X-ray Morphology Figure 1 displays a merged 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5−2 keV image based on all three cam- paigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Similar images can be found in Finoguenov & Jones (2001) and Finoguenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2008) from the first two sets of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using only limited Chandra data, Finoguenov & Jones (2001) were able to identify the salient features of the galaxy’s X-ray emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead of a featureless X-ray halo, M84 hosts depressions in emissivity North and South of the central AGN, coincident with radio lobes produced by Fanaroff & Riley (1974) Type I jet activity (Laing & Bridle 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These cavities create an H-shaped structure in the halo gas, which extends ≈ 150′′ (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 kpc) from the Northernmost edge of the emission to the faint rim in the Southwest of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The crossbar of the H spans ≈ 46′′ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 kpc) and is approximately aligned with optical dust lanes (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1985), although the dust lanes are on a much larger scale, cutting across the X-ray image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As argued by Finoguenov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2008), these cavities may actually be comprised of at least two bubbles each, with bright rims (viewed in projection) demarcating the bubble boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Indeed, our deep observation is able to clearly detect a tenuous bubble rim extending toward the Southwest in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The Northern bubble is com- pressed, likely by the ram pressure of ICM gas as the galaxy moves through the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While the bubbles are located just to the North and South of the crossbar, the jet is aligned with the West filament;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' the galaxy has drifted over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Subsequently, ram pressure has swept the Northern bubble back and “bent” the radio jet—a signature of radio galaxies moving through clusters (Miley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Owen & Rudnick 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Begelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Morsony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' McBride & McCourt 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Intriguingly, the Southern bubble has not been swept in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' There may be a large-scale shear flow across M84, or the jet may have reoriented itself over the course of the episodes recorded in the radio lobes, possibly through precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 Spectral Fitting Unless stated otherwise, we fit spectra simultaneously with the XSPEC spectral fitting package (Arnaud 1996) using all Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' IDs listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Spectra are extracted using CIAO’s specextract function and grouped such that at least one count is present in all energy bins over the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5−7 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fits are performed using the modified C- statistic (Cash 1979) with elemental abundances taken from Anders & Grevesse (1989) for comparison with past results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' All spectral models are fit with galactic absorption included via a photoelectric absorption (phabs) model, with a constant galactic column density of 𝑁H,gal = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 × 1020 cm2 as measured by the HI4Pi Survey (HI4PI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 Virgo Cluster Spectral Model M84 is embedded in the Virgo Cluster, so we must peer through a “screen” of hard X-ray emission ≳1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the Virgo Cluster occupies the entire field of view, we follow HRR15 and choose to use blank sky backgrounds for all spectral fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The appropriate blank- sky background dataset was processed identically to the event file, reprojected to the same sky position, and normalized so that the count rate matched that of the event file for the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 − 12 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We model the bremsstrahlung emission from the Virgo ICM with a single temperature APEC plasma emission model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The parameters for this model are determined by fitting a spectrum extracted from a large region, a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6′ × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9′ box around M84, excluding point sources (as detected by wavdetect) and the galaxy, whose X-ray emission is confined within a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9′ × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6′ box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This Virgo spectrum is fit with temperature (𝑇), metallicity (𝑍), and normalization free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The fit yields reasonable values, MNRAS 000, 000–000 (0000) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content="kpc 20'arcsec 2 5 12 24 49 98 197 3931." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='63 kpc 20 arcsec 2 4 9 18 37 74 150 2994 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID Date Exposure 𝑁H Γ Flux (2-10 keV) C-Stat/DOF (ks) (1022 cm−2) (10−13 erg cm−2 s−1) 803 19/05/2000 28.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='50+0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 3055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9/ 3328 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Summary of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fits to AGN are obtained using the “M84 Model” as presented in the text, with a VAPEC model for the galactic emission and an APEC model for the Virgo Cluster “screen.” The photon index Γ remains close to 2 as expected for Comptonized emission from an ADAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We include local absorption with column density 𝑁H to account for intervening dust lanes or a dusty torus around the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 𝑇 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='06 keV and 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='429 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 𝑍⊙, consistent with 𝑇 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 keV obtained by Urban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2011) and Ehlert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2013) using the much higher spectral resolution of XMM-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 M84 Galactic Gas Spectral Model The earliest Chandra measurements of M84 by Finoguenov & Jones (2001) showed an overabundance of metals relative to solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This overabundance could be contributed both by iron-peak elements (Fe and Ni) originating from Type Ia supernovae, or 𝛼 elements (C, N, O, Al, Si, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=') produced by Type II supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Since XMM- Newton lacks the spatial resolution of Chandra, abundance measure- ments performed by XMM probe larger length scales than we are studying;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' we must constrain 𝛼 element metallicities ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We fit the spectrum of the full 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6′ × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9′ box with the galaxy included using a VAPEC model for M84’s galactic gas emission, an APEC component for the Virgo ICM, and an extra power law component for unresolved point sources (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Since the helium (He) abundance of the VAPEC component cannot be constrained in the X-ray band, we set the He abundance to solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' VAPEC iron-peak element abundances 𝑍Fe are tethered together in the fits, as are all remaining 𝛼 element metallicities 𝑍𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The APEC temperature and metallicity are fixed based on §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4, but the normalization is left free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 Unresolved Point Sources M84 is known to host a substantial number of X-ray binary (XRB) point sources (Finoguenov & Jones 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While many of these XRBs can be masked out, unresolved XRBs and AB/CV stars rep- resent a source of hard emission which can affect temperature and abundance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We follow the common practice of mod- eling these unknown populations using a simple power law model with fixed photon index ΓXRB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 (Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The nor- malization of this power law is left free in the fits to the VAPEC+APEC model, which are designed to provide adequate statistics for con- straining the 𝛼 element metallicity, 𝑍𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Unfortunately, the contaminant build-up which has degraded Chandra’s soft energy response prevents us from constraining the 𝛼 element metallicity from the new extended campaign, even with ample source counts available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, the only Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' IDs used for de- termining 𝑍𝛼 come from legacy observations: Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' IDs 803, 5908, and 6131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We find a reasonable constraint on 𝑍𝛼, approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='45 times solar abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This value changes only slightly to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='51 solar when the power law component is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For all remaining spec- tral fits in this paper, we fix 𝑍𝛼 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='45 and fit only the temperature, normalization, and the iron-peak metallicities in the VAPEC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 5 0 500 1000 1500 2000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 log(Flux) (erg cm−2 s−1) 0 500 1000 1500 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 Photon Index Γ 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 Source NH (1022 cm2) 6850 6860 6870 6880 6890 6900 6910 6920 6930 6850 6860 6870 6880 6890 6900 6910 6920 6930 6850 6860 6870 6880 6890 6900 6910 6920 6930 Days since First Observation Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Time variability of AGN over all three observational campaigns used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The observation corresponding to Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID 5908 (red) has a much higher flux than the others, violating an assumption that the AGN has a constant luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We remove this observation from our analysis of Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 Accounting for AGN Contamination While the AGN is a point source, Chandra’s point spread function (PSF) distributes AGN photons across a number of pixels, including those containing photons produced by gas at Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Therefore, we must account for AGN contamination when fitting spectra extracted from these sub-kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The AGN and galactic gas are spectrally distinct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, they are not spectrally separable with Chandra’s spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We find that when fitting the AGN and galactic gas together, we are unable to adequately constrain the parameters for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In response to this limitation, we follow the example of HRR15 and forward-model the AGN contamination by simulating Chandra’s energy-dependent PSF’s effect on the measured AGN spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We first fit the spectrum of the AGN based on fixing the parameters of the galactic gas spectral model (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using the parameters obtained from this fit, we produce an AGN spectrum free of contributions from the galactic gas or the Virgo screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This spectrum is fed into the Chandra Ray Tracer tool (ChaRT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003) which simulates the detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The MARX software package (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2012) is used to produce a second-level event file of the simulated AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 AGN Spectral Model Previous estimates based on the most recent SMBH mass measure- ments by Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2010) place the angular scale of the SMBH Bondi radius at approximately 1′′ (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, measurements in the literature differ by as much as a factor of ≈4 (Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Maciejewski & Binney 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, we choose to define the region of AGN contamination as a circle with a radius of 1′′ centered on the peak of the AGN surface brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When we refer to the “AGN spectrum” in this paper, we are referring to the spectrum extracted from this region which includes the AGN, galactic gas emission, Virgo Cluster emission, and unresolved XRBs and AB/CV stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The Comptonized emission from the AGN is modeled as a red- shifted power law (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Siemiginowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Due to the presence of the intervening dust lanes as well as a possible “dusty torus” surrounding the AGN, we allow for lo- cal photoelectric absorption through a zphabs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, the spectrum of the 1′′ AGN region is fit using the “M84 model,” phabs(zphabs(zpowerlw)+VAPEC+APEC+powerlaw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While this model seems complicated, the only free parameters are the local column density and the photon index Γ and normalization of the zpowerlw component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Appendix A2 of HRR15 presents the method for obtaining the parameters for the remaining components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Spectra are extracted from a 2′′ − 4′′ annulus circumscribing the AGN and fit with a phabs(VAPEC+APEC+powerlaw) model for the galactic gas, Virgo ICM, and unresolved point source emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The APEC component is fixed based on fits in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 and the 𝑇 and 𝑍 from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 (note that all normalizations are scaled to the appropriate region areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We then fit the annulus spectrum with a free VAPEC temperature, iron-peak metallicity, and normalization as well as a free powerlaw normal- ization with fixed photon index ΓXRB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The VAPEC normalization used in the AGN fits is determined by boosting the normalization from the annulus fit centered at 3′′ to the 1′′ AGN circle based on a power law extrapolation of the 1′′ − 40′′ surface brightness profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For the powerlaw component, we assume the normalization is constant from 1′′ − 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 AGN Variability We perform the fit described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 for all Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' IDs individually in addition to a simultaneous fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The measured parameters for the AGN model are shown in Table 1 and plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Note that the AGN, unlike the galactic gas emission, is highly variable, with nearly an order of magnitude variation in flux over the three campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Serendipitously, the AGN was relatively quiescent MNRAS 000, 000–000 (0000) 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (a) 10−8 10−7 10−6 (b) 1 2 5 10 Radius (arcsec) 10−7 10−6 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 Radius (kpc) Surface Brightness (counts/s/cm2/arcsec2) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Left: Sectors (10′′ in radius) overlaid on merged 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 − 2 keV image of M84 central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The East (magenta) and West (green) regions are aligned perpedicular with the AGN jet which is approximately aligned with the Western filament (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Right: Background-subtracted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 − 7 keV surface brightness profiles in sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The upper panel shows profiles of the AGN (triangles) as well as the broadband data, while the lower panel shows the AGN-subtracted surface brightness profile, used to compute the deprojected density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' during 2019 (observations following the break in Figure 2), implying that the vast majority of data is subject to minimal AGN contamina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Unfortunately, one observation, Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID 5908, caught a state of outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because we are modeling the AGN based on the statistically powerful simultaneous fit to all usable observations, we necessarily make the assumption that the AGN flux is constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID 5908 (shown in red in Figure 2) breaks this assumption and is thus omitted from the remaining analysis of Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='10 Simulating the AGN Spectrum The fit to the AGN spectrum is consistent with a photon index of Γ ≈ 2, in accord with other similar ADAF spectra (Gu & Cao 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Younes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Younes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' With this result, we simulate how the AGN spectrum free of contributions from the galactic gas, Virgo, and XRB/AB/CVs would appear to Chandra’s ACIS-S detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Appendix A3 of HRR15 describes this process in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We boost the normalization of the spectrum input to ChaRT and use the aspect solution file for Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ID 20543 (the longest exposure observation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pile-up is negligible in M84 and not included in our modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The MARX tool is used to produce a second level event file and exposure map for the simulation, and we reproject the simulation to the coordinate system of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We have accounted for galactic emission in the AGN spectrum by measuring the galactic background from 2′′ − 4′′ and modeling the surface brightness (SB) profile to account for the increase in background into 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, our choice of model introduces a systematic error which may cause us to inadvertently over or under-subtract the AGN by a small amount— enough to impact the measured temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Subtraction of the AGN can be tested by comparing the hard band (4 − 7 keV) SB profile of the simulation with that of the data (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bright, lumpy, soft emission from galactic gas, which may vary on scales of the PSF, is entirely subdominant in the 4 − 7 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, AGN, Virgo ICM, and unresolved point source emission dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If the AGN is under-subtracted, a hard band ex- cess will manifest itself at the scales of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, if the AGN is over-subtracted, there should be a drop in hard emission at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the Virgo screen is uniform over these small scales, any discontinuities in the AGN-subtracted SB profile is evidence of spatial variation in the unresolved point source flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our AGN simulation leaves a modest hard band excess of 7% at PSF scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We compensate for this excess by boosting the AGN simulation 7% such that the AGN-subtracted 4 − 7 keV band SB profile flattens from 1′′ − 2′′ (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Continuity of the AGN- subtracted SB at these small scales indicates that the unresolved point source emission is relatively constant with radius, and our assumption that the XRB/AB/CV normalization is constant with radius obtains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We extract spectra for both the observations and AGN simulation in 1′′ radial bins extending from 1′′ − 10′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Simulated spectra are fit using an absorbed, redshifted power law model, and the fit pa- rameters are fixed for the AGN components in the combined “M84 model” used in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We compute the error bars on temperature and metallicity for the two points in the innermost 2′′ of each sector by boosting or diminishing the AGN normalization by 5%, marginaliz- ing over uncertainties in the AGN flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The AGN normalization is set to 0 beyond the innermost three annuli in each sector since the AGN’s PSF is insignificant beyond 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='11 Profile Deprojection To obtain density profiles, we follow HRR15 and first compute background-subtracted surface brightness (SB) profiles of the inner 1′′ −10′′ (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 kpc) in sectors, referred to as North, East, West, and South respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The North and South sectors are aligned with the radio jet, while the East and West sectors are anti-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We sub- tract off the Virgo and X-ray background from the SB profiles based on a measurement of SB taken far away from the galaxy and free of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The sectors and SB profiles are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) N 410 pc 5 arcsecFeeding and Feedback at the Bondi Radius of M84 7 Sector 𝑅B 𝑇 (𝑅B) 𝑍 (𝑅B) 𝑛𝑒 (𝑅B) Index �𝑀B 𝜂 𝐿B/𝐿Edd pc keV Z⊙ cm−3 𝛼 10−3 M⊙yr−1 ×10−6 ×10−4 North 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='57+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='01 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='62+31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='64 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='54 East 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='72+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='35+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='58 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='37+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='78 West 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='52+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='21 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='75 South 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='33 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='88 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='68 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='60 All 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='89 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='42+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='46 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='50 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Summary of measurements at 1′′, density profile index 𝛼, accretion rates �𝑀B, efficiencies 𝜂, and Eddington ratios 𝐿B/𝐿Edd for Bondi accretion in each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because of small-scale cavities evident in the AGN-subtracted SB profiles in the South sector (Figure 3), we are only able to obtain limits on the density 𝑛𝑒 and quantities derived from density in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' SB is a projection of a 3D distribution of X-ray emission onto a 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' By assuming spherical symmetry, we can deproject each sec- tor’s SB profile, “peeling back” shells of X-ray emission to determine a volumetric emissivity at each profile radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Spherical symmetry is a poor assumption for M84’s highly-structured H-shape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the assumption may obtain more readily around the quasi-spherical halo in the inner 10′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We apply the deprojection method of Kriss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (1983) to the AGN-subtracted SB profiles, panel (c) in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This method only strictly applies when the SB is monotonically increasing inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Cav- ities, evident in the significant drop in the AGN-subtracted surface brightness profiles for the North and South sectors at a radius of 1′′, violate this assumption and prevent us from obtaining anything more than an upper limit on 𝑛𝑒 in the South sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Even though the AGN simulation is always sub-dominant to the observed SB (panel (b) in Figure 3), the simulation is brightest in the South where the observed profiles show a depression in SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We note that in the jet-aligned sectors, the average SB of the innermost radial bin is ∼20% less than that in the off jet-axis sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, the AGN simulation tends to favor more photons in the jet-aligned sectors, with ∼26% more photons in the North and South compared to the East and West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Chandra’s PSF at sub-arcsecond scales is subject to a hook feature which is captured in the ChaRT-MARX simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the simulation PSF is asymmetric, a simple re-alignement of the simulation is insufficient to eliminate the cavities from the AGN-subtracted SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our emissivity profiles, temperatures, and metallicities are all mea- sured in the same radial bins/ sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, we are able to use the temperatures and metallicities to determine the number density of X-ray emitting electrons 𝑛𝑒 from the emissivity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this way, we obtain profiles of 𝑛𝑒 in each sector separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='12 Contour Binning The large signal-to-noise afforded by our deep observations allows us to produce maps tracing the large-scale temperature and metallicity structure in M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For this task, we use the contour binning method presented in Sanders (2006) and made possible through the contbin software package (Sanders 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Contour binning groups adjacent pixels of similar surface bright- ness to achieve a requested signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The method groups gas expected to be spectrally similar, allowing us to extract spectra with high signal-to-noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, contour binning produces accurate temperature maps of spatially-resolved extended sources with non- smooth surface brightness distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We use a signal-to-noise ratio of 32 and set the smooth signal- to-noise parameter to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because of M84’s H-shaped emissivity distribution, contours tend to be elongated along the filaments, con- necting regions which are too spatially separated to be causally con- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We thus constrain the shape of the contours using contbin’s constrainval parameter, which we set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The spectra extracted from the regions defined by contbin are fit using the spectral model defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, we do not include an XRB/AV/CV com- ponent in our fits as this component tends to be negligible on the kiloparsec (kpc) scales relevant for the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3 RESULTS In this section, we present profiles of temperature, metallicity, and deprojected density measured in four separate sectors—two aligned with the jet axis and two anti-aligned (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We use these mea- surements to calculate the accretion rates �𝑀B and efficiencies 𝜂 for Bondi accretion in each of the sectors, where 𝐿Jet + 𝐿X = 𝜂 �𝑀𝑐2 and 𝐿X is the X-ray luminosity of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our main results are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These measurements allow us to explore the large-scale structure of the accretion flow and compare timescales which dictate the flow’s dynamics, namely the cooling time 𝑡cool and inflow time 𝑡inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We conclude this section with maps of tem- perature, metallicity, and pseudo-pressure to connect the small-scale physics of accretion with the kpc-scale structure of the X-ray halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 Density Profile The top panel of Figure 4 displays the density profile for each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Density increases monotonically toward smaller radii in all sectors from 7′′ to 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The scaling of density with radius provides a direct comparison between our data and the theoretical prediction from the adiabatic Bondi solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We can model the observed density profile as a simple power law, 𝑛𝑒(𝑟) = 𝑛𝑒,0 � 𝑟 𝑅B � 𝛼 , (1) where 𝑛𝑒,0 is the number density at the Bondi radius, 𝑟 = 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using a Markov chain Monte Carlo (MCMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013) method, we fit this power law model to the density data for each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We omit the innermost data points at 1′′ in the North and South sector fits as these points are strongly affected by the presence of Bondi radius-scale cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Values of 𝛼 are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 Temperature Profiles We see evidence for shock heated gas in the temperature pro- file of the North sector (middle panel of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A jump from 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 keV at 4′′ to 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 keV at 3′′ in the North sector points to the influence of the radio jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, this feature MNRAS 000, 000–000 (0000) 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' appears to be exceptional rather than commonplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We find that the temperature profiles are relatively flat with radius, increasing only gradually inward from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 keV at 800 pc to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 keV at 100 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Temperatures at the innermost points, with radial error bars crossing through the Bondi radius, show substantial scatter from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We see that temperature in the South begins to de- crease inward at the radii where M84’s quasi-spherical central halo begins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' gas may be cooling more efficiently in these denser regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, we note that the South sector, especially the innermost point, is certainly affected by cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Note that points obtained in the South sector lack the constraining power of the other sectors due to a clear point source throughout much of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In all but the South sector, there are at least 700 source counts per radial bin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, from 3′′ − 6′′ in the South, the number of counts drops below 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, while the temperature of the innermost point in the South sector may be reliable (although certainly a cavity is present), the decreasing trend in temperature may not be physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, based on the other sectors, one may reasonably conclude that temperature is relatively constant over the inner kpc, with a gradual increase toward Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 Bondi Accretion Rate The adiabatic Bondi accretion rate is given by, �𝑀B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='012 � 𝑇 keV �−3/2 � 𝑛𝑒 cm−3 � � 𝑀BH 109 M⊙ �2 M⊙yr−1, (2) where we have measured 𝑛𝑒 and 𝑇 at the Bondi radius (Rafferty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2006), and we use the most recent measurement of the SMBH mass from Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2010), 𝑀BH = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 × 108𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Central values for �𝑀B are calculated through Equation 2 and the data in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our method for computing errors is described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We choose to use a Monte Carlo method, drawing sam- ples from distributions of 𝑛𝑒, 𝑇, and 𝑀BH and applying Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The 1𝜎 errors on �𝑀B are then the 16th and 84th percentiles of the resulting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Cavities formed by the radio jet and uncertain subtraction of the AGN PSF have a significant impact on measurements of the Bondi accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Averaging together the East and West sectors to yield �𝑀B = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='94) × 10−3 𝑀⊙yr−1 perpendicular to the jet, we find that the accretion rate in the North sector parallel to the jet, �𝑀B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='57+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='01 × 10−3 𝑀⊙yr−1, is discrepant at the level of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Given the dearth of photons in the South sector, the upper limit ob- tained from this sector is likely a vast over-estimate, with the true inferred �𝑀B lying even below that in the North sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These discrep- ancies point not only to the difficulty of measuring �𝑀B, but also the importance of carefully accounting for cavities, which will systemat- ically suppress the measured �𝑀B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Traditional methods which assume spherical symmetry to compute a deprojected 𝑛𝑒 in a full annulus around the AGN rather than in sectors are possibly under-estimating the true Bondi accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We close this section by noting that recent measurements by the Event Horizon Telescope (EHT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Event Horizon Telescope Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019a) may indicate that the gas dynamical measurement of M84’s SMBH mass is an underestimate of the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The EHT employs an emission modeling technique for assessing SMBH masses which, in the case of Sgr A∗ (Event Horizon Telescope Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2022) yields a value completely consistent with stel- lar dynamical measurements (Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Gillessen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, when applied to M87∗, the EHT measurement (Event Hori- zon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019b) is discrepant with previous 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 ne (cm−3) North East West South 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 T (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 1 2 5 10 Radius (arcsec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 Z (Z⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 Radius (kpc) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Deprojected electron number density 𝑛𝑒, gas temperature 𝑇 , and iron-peak element metallicity 𝑍 as a function of radius for all four sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Dashed lines indicate the range of our measured Bondi radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The presence of cavities in the South sector precludes a measurement of 𝑛𝑒 for the innermost point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, because the emissivity profile drops sharply at this radius, we use the data point at 2′′ as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' gas dynamical measurements (Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, the Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2010) measurements of M84’s SMBH and subsequently our measurements of the Bondi radius and Bondi accretion rate may also be underestimates of the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 The Inefficiency of Bondi Accretion Using the central values and distributions of �𝑀B, we can compute the efficiency of Bondi accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We define this efficiency factor through MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 9 𝐿Jet + 𝐿X = 𝜂 �𝑀𝑐2, where our combined fit to the AGN gives an X- ray luminosity 𝐿X = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='14 × 1039 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The jet power 𝐿Jet is obtained by measuring the enthalpy of M84’s cavities assuming they are in pressure equilibrium with their surroundings, and dividing by the characteristic timescale of the bubbles, either the sound crossing time or buoyancy timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using this method, Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2013) found the jet power to be 𝐿Jet = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 × 1042 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For determining the errors in 𝜂, we assume dimidiated Gaussians for 𝐿X and 𝐿Jet, but instead of assuming distributions for �𝑀B, we use the distributions computed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While typically ∼10% of the �𝑀𝑐2 power is released by gravita- tional infall through an accretion disk, Bondi accretion onto M84’s SMBH is far less efficient, with 𝜂 ∼ 10−6 in the East and West sectors unaffected by cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These low efficiencies imply that M84 hosts a radiatively inefficient accretion flow (RIAF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A similar conclusion follows from the Eddington ratios for Bondi accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Using the definition 𝐿B = �𝑀B𝑐2, and the Eddington luminosity for the SMBH (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013), 𝐿Edd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='26 × 1047 � 𝑀BH 109𝑀⊙ � erg/s, (3) we can determine the Eddington ratio for Bondi accretion, 𝐿B/𝐿Edd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In all sectors, M84’s AGN displays Eddington ratios around a few ×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When the Eddington ratio is based on the true accretion rate onto the hole �𝑀, the value is much lower, 𝜂𝐿B/𝐿Edd ∼ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Accretion proceeds not through a thin, radiatively efficient disk (Shakura & Sunyaev 1973), but rather via a hot RIAF (Yuan & Narayan 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 Timescales for Accretion We can better elucidate the structure of the flow by measuring profiles of the relevant timescales for accretion, namely the cooling time 𝑡cool, free-fall time 𝑡ff, and Bondi inflow time 𝑡inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The cooling time is the timescale for a gas with thermal energy density 3 2𝑛𝑒𝑇 to radiate its energy away, 𝑡cool = (3/2)𝑛𝑒𝑇/𝑛2𝑒Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Here, Λ represents the cooling function for the X-ray gas and consists of bremsstrahlung continuum as well as significant line cooling in 𝑇 ≲ 1 keV gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The free-fall time is the dynamical timescale for gas to free fall from a radius 𝑟 under the gravitational influence of M84’s SMBH and dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In line with the Bondi (1952) solution, we assume that M84’s dark matter distribution is described by a spherically- symmetric Hernquist (1990) profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We add an additional point mass potential with mass 𝑀BH to the dark matter potential, which is used to determine the gravitational acceleration as a function of radius, 𝑔(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The free-fall time can then be computed using the simple expression, 𝑡ff = √︁ 2𝑟/𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Traditionally, the ratio 𝑡cool/𝑡ff has been used as a probe of thermal instability in galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' At the small ≲ 1 kpc scales where the Bondi flow originates, the free fall time is far too short to be relevant for the structure of the flow (see Discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather, the dynamical timescale for the accretion flow is the Bondi inflow time, 𝑡inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For steady-state Bondi accreton, �𝑀B = 4𝜋𝑟2𝜌(𝑟)𝑢𝑟 (𝑟) = constant, (4) where we have introduced the mass density 𝜌(𝑟) and the radial inflow velocity 𝑢𝑟 (𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The mass density is related to our mea- sured number density by assuming quasi-neutrality and introducing the mean-molecular weight 𝜇, which we take to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 such that, 𝜌(𝑟) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='15 𝑚 𝑝𝑛𝑒(𝑟), and 𝑚 𝑝 is the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We can approxi- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 tcool (Gyr) North East West South 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 tinflow (Gyr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 1 2 5 10 Radius (arcsec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0 tcool/tinflow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 Radius (kpc) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Profiles of cooling time and inflow time in the inner 10′′ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The cooling time 𝑡cool gradually decreases inward from 200 Myr at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 kpc to 60 Myr near the Bondi radius (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Inflow times show a similar trend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the inflow time is ∼10 Myr at the Bondi radius in all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The ratio of timescales shows a clear transition above 𝑡cool/𝑡inflow = 1 at 2′′ as the flow approaches the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' mate the inflow time as 𝑡inflow = 𝑟 𝑢𝑟 (𝑟) = 4𝜋𝑟3𝜌(𝑟) �𝑀B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (5) Note that this expression is particularly simple as the number density 𝑛𝑒 drops out when substituting in the expression for �𝑀B (Equation 2), 𝑡inflow ≈ 30 � 𝑟 kpc �3 � 𝑇 keV �3/2 � 𝑀BH 109 𝑀⊙ �2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (6) Figure 5 shows profiles of the timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The free-fall timescale is short (𝑡ff < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 Myr) within the inner kpc and does not dictate the gas flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather, cooling is the dominant process from 200−800 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We emphasize that this flow structure is different from a “cooling flow” (Fabian 1994) which involves catastrophic levels of star formation and a deluge of cold gas from the cluster ICM onto the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our observations probe much smaller scales than would be relevant for MNRAS 000, 000–000 (0000) 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Left: Temperature map of M84, where brighter colors indicate higher temperature, measured in keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Right: Metallicity map, where darker colors indicate higher metallicity, measured relative to solar metallicity 𝑍⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The same 2 × 10−3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 mJy VLA radio contours from Figure 1 are overlaid to emphasize the structure of the H-shaped filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' cooling flows, which are suppressed by AGN feedback (Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' At these small scales, gas slowly condenses and begins to flow inward as the atmosphere cools and loses thermal pressure support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As material approaches the Bondi radius, the dominant (shorter) timescale becomes the Bondi inflow time 𝑡inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thermal pressure support loses relevance and the flow begins to experience the in- fluence of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this way, detection of the transition from 𝑡cool/𝑡inflow < 1 to 𝑡cool/𝑡inflow > 1 may indicate that we are probing the beginning of the large-scale accretion flow onto the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 Temperature and Metallicity Maps The temperature and metallicity profiles presented in Figure 4 pro- vide insight into the structure of the beginnings of the accretion flow feeding the AGN in M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this section, we “zoom out” from these small, sub-kpc scales to explore the temperature and metallicity structure of the galactic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 6 shows the temperature and metallicity maps produced us- ing the methods described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The maps are fit using the com- bined VAPEC+APEC model described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 whenever the surface brightness of the region is larger than 10−7 counts/s/cm2/arcsec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Otherwise, we use an APEC model since low surface brightness re- gions are likely dominated by emission from the Virgo ICM rather than from M84’s galactic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This choice does not have a significant effect on the structure of the maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' the use of an APEC+APEC model for all regions yields similar maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The temperature map shows that the galactic gas is remarkably isothermal, with the vast majority of gas occupying a narrow range of temperatures from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 keV, similar to what is seen in the temperature profiles of the inner kpc around the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' There ap- pears to be a large-scale, although weak, temperature gradient, with colder material located West of the radio jet and warmer material sandwiched between the radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Gas is generally hotter within the radio lobes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however this trend may be due to the fact that the cavities are dominated by dim Virgo ICM emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In the Northern radio lobe, we see rich temperature structure, with a colder filament bridging through the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This feature may be a projection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, a cold filament could be wrapping around in front of or behind the bubble in three dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Colder temperatures trace out filaments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the H-shape is far more apparent in the metallicity map in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this map, higher metallicities, with values around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 𝑍⊙ define the H, and metallicity appears relatively symmetric about the radio jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We see evidence for a drop in metallicity approaching the interior of the galaxy, and this trend is clearly apparent in the metallicity profiles in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Radio cavities appear to clear out the X-ray halo of metal-enriched material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, rather than pulling high metallicity material into the wake of the bubbles, the jet appears to simply push metal enriched material aside into the H-shaped filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While jets shape the filaments, feedback is gentle and does not lead to a substantial over-pressurization of the filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We can study this process via the pseudo-pressure map (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pseudo-pressure is calculated by multiplying the gas temperature in each region with the square root of the normalization per unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For bremsstrahlung emission, which has a weak 𝑇1/2 temperature dependence, the nor- malization is proportional to the emission measure, ∫ 𝑛2𝑒𝑑ℓ, where 𝑑ℓ is the differential path length through the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pseudo-pressure increases only by a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 between the Virgo ICM regions located beyond about 55′′ East of the SMBH and the outer halo of M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The filaments, which begin 30′′ along the same direction, show another factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Such an increase can- not be explained by the temperature, which instead decreases inward along the same East-pointing ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The increase is also too steep to be attributable simply to an increase in the path length (and thus emission measure) if we assume M84’s galactic gas is distributed quasi-spherically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, gas density is enhanced in the filaments, likely mediated by cooling in the dense, metal-rich gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='63 kpc 20 arcsec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='63 kpc 20 arcsec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='45Feeding and Feedback at the Bondi Radius of M84 11 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pseudo-pressure map, formed by multiplying the temperature map in Figure 6 with a map of the square root of area-corrected normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The scale shown is logarithmic and spans 2 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Filaments are not substantially over-pressurized relative to the surrounding cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The crossbar of the H can be interpreted as a disk of dense gas around the AGN, viewed in projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Cooling in the dense disk leads to a collapse into a thin, over-pressurized structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this way, rather than being squeezed by the radio lobes, the crossbar may simply be condensing through the cooling and gravitational collapse of a large-scale (35′′ or ∼3 kpc) centrifugally-supported disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Similarly, features of the pressure map which seem to extend into the radio lobes, such as a “fish-tail” like structure visible in the Southeast in Figure 7, can be attributed to filaments wrapping around the radio lobes, viewed in projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If such filaments are uplifted with the bubbles, cooling may be encouraged, leading to the formation of the dense, metal rich structures clinging to the bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4 DISCUSSION The density, temperature, and metallicity profiles in Figure 4 provide a direct comparison to the spherically-symmetric Bondi (1952) so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Similarly, our measurements of the profile index 𝛼 and Bondi accretion rate �𝑀B (Table 2) quantify the degree of asymmetry in the flow for sectors aligned with the jet axis and those anti-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this section, we discuss the origins of the observed deviations from the Bondi solution, namely the influence of the jet on the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We include a brief discussion of multiphase structure and ther- mal instability in M84, presenting the entropy profiles based on our temperature and density measurements, and discuss the lack of an observed temperature rise at the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Finally, we close with an analysis of the “hot blob” of material noted in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 Asymmetry Imposed by the Jet Within the 1𝜎 error bars, all density profiles are just slightly flatter than 𝑛𝑒 ∝ 𝑟−1, which is consistent with findings by HRR15 in M87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This profile however is completely inconsistent with the density pro- file predicted by the Bondi (1952) solution for an adiabatic gas, which instead would predict a density profile of 𝑑 ln 𝜌/𝑑 ln 𝑟 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='373 at 𝑟 = 𝑅B—a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5𝜎 discrepancy from the “All” sectors value in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This discrepancy points to the fact that the flow may be strongly in- fluenced by the galactic gravitational potential rather than the SMBH point mass alone (Quataert & Narayan 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Both jet-aligned sectors show steeper radial profiles compared with the shallow profiles perpendicular to the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the points most affected by the presence of cavities were removed when fitting for 𝛼, the cavities do not account for this steepening in the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, jet-inflated bubbles may entrain dense material from the core of the galaxy in their wakes, buoyantly lifting this gas to larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The result is a dearth of material at the Bondi radius along the jet and density enhancement at larger radii (Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Alternatively, because the highest densities in the North sector (𝑛𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='009 cm−3 at 3′′) are coincident with the “hot blob” of shocked gas, the density enhancement may be due to compression in the shocked region itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 Mechanical Feedback on the Accretion Flow The Bondi accretion rates �𝑀B aligned and anti-aligned with the jet axis are discrepant at a level > 4𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This discrepancy may be at- tributed to small-scale cavities formed by radio jets blasting through and clearing out halo gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, whether or not this discrepancy indicates that jets modify the true accretion rate of material through the Bondi radius remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Certainly, cavities com- plicate measurements of �𝑀B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Spherical symmetry does not apply in the presence of a jet and deprojection is no longer well-posed if surface brightness decreases inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Still, the radio jet may have a negligible impact on the accretion flow itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Relativistic jets with small opening angles can impart substantial energy to the accretion flow via shock heating (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6) which impedes gas cooling and introduces further asymmetry to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, these jets impact a relatively small fraction of the accreting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While a large number of simulations have been able to explore self-regulation of AGN in cluster environments (Cattaneo & Teyssier 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Gaspari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Li & Bryan 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yang & Reynolds 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bourne & Sijacki 2017), these simulations lack the dynamic range to study black hole feeding at scales below 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Recently, Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018) and Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2020) demon- strated a calculation of black hole feeding for the RIAF in Sgr A∗ which evolved the origins of the flow fed by stellar winds down to the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A similar procedure has been undertaken by Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2022) for the AGN in M87, with a heating prescription standing in for jetted AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In all of these works, angular momentum plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thermal instability, turbulence, stellar winds, and cloud-cloud or cloud-filament interactions set the angular momen- tum distribution of accreting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' High angular momentum gas which is unable to shed angular momentum through collisions of turbulent transport (Narayan & Fabian 2011), is flung away as it encounters the centrifugal barrier of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yet, low angular momentum gas has the possibility of settling into the observed accretion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These works predict a suppression of the Bondi accretion rate with the scaling �𝑀 ∼ (𝑟/𝑅B)1/2 �𝑀B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' M84’s 𝑀BH = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 × 108 𝑀⊙ black hole has an innermost stable circular orbit (ISCO) with a radius 𝑅ISCO ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6×1014 cm (assuming no black hole spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For the Bondi radius based on all sectors, 𝑅B(All) = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 pc, the predicted accretion rate at the ISCO using the scaling inferred from simulations is �𝑀 = (𝑅ISCO/𝑅B)1/2 �𝑀B ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5×10−6𝑀⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If the flow liberates MNRAS 000, 000–000 (0000) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='63 kpc 20 arcsec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='00134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='0050612 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 𝜂 ∼ 10% of the �𝑀𝑐2 energy which reaches the hole, the inferred power is 𝐿ISCO ≈ 5 × 1040 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This power is well short of the Gyr-averaged jet power 𝐿Jet = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 × 1042 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, if the 𝑟1/2 scaling obtains in M84, there must be additional sources of accreting gas beyond the hot phase material inferred from X-ray observations alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Understanding the interaction between jets and the accretion flows powering them remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Self-consistently evolving the sub-parsec scales responsible for launching jets with the ∼50 pc scales of the Bondi radius requires resolving gas thermodynamics, inflows, and outflows over 5 orders of magnitude in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We ex- pect that a combination of increased computational power and deep observations of molecular gas, enabled by observatories like the At- acama Large Millimeter Array (ALMA), will serve to better elucidate how jets and bubbles affect the distribution of mass and angular momentum in the gas fueling RIAFs in massive elliptical galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 Cold vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Hot Mode Accretion Large-scale accretion at scales comparable to the Bondi radius can be broadly divided into two classes, similar to those invoked in the galaxy formation community (Kereš et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2005): cold mode and hot mode accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bondi accretion of ∼keV X-ray gas represents the hot mode of accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As we have shown by our measurements of density and temperature at the Bondi radius, Bondi accretion alone is more than sufficient to power the central AGN in M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, if multiphase gas, particularly components much colder than what we are studying in the X-rays, is present, the cold mode of accretion may be equally if not more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In cold mode accretion, thermally unstable (Field 1965) gas cools, condenses, and precipitates out of the hot medium, forming dense structures such as “clouds” (or “blobs”) and filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As long as these cold structures possess a minimal amount of angular momen- tum, or can shed angular momentum via cloud-cloud, cloud-filament, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' collisions, they can chaotically “rain down” onto the central SMBH, providing a gas supply even in excess of that provided by Bondi accretion alone (Pizzolato & Soker 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Gaspari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While this picture of accretion is straightforward in principle, in practice, a number of challenges remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In cluster environments, buoyancy acts to negate thermal instability, at least at the level of linear theory (Defouw 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Nulsen 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bal- bus & Soker 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Idealized nonlinear simulations with heating and cooling globally balanced, as carried out by McCourt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2012) in plane parallel geometry, Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2012) in spherical coordi- nates, and with jetted feedback as in the simulations by Gaspari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2012), argue that the existence of multiphase gas depends sensi- tively on the minimum of the cooling to free-fall time ratio, 𝑡cool/𝑡ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Subsequently, simulations by Li & Bryan (2014) and Meece et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2015), as well as observational efforts by Voit & Donahue (2015) and Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2015) have further solidified the importance of the ratio of these timescales in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2017) adds motivation for the minimum 𝑡cool/𝑡ff ratio in clusters, arguing that the ratio sets a critical slope for the entropy profiles in clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For entropy profiles steeper than this threshold, multiphase gas cannot precipitate from the hot phase since it is subject to buoyant oscillations and thus strong buoyancy damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yet, when the slope is flattened by an injection of high entropy material into the center of the cluster via feedback, thermal instability can proceed and cold mode accretion is once again relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this way, the minimum 𝑡cool/𝑡ff ratio alone is not the only crucial parameter in clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather, this ratio must be compared to the entropy gradient to predict the presence of multiphase gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 1 2 5 10 Radius (arcsec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 1 Tn−2/3 e � keV cm2� North East West South 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8 Radius (kpc) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Entropy profiles of all 4 sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' There is a clear inward decrease in entropy within the inner kpc of M84, which should indicate a dearth of multiphase gas based on the 𝑡cool/𝑡ff ratio at these small radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We note that while there is some observational support for these models, the idea of a critical 𝑡cool/𝑡ff ratio setting the conditions for the formation of multiphase gas is by no means settled physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Buoyantly rising bubbles may stimulate cooling and multiphase gas formation via adiabatic uplift (McNamara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2016), as may be indicated by our temperature, metallicity, and psuedo-pressure maps (Figures 6-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In addition, observational biases may over-emphasize the importance of 𝑡cool/𝑡ff (Hogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pulido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our work is focused on AGN fueling at the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, rather than wade headlong into rather subtle questions of thermal instability in galaxy clusters, we present a simple test for multiphase gas based on the measured entropy profiles in M84’s X-ray halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 8 presents the radial entropy profiles in each of the four sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Following Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2017), we define the dimensionless entropy gradient as 𝛼𝐾 ≡ 𝒓 · ∇ ln (𝐾), where 𝐾 ≡ 𝑇𝑛−2/3 𝑒 is the gas entropy in units of keV cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Equation 22 of Voit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2017) provides a condition on the entropy gradient for nonlinear condensation, 𝛼𝐾 < 𝛼𝐾 ,crit ≡ 3(2 − 𝜆)2 40 � 𝑡ff 𝑡cool �2 , (7) where 𝜆 ≡ 𝑑 ln Λ/𝑑 ln𝑇 parameterizes the cooling function Λ, and for the relevant cooling mechanisms in clusters lies in the range −1 ≲ 𝜆 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, entropy profiles steeper than the critical value 𝛼𝐾 ,crit should result in gradually damped buoyant oscillations of cooling gas, rather than the condensation necessary to fuel cold mode accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because of the short free fall times so close to the Bondi radius (𝑡ff ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 Myr) and comparatively long cooling times (𝑡cool ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 5) at these small scales, the ratio 𝑡ff/𝑡cool ∼ 2 × 10−3 implies that the critical entropy gradient is essentially flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 8 indicates that in all sectors, the entropy gradient is far too steep to admit condensation and the formation of multiphase gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If multiphase gas is in fact present, this material must have been sent toward the Bondi radius from much larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While we searched for multi-temperature gas in the central kpc as an indication of gas cooling out of the ionized phase of the X-ray emit- ting plasma, we were unable to find evidence of a second temperature component in the X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Adding in a second VAPEC component provided no constraint on a second temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The likely reason is that Chandra can only distinguish temperatures separated by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 13 keV in energy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because M84’s gas is cold (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 keV), a colder component would appear at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 keV, below the detector’s sensitive energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' A hotter component may be detectable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' how- ever, if the AGN was over-subtracted rather than under-subtracted, this potentially weak signal may be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, consistent with the conclusion from the entropy profiles, we find no evidence for the presence of multiphase X-ray emitting gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We note that the strict criterion presented in Equation 7 may not be applicable to Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The central assumption under- pinning the importance of 𝑡ff/𝑡cool in cluster environments is that heating and cooling is globally balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While such a balance may apply globally within the Virgo Cluster, locally, at the small sub-kpc scales probed in our analysis, heating cannot offset cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' AGN jet energy is thermalized on 10s of kpc length scales comparable to or larger than the bubbles, via weak shocks and sound waves (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Sanders & Fabian 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic & Reynolds 2019), turbulence driven by g-modes (Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2002, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Zhuravleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018), mixing of high entropy bubble material with cluster gas (Hillel & Soker 2016, 2018), cosmic rays (Guo & Oh 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Pfrommer 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Ruszkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Ehlert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Kempski & Quataert 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Near the Bondi radius of M84 where the cooling time is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 Gyr, these heating mechanisms operate inefficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For this reason, the traditional comparison of the free-fall and cooling timescales should give way to a comparison of the cooling and Bondi inflow timescales (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Figure 5 indicates a transition from a “cooling-dominated” flow to an “inflow-dominated” flow at scales comparable to 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this way, we see that the accretion flow around the Bondi radius should not be regarded as a static equilibrium defined by the interplay of AGN heating and radiative cooling, but a dynamic inflow of material under the influence of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 Metallicity Structure When the iron-peak elementmetallicity isfree tovary inthe fit, we see a clear metallicity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Metallicity decreases inwards in all sec- tors, with the exception being the cavity-affected point in the South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This gradient (referred to in the literature as a central abundance drop) is common in cluster environments and has been observed in more than 8 objects (Panagoulia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Lakhchaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Though dust from old stars should be increas- ing the central metallicity, if this dust is locked into filaments, it can be lifted buoyantly to larger radii by jet-inflated bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, the same processes which shape the density gradients along the jet axis may be responsible for the metallicity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While buoyancy may explain some of the metallicity gradient along the jet axis, the challenge remains to explain the central abun- dance drops in sectors perpendicular to the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Without a mod- ulation of the SNe Ia rate with radius in the galaxy, this gradient is difficult to account for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather than arising from a physical process, the drop in metallicity may point to an unresolved second temperature component and thus, gas cooling en route to a multiphase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While we found no evidence for such multi-temperature structure (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3), future, deeper observations free of the constraints on soft en- ergy response which afflict Chandra are necessary to tease out the existence of this cooler material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We close by noting that absorption, specifically “intrinsic” absorp- tion due to interlaced cold and hot phase gas may be obstructing our view of the gas cooling which is responsible for fueling M84’s AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Such a “hidden” cooling flow (Fabian 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2022) may be present within M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Indeed, there is evidence from XMM-Newton observations that an intrinsic absorption model may describe M84’s galactic gas (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2022 in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, high spectral res- olution, far beyond what can be achieved by Chandra, is required to tease out the parameters of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, XMM-Newton obser- vations, which probe much larger scales than Chandra (comparable with the extent of M84’s H-shaped filaments) are unable to constrain an accretion rate for a “hidden” cooling flow at Bondi radius scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 No Temperature Rise at the Bondi Radius We find no evidence for a temperature rise approaching the Bondi radius, a phenomenon that has been proposed as evidence for the transition from the galactic potential to that of the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This conclusion may be a consequence of the changing metallicity, which decreases by nearly a factor of 4 over the inner few hundred pc in all but the South sector (although the cavity and limited numbers of counts may be playing a role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When we fix the metallicity to the radially averaged value (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 𝑍⊙), we see signs of a temperature jump, with the inner- most points in the East and West sectors reaching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2 keV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While temperature does increase with fixed iron-peak element metallicity, so also does the reduced C-statistic, indicating that a temperature rise may not truly be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Instead, there may have been an under-subtraction of the AGN which provides an ex- cess of hard photons, enough to over-estimate the temperature when metallicity is not a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The lack of an observed temperature rise may not be surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Observations of the temperature profiles in M87 by HRR15 find a similar absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In some respects this is to be expected: the analytical Bondi solution at radii comparable to 𝑅B shows a relatively flat tem- perature profile, with the majority of the adiabatic heating occurring at small scales, well below the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, the notable absence of the temperature rise may be a result of biases inherent to observations of multiphase gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Indeed, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2022) have recently performed simulations of Bondi accretion from tens of kpc scales down to accretion flow scales well below the Bondi radius which include gas at a wide range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' They find that their simulations tend to predict a flat emission-weighted temperature profile in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 − 7 keV band, even at scales an order of magnitude below 𝑅B, where adiabatic heating of hot phase gas becomes significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Our X-ray observations may be biased by the energy band accessible to Chandra, and even future missions which probe scales below 𝑅B may similarly never detect a temperature rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 Shock Heating by the Jet or Nonthermal Emission?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While there is no evidence for a temperature rise at the Bondi radius, we do see a clear (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8𝜎) temperature increase from 4′′ to 3′′ in the North sector which we refer to as a “hot blob.” This temperature increase at 2′′ − 3′′ from the AGN is at the same angular separation from the AGN as a knot of radio emission detected by VLA in the 5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 GHz bands, and ALMA in the 97 and 236 GHz bands (see Knot B in Figures 1-3 of Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This “blob” or “knot” of X-ray emission, first detected by Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2002), stands out clearly in X-rays, even with the limited exposure time (≈29 ks) of Chandra’s first observation of M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 − 7 keV X-ray spectrum of the region containing the “hot blob” is well described (reduced C-statistic of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='06 for the North Sector at 2′′ and 3′′ respectively) by a VAPEC model in our analysis, indicating thermal X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, the radio emission MNRAS 000, 000–000 (0000) 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bambic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' is far more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Early results favored a synchrotron origin for the emission (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018) argue that the radio and X-ray spectra of the knots cannot both be explained by standard models for jet emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In their analysis, the X-ray emission is modeled as both a power law representing the jet and an APEC component representing the thermal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Motivated by these works, we re-fit the spectra from the North sector at 2′′ and 3′′ using our M84 model (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='8), with the addition of a red-shifted power law (zpowerlw) component meant to represent the X-ray jet detected by Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When fitting the 2′′ and 3′′ North spectra with the extra zpowerlw “jet” component, the resulting temperature and metallicity of the VAPEC component are unconstrained (in the case of the 3′′ point, temperature is constrained but metallicity is not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We thus proceeded to leave the temperature and zpowerlw normalization free in the fit, but fix the VAPEC metal- licity to three different values of metallicity: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 (consistent with our measurements), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='2, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 solar (consistent with that used by Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For each metallicity, we scan zpowerlw photon indices from Γ = 1 − 3, which we fix in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This procedure yields improved C-statistics over the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='08 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='06 found initially, in some cases comparable to or better than the fit to the full annulus spectrum including all sectors at 2′′ and 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rea- sonable photon indices near Γ ≈ 2 yield good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As expected, the corresponding VAPEC temperature is lower when the jet component is included;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, rather intriguingly, the fit temperature is lower than all other sectors save that in the West at the same radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The temperature never exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 keV for all photon indices and metal- licities tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For the metallicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1 solar consistent with what was found in our profiles (Figure 4), the fit finds VAPEC temperatures below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 keV for the point at 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When using the metallicitity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='3 solar assumed in Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2018), the fits settle around 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='65 keV, which is just below the temperature at a radius of 2′′ in the East Sector (𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='03 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Note that all of these temperatures are well below the 3 keV thermal model for the X-ray emission proposed as an alternative to synchrotron emission in Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The takeaway message from this analysis is clear: a model with nonthermal emission from an X-ray jet and colder (𝑇 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='65 keV) galactic gas describes the “hot blob” as well as a purely thermal emission model of 𝑇 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 keV gas shock heated by the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because of Chandra’s limited spectral resolution, we cannot discriminate between these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' However, given the complexity of M84’s jet emission in the radio band and the co-spatial temperature increase observed in our X-ray observations, a deeper study of M84’s jet which can harness the full power of our 840 ks data set is merited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 Comparison to Other Measurements Bondi accretion is incredibly inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The Bondi accretion rate �𝑀B is measured to be a few × 10−3𝑀⊙yr−1 in each of the sectors in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, the accretion flow in M84 need only liberate 𝜂 ∼ 10−6 of the �𝑀B𝑐2 fuel provided by Bondi accretion to power the galaxy’s relativistic jets and X-ray AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We are not the first group to arrive at this conclusion in M84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The first measurement of the Bondi accretion rate can be at- tributed to Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2006), who found an accretion rate of �𝑀B = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1×10−3𝑀⊙yr−1 by measuring the temperature and den- sity of the full annulus around the AGN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' including all sectors at the innermost “Bondi radius” point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While this work was in preparation, another measurement of �𝑀B was performed by Plšek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (2022) which leveraged the data from our new campaign, presented here and publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' They found �𝑀B = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 × 10−3𝑀⊙yr−1, again using all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If we compare these values with our “All” sector measurement of �𝑀B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='89 ×10−3𝑀⊙yr−1, then there is strong agreement among all published values of the Bondi accretion rate in M84, with notably tighter error bars in the more recent values enabled by an extra ≳ 750 ks provided by the new campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 5 CONCLUSION We have presented the deepest Chandra X-ray observations to date of M84, a jetted elliptical galaxy in the Virgo Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' These obser- vations, which comprise over 840 ks of Chandra data, have enabled a detailed study of the temperature, density, and metallicity structure of the galaxy, from kiloparsec scales to ≈ 50 pc scales just inside the Bondi radius of the galaxy’s SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' New images of M84 have been presented, emphasizing the intricate structure of the soft X-ray filaments, formed into an H morphology by the action of powerful (𝐿Jet = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 × 1042 erg/s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2013) radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Density and temperature measurements obtained through spectra extracted from the innermost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5′′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5′′ bin allowed us to compute Bondi accretion rates for each of 4 sectors around the central AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' “All” sectors are fit together to allow comparison to previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The main conclusions of our analysis are as follows: (i) Radial profiles of deprojected electron number density 𝑛𝑒 are consistent with 𝑛𝑒 ∝ 𝑟−1, but slightly flatter (Figure 4 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This profile is in tension at the level of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5𝜎 with the expectation of Bondi accretion, which predicts 𝑑 ln 𝜌/𝑑 ln 𝑟 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='373 at 𝑟 = 𝑅𝐵 and an 𝑟−3/2 scaling at 𝑟 ≪ 𝑅𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (ii) The radial profile indices 𝛼 are statistically consistent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' how- ever, we see that the profiles are steeper along the jet axis than per- pendicular to the jet (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This violation of spherical symmetry is counter to the assumptions of the Bondi solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (iii) There is a discrepancy in the Bondi accretion rate depending upon which sector is used to measure �𝑀𝐵 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This discrepancy between jet-aligned and mis-aligned sectors is at the level of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6𝜎, which is statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While the discrepancy may point to the influence of the jet on the large-scale accretion flow, the disparity likely arises due to the presence of cavities (Figures 3 and 4) or uncertainties in modeling the AGN emission at Bondi radius scales (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (iv) Temperatures do not vary widely throughout the galaxy (Fig- ure 6) and only increase gradually from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 keV to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='7 keV over the inner kpc approaching the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The exception is that in the North sector, we see evidence for a temperature increase at points 2′′ − 3′′ from the AGN (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We refer to this feature as a “hot blob” of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The physical origin of this “hot blob” remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Shock heating by the radio jet or nonthermal emission from knots in an unresolved X-ray jet (Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' 2018) are both plausible explanations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, Chandra lacks the spectral resolution to rule out either of these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (v) We detect no temperature rise at the Bondi radius, consistent with findings by HRR15 in M87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' (vi) By comparing the Bondi inflow time 𝑡inflow to the cooling time as a function of radius, we observe evidence for a transition from a “cooling-dominated” flow to an “inflow-dominated” flow at scales of 1′′ − 2′′, providing support to the conclusion that we have resolved M84’s Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The data presented in this work was obtained through Chandra Pro- posal #19800344: “Fueling and self-regulation of AGN feedback at MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 15 the Bondi radius of M84” and is publically available on the Chandra archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We are grateful to the Chandra X-ray Center for support, not only with the initial observations and pointing of the telescope, but also with use of the ChaRT tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This manuscript was improved thanks to the careful reading and suggestions of an anonymous ref- eree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' CJB thanks Andy Goulding and Jeremy Sanders for technical support throughout this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This work benefited from stimulating discussions with Eliot Quataert and Minghao Guo at Princeton and Eileen Meyer at the Texas Symposium on Relativistic Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' CJB thanks the graduate students of the Institute of Astronomy, Cam- bridge and Princeton University for their many insights as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This work is possible through the financial support of the Churchill Foun- dation of the United States and CJB continues to be supported by a National Science Foundation (NSF) Graduate Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Early stages of this work were performed at the Multiscale Phe- nomena in Plasma Astrophysics program at KITP in Santa Barbara, CA, research supported in part by the NSF under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' NSF PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' HRR acknowledges support from an STFC Ernest Rutherford Fellowship and an Anne McLaren Fellowship provided by the University of Nottingham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' CSR thanks the STFC for sup- port under the Consolidated Grant ST/S000623/1, as well as the European Research Council (ERC) for support under the European Union’s Horizon 2020 research and innovation programme (grant 834203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' DATA AVAILABILITY The Chandra data described in this work are available in the Chandra data archive (https://cxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='harvard.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=', 2014, Nature, 515, 85 APPENDIX A: MODELING AGN CONTAMINATION ChaRT and MARX simulations of the AGN require an input energy spectrum, which was obtained by fitting the spectrum extracted from a 1′′ circle about the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' While this spectrum is dominated by the AGN “source,” it includes “background” contributions from M84’s galactic gas, the Virgo screen, and unresolved XRB/AB/CV stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The Virgo and unresolved point source backgrounds were determined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='4 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='6 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' however, the treatment of the underlying galactic gas emission required more care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' To set the background galactic gas component at 1′′, we fit the spectrum of an annulus from 2′′−4′′ circumscribing the AGN where AGN emission is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This procedure provided a normalization for the VAPEC background component at 3′′ from the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Then, we fit the surface brightness (SB) distribution with the assumption of spherical symmetry using a simple power law in radius, assuming that the power law extrapolation from 3′′ to 1′′ accurately described the SB at 1′′ from the central AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The ratio of the SB determined at 1′′ and 3′′ from this power law was taken as a “boost” factor multiplied on to the previously fit-for VAPEC normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In the case of M84, we found this boost factor to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Then, the parameters of the input energy spectrum passed to ChaRT were determined by fitting the full spectrum (source + background) extracted from the 1′′ circle with a fixed background VAPEC compo- nent and only the zpowerlw parameters (meant to model the AGN source) left free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The normalization of this fixed VAPEC compo- nent is simply the normalization of the 2′′ − 4′′ annulus, multi- plied by the boost factor (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='98) and corrected by the ratio of the annulus to 1′′ circle areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Once the parameters for the zpowerlw source component were determined, a clean spectrum including only phabs(zphabs(zpowerlw)) parameters was produced and passed to the ChaRT tool as the input spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This source spectrum is rep- resentative of the AGN without contributions from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) Feeding and Feedback at the Bondi Radius of M84 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 1 2 5 10 20 30 40 Radius (arcsec) 10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 Surface Brightness (cts s 1 cm 2arcsec 2) Hard Band Simulation (4-7 keV) Hard Band Data (4-7 keV) Broad Band Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='5 1 2 5 10 20 30 40 Radius (arcsec) 10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 Surface Brightness (cts s 1 cm 2arcsec 2) 5% Boost in Simulation 5% Drop in Simulation Simulation Subtracted Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Top: Hard band (4 − 7 keV) surface brightness profiles extracted from merged event file and ChaRT+MARX AGN simulation, with 7% boost applied to simulated profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Bottom: Subtraction of hard band simulation profile from hard band data profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The hard band, where only AGN emis- sion and background from the Virgo screen and unresolved point sources remains, is more than an order of magnitude subdominant to the total SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' By applying a 7% boost to the simulation, we achieve a flattening of the AGN-subtracted profile within the inner 2′′, indicating that the AGN has been properly subtracted from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' All that remains at PSF scales are negligible contributions from the spatially uniform Virgo screen and unre- solved point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The point source emission appears to be subdominant given the smoothness of the AGN-subtracted hard band SB from 1′′ − 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because the normalization of the input spectrum was determined by assuming a model for the galactic gas spectrum, the flux of the simulation may not be an accurate representation of the true AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' To test the accuracy of the AGN modeling, we choose an energy band where the AGN completely dominates and which is free of the small-scale variations (on scales comparable to the PSF) imposed by bright, lumpy, soft emission from galactic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' In this case, following HRR15, we choose the hard 4 − 7 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' At these energies, the only contributions to the hard band SB should be from the AGN source, Virgo ICM background (which should be spatially uniform), and unresolved point source background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If point source emission is substantial, the hard band SB profile should display discontinuities and rapid spatial variations on scales of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The top panel of Figure 9 shows a comparison of the hard band profiles for the data (black) and simulation (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' By forward mod- eling the AGN, we are working to subtract the AGN contribution at Bondi radius scales from the spectra extracted in each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' The blue points in the lower panel of Figure 9 show a subtraction of the simulation’s hard band SB profile from that of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' If we were to overestimate the AGN flux and thus over-subtract the AGN from the data, we should expect a drop in hard band SB at 1′′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' the scales of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Alternatively, if we under-subtract the AGN, we should expect a hard band excess in the bottom panel of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We find that our AGN simulation based on modeling the galactic gas background leaves a 7% excess in the hard (4 − 7 keV) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Without compensating for this excess, we would under-subtract the AGN and possibly bias our temperature measurements with excess hard AGN photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Thus, we boost the overall AGN simulation nor- malization by 7%, which means that the difference profile in the bottom panel of Figure 9 flattens at the scales of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' When computing errors on temperature, we do not simply report the statis- tical uncertainties determined by XSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Rather, we boost the AGN normalization by 5% (on top of the 7% compensation) and decrease the normalization by 5% (shown as the red and black points in the bottom panel of Figure 9 respectively), to marginalize over uncer- tainties in the AGN modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' As a result, errors in temperature and metallicity are larger at the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Finally, because the hard band SB is relatively continuous at PSF scales, we conclude that unresolved point sources are properly accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' APPENDIX B: ASCERTAINING ERRORS ON �𝑀B AND 𝜂 For computing errors on �𝑀B, we use a Monte Carlo method, draw- ing 107 samples from distributions of 𝑛𝑒, 𝑇, and 𝑀BH and applying Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' Because of asymmetric error bars in 𝑇 and 𝑀BH, we model the distributions of these variables as “dimidiated Gaussians” (Barlow 2003), two Gaussians centered on the same mean with dif- ferent standard deviations above and below the mean based on the 1𝜎 upper and lower error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' For equal positive and negative error bars, the dimidiated Gaussian is equivalent to a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We model the underlying distribution of 𝑛𝑒 as a log-normal with mean and standard deviation based on the central value and 1𝜎 error bar respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' This choice ensures strict positivity of 𝑛𝑒 but only has significance for the point in the North—a Gaussian yields a similar error bar for all other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' We take the 1𝜎 errors on �𝑀B to be the 16th and 84th percentile of the resulting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFKT4oBgHgl3EQf5S7u/content/2301.11937v1.pdf'} diff --git a/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/2301.00250v1.pdf.txt b/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/2301.00250v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1386ae6af9847e3e06482d5055452a9682443375 --- /dev/null +++ b/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/2301.00250v1.pdf.txt @@ -0,0 +1,1228 @@ +DensePose From WiFi +Jiaqi Geng +jiaqigen@andrew.cmu.edu +Carnegie Mellon University +Pittsburgh, PA, USA +Dong Huang +donghuang@cmu.edu +Carnegie Mellon University +Pittsburgh, PA, USA +Fernando De la Torre +ftorre@cs.cmu.edu +Carnegie Mellon University +Pittsburgh, PA, USA +ABSTRACT +Advances in computer vision and machine learning techniques have +led to significant development in 2D and 3D human pose estimation +from RGB cameras, LiDAR, and radars. However, human pose esti- +mation from images is adversely affected by occlusion and lighting, +which are common in many scenarios of interest. Radar and LiDAR +technologies, on the other hand, need specialized hardware that is +expensive and power-intensive. Furthermore, placing these sensors +in non-public areas raises significant privacy concerns. +To address these limitations, recent research has explored the use +of WiFi antennas (1D sensors) for body segmentation and key-point +body detection. This paper further expands on the use of the WiFi +signal in combination with deep learning architectures, commonly +used in computer vision, to estimate dense human pose correspon- +dence. We developed a deep neural network that maps the phase +and amplitude of WiFi signals to UV coordinates within 24 human +regions. The results of the study reveal that our model can estimate +the dense pose of multiple subjects, with comparable performance +to image-based approaches, by utilizing WiFi signals as the only +input. This paves the way for low-cost, broadly accessible, and +privacy-preserving algorithms for human sensing. +CCS CONCEPTS +• Computing methodologies → Neural networks; Artificial +intelligence; Machine Learning; • Hardware → Communica- +tion hardware, interfaces and storage; Robustness. +KEYWORDS +Pose Estimation, Dense Body Pose Estimation, WiFi Signals, Key- +point Estimation, Body Segmentation, Object Detection, UV Coor- +dinates, Phase and Amplitude, Phase Sanitization, Channel State +Information, Domain Translation, Deep Neural Network, Mask- +RCNN +1 +INTRODUCTION +Much progress has been made in human pose estimation using +2D [7, 8, 12, 22, 28, 33] and 3D [17, 32] sensors in the last few +years (e.g., RGB sensors, LiDARs, radars), fueled by applications +in autonomous driving and augmented reality. These traditional +sensors, however, are constrained by both technical and practical +considerations. LiDAR and radar sensors are frequently seen as +being out of reach for the average household or small business due +to their high cost. For example, the medium price of one of the +most common COTS LiDAR, Intel L515, is around 700 dollars, and +the prices for ordinary radar detectors range from 200 dollars to +600 dollars. In addition, these sensors are too power-consuming for +daily and household use. As for RGB cameras, narrow field of view +and poor lighting conditions, such as glare and darkness, can have +a severe impact on camera-based approaches. Occlusion is another +obstacle that prevents the camera-based model from generating +reasonable pose predictions in images. This is especially worrisome +for indoors scenarios, where furniture typically occludes people. +More importantly, privacy concerns prevent the use of these +technologies in non-public places. For instance, most people are +uncomfortable with having cameras recording them in their homes, +and in certain areas (such as the bathroom) it will not be feasible +to install them. These issues are particularly critical in healthcare +applications, that are increasingly shifting from clinics to homes, +where people are being monitored with the help of cameras and +other sensors. It is important to resolve the aforementioned prob- +lems in order to better assist the aging population, which is the +most susceptible (especially during COVID) and has a growing +demand to keep them living independently at home. +We believe that WiFi signals can serve as a ubiquitous substitute +for RGB images for human sensing in certain instances. Illumination +and occlusion have little effect on WiFi-based solutions used for +interior monitoring. In addition, they protect individuals’ privacy +and the required equipment can be bought at a reasonable price. In +fact, most households in developed countries already have WiFi at +home, and this technology may be scaled to monitor the well-being +of elder people or just identify suspicious behaviors at home. +The issue we are trying to solve is depicted in Fig. 1 (first row). +Given three WiFi transmitters and three aligned receivers, can +we detect and recover dense human pose correspondence in clut- +tered scenarios with multiple people (Fig. 1 fourth row). It should +be noted that many WiFi routers, such as TP-Link AC1750, come +with 3 antennas, so our method only requires 2 of these routers. +Each of these router is around 30 dollars, which means our entire +setup is still way cheaper than LiDAR and radar systems. Many +factors make this a difficult task to solve. First of all, WiFi-based +perception[11, 30] is based on the Channel-state-information (CSI) +that represents the ratio between the transmitted signal wave and +the received signal wave. The CSIs are complex decimal sequences +that do not have spatial correspondence to spatial locations, such +as the image pixels. Secondly, classic techniques rely on accurate +measurement of time-of-fly and angle-of-arrival of the signal be- +tween the transmitter and receiver [13, 26]. These techniques only +locate the object’s center; moreover, the localization accuracy is +only around 0.5 meters due to the random phase shift allowed by +the IEEE 802.11n/ac WiFi communication standard and potential +interference with electronic devices under similar frequency range +such as microwave oven and cellphones. +To address these issues, we derive inspiration from recent pro- +posed deep learning architectures in computer vision, and propose +a neural network architecture that can perform dense pose estima- +tion from WiFi. Fig 1 (bottom row) illustrates how our algorithm +is able to estimate dense pose using only WiFi signal in scenarios +with occlusion and multiple people. +arXiv:2301.00250v1 [cs.CV] 31 Dec 2022 + +Figure 1: The first row illustrates the hardware setup. The second and third rows are the clips of amplitude and phase of the +input WiFi signal. The fourth row contains the dense pose estimation of our algorithm from only the WiFi signal. +2 +RELATED WORK +This section briefly describes existing work on dense estimation +from images and human sensing from WiFi. +Our research aims to conduct dense pose estimation via WiFi. In +computer vision, the subject of dense pose estimation from pictures +and video has received a lot of attention [6, 8, 18, 40]. This task +consists of finding the dense correspondence between image pixels +and the dense vertices indexes of a 3D human body model. The +pioneering work of Güler et al. [8] mapped human images to dense +correspondences of a human mesh model using deep networks. +DensePose is based on instance segmentation architectures such as +Mark-RCNN [9], and predicts body-wise UV maps for each pixel, +where UV maps are flattened representations of 3d geometry, with +coordinate points usually corresponding to the vertices of a 3d +dimensional object. In this work, we borrow the same architecture +as DensePose [8]; however, our input will not be an image or video, +but we use 1D WiFi signals to recover the dense correspondence. +Recently, there have been many extensions of DensePose pro- +posed, especially in 3D human reconstruction with dense body parts +[3, 35, 37, 38]. Shapovalov et al.’s [24] work focused on lifting dense +pose surface maps to 3D human models without 3D supervision. +Their network demonstrates that the dense correspondence alone +(without using full 2D RGB images) contains sufficient information +to generate posed 3D human body. Compared to previous works on +reconstructing 3D humans with sparse 2D keypoints, DensePose +annotations are much denser and provide information about the +3D surface instead of 2D body joints. +While there is a extensive literature on detection [19, 20], track- +ing [4, 34], and dense pose estimation [8, 18] from images and +videos, human pose estimation from WiFi or radar is a relatively +unexplored problem. At this point, it is important to differentiate +the current work on radar-based systems and WiFi. The work of +Adib et.al. [2] proposed a Frequency Modulated Continuous Wave +(FMCW) radar system (broad bandwidth from 5.56GHz to 7.25GHz) +for indoor human localization. A limitation of this system is the +specialized hardware for synchronizing the transmission, refrac- +tion, and reflection to compute the Time-of-Flight (ToF). The system +reached a resolution of 8.8 cm on body localization. In the following +work [1], they improved the system by focusing on a moving per- +son and generated a rough single-person outline with depth maps. +Recently, they applied deep learning approaches to do fine-grained +human pose estimation using a similar system, named RF-Pose [39]. +These systems do not work under the IEEE 802.11n/ac WiFi com- +munication standard (40MHz bandwidth centered at 2.4GHz). They +rely on additional high-frequency and high-bandwidth electromag- +netic fields, which need specialized technology not available to the +general public. Recently, significant improvements have been made +to radar-based human sensing systems. mmMesh [36] generates + +3ReceiverAntennas +6DBiRP-SMA +3TransmitterAntennas +CameraforAnnotation +2.4GHz +Copperdipoleantenna +6.8 +inches +1.5 +inches +20.0 +20.0 +18 +17.5 +17.5 +17.5 +16 +15.0 +15.0 +15.0 +14 + 12.5 +12.5 + 12 + 10.0 +10.0 +10 +7.5 +5.0 +05 +5.0 +2.5 +2.5 +0.0 +%0 +Subcarrier Index +10. +20 +25 +10. +20 +25 +Subcarrier Index +20 +rrier Index + Subcarrier Index +Subcarrier Index3D human mesh from commercially portable millimeter-wave de- +vices. This system can accurately localize the vertices on the human +mesh with an average error of 2.47 cm. However, mmMesh does +not work well with occlusions since high-frequency radio waves +cannot penetrate objects. +Unlike the above radar systems, the WiFi-based solution [11, 30] +used off-the-shelf WiFi adapters and 3dB omnidirectional antennas. +The signal propagate as the IEEE 802.11n/ac WiFi data packages +transmitting between antennas, which does not introduce addi- +tional interference. However, WiFi-based person localization using +the traditional time-of-flight (ToF) method is limited by its wave- +length and signal-to-noise ratio. Most existing approaches only +conduct center mass localization [5, 27] and single-person action +classification [25, 29]. Recently, Fei Wang et.al. [31] demonstrated +that it is possible to detect 17 2D body joints and perform 2D se- +mantic body segmentation mask using only WiFi signals. In this +work, we go beyond [31] by estimating dense body pose, with +much more accuracy than the 0.5m that the WiFi signal can pro- +vide theoretically. Our dense posture outputs push above WiFi’s +signal constraint in body localization, paving the road for complete +dense 2D and possibly 3D human body perception through WiFi. +To achieve this, instead of directly training a randomly initialized +WiFi-based model, we explored rich supervision information to +improve both the performance and training efficiency, such as uti- +lizing the CSI phase, adding keypoint detection branch, and transfer +learning from the image-based model. +3 +METHODS +Our approach produces UV coordinates of the human body sur- +face from WiFi signals using three components: first, the raw CSI +signals are cleaned by amplitude and phase sanitization. Then, a +two-branch encoder-decoder network performs domain translation +from sanitized CSI samples to 2D feature maps that resemble im- +ages. The 2D features are then fed to a modified DensePose-RCNN +architecture [8] to estimate the UV map, a representation of the +dense correspondence between 2D and 3D humans. To improve the +training of our WiFi-input network, we conduct transfer learning, +where we minimize the differences between the multi-level fea- +ture maps produced by images and those produced by WiFi signals +before training our main network. +The raw CSI data are sampled in 100Hz as complex values over +30 subcarrier frequencies (linearly spaced within 2.4GHz±20MHz) +transmitting among 3 emitter antennas and 3 reception antennas +(see Figure 2). Each CSI sample contains a 3 × 3 real integer matrix +and a 3 × 3 imaginary integer matrix. The inputs of our network +contained 5 consecutive CSI samples under 30 frequencies, which +are organized in a 150×3×3 amplitude tensor and a 150×3×3 phase +tensor respectively. Our network outputs include a 17 × 56 × 56 +tensor of keypoint heatmaps (one 56 × 56 map for each of the 17 +kepoints) and a 25 × 112 × 112 tensor of UV maps (one 112 × 112 +map for each of the 24 body parts with one additional map for +background). +3.1 +Phase Sanitization +The raw CSI samples are noisy with random phase drift and flip (see +Figure 3(b)). Most WiFi-based solutions disregard the phase of CSI +(a) Layout of WiFi devices +and human bodies +(b) The 3 × 3 dimensions +of the CSI tensor +Figure 2: CSI samples from Wifi. (a) the layout of WiFi de- +vices and human bodies, and (b) the 3 × 3 tensor dimen- +sion corresponds to the 3 × 3 transmitter-receiver antenna +pairs. For instance, 𝐸1 denotes the first emitter and 𝑅1 de- +notes the first receiver, etc. By incorporating the 5 consecu- +tive complex-valued CSI samples (100 samples/second) un- +der 30 subcarrier frequencies, the two input tensors to our +network are a 150 × 3 × 3 amplitude tensor and a 150 × 3 × 3 +phase tensor. +signals and rely only on their amplitude (see Figure 3 (a)). As shown +in our experimental validation, discarding the phase information +have a negative impact on the performance of our model. In this +section, we perform sanitization to obtain stable phase values to +enable full use of the CSI information. +In raw CSI samples (5 consecutive samples visualized in Fig- +ure 3(a-b)), the amplitude (𝐴) and phase (Φ) of each complex ele- +ment 𝑧 = 𝑎+𝑏𝑖 are computed using the formulation 𝐴 = +√︁ +(𝑎2 + 𝑏2) +and Φ = 𝑎𝑟𝑐𝑡𝑎𝑛(𝑏/𝑎). Note that the range of the arctan function is +from −𝜋 to 𝜋 and the phase values outside this range get wrapped, +leading to a discontinuity in phase values. Our first sanitization +step is to unwrap the phase following [10]: +Δ𝜙𝑖,𝑗 = Φ𝑖,𝑗+1 − Φ𝑖,𝑗 +if Δ𝜙𝑖,𝑗 > 𝜋, Φ𝑖,𝑗+1 = Φ𝑖,𝑗 + Δ𝜙𝑖,𝑗 − 2𝜋 +if Δ𝜙𝑖,𝑗 < −𝜋, Φ𝑖,𝑗+1 = Φ𝑖,𝑗 + Δ𝜙𝑖,𝑗 + 2𝜋, +(1) +where 𝑖 denotes the index of the measurements in the five consecu- +tive samples, and 𝑗 denotes the index of the subcarriers(frequencies). +Following unwrapping, each of the flipping phase curves in Fig- +ure 3(b) are restored to continuous curves in Figure 3(c). +Observe that among the 5 phase curves captured in 5 consecutive +samples in Figure 3(c), there are random jiterings that break the +temporal order among the samples. To keep the temporal order of +signals, previous work [23] mentioned linear fitting as a popular +approach. However, directly applying linear fitting to Figure 3(c) +further amplified the jitering instead of fixing it (see the failed +results in Figure 3(d)). +From Figure 3(c), we use median and uniform filters to eliminate +outliers in both the time and frequency domain which leads to +Figure 3(e). Finally, we obtain the fully sanitized phase values by +applying the linear fitting method following the equations below: +𝛼1 = Φ𝐹 − Φ1 +2𝜋𝐹 +𝛼0 = 1 +𝐹 +∑︁ +1≤𝑓 ≤𝐹 +𝜙𝑓 +ˆ𝜙𝑓 = 𝜙𝑓 − (𝛼1𝑓 + 𝑎0), +(2) + +R1 +R2 +R3 +E1 +E2 +E3(a) Original CSI Amplitude +(b) Original CSI Phase +(c) Phase after unwrapping +(d) Phase after unwrapping + linear fitting +(e) Phase after unwrapping + filtering +(f) Phase after unwrapping + filtering + linear +fitting +Figure 3: Sanitization steps of CSI sequences described in Section 3.1. In each subfigure, we plot five consecutive samples (five +colored curves) each containing CSI data of 30 IEEE 802.11n/ac sub-Carrier frequencies (horizontal axis). +Figure 4: Modality Translation Network. Two encoders extract the features from the amplitude and phase in the CSI domain. +Then the features are fused and reshaped before going through an encoder-decoder network. The output is a 3 × 720 × 1280 +feature map in the image domain. +where 𝐹 denotes the largest subcarrier index (30 in our case) and +ˆ𝜙𝑓 is the sanitized phase values at subcarrier 𝑓 (the 𝑓 th frequency). +In Figure 3(f), the final phase curves are temporally consistent. +3.2 +Modality Translation Network +In order to estimate the UV maps in the spatial domain from the +1D CSI signals, we first transform the network inputs from the +CSI domain to the spatial domain. This is done with the Modality +Translation Network (see Figure 4). We first extract the CSI latent +space features using two encoders, one for the amplitude tensor + +Encoder +Encoder- +Decoder +Flatten +1280 +Phase +Feature +Fusion +24 +Reshape +24 +720 +Flatten +3 +Amp +Encoder30 +25 +Amplitude +20 +15 +10 +5 +0 +5 +10 +15 +20 +25 +30 +Subcarrier Indexm +2 +1 +Phase +0 +-1 +-2 +-3 +0 +5 +10 +15 +20 +25 +Subcarrier Index0 +-5 +hase +-10 + Ph +-15 +-20 +0 +5 +10 +15 +20 +25 +Subcarrier Index4 +2 +Phase +1 +0 +-1 +-2 +0 +5 +10 +15 +20 +25 +Subcarrier Index0 +-5 +Phase +-10 +-15 +-20 +0 +5 +10 +15 +20 +25 +Subcarrier Index1.5 +1.0 +Phase +0.5 +0.0 +0.5 +0 +5 +10 +15 +20 +25 +Subcarrier Indexand the other for the phase tensor, where both tensors have the +size of 150 × 3 × 3 (5 consecutive samples, 30 frequencies, 3 emitters +and 3 receivers). Previous work on human sensing with WiFi [30] +stated that Convolutional Neural Network (CNN) can be used to +extract spatial features from the last two dimensions (the 3 × 3 +transmitting sensor pairs) of the input tensors. We, on the other +hand, believe that locations in the 3×3 feature map do not correlate +with the locations in the 2D scene. More specifically, as depicted +in Figure 2(b), the element that is colored in blue represents a 1D +summary of the entire scene captured by emitter 1 and receiver 3 (E1 +- R3), instead of local spatial information of the top right corner of +the 2D scene. Therefore, we consider that each of the 1350 elements +(in both tensors) captures a unique 1D summary of the entire scene. +Following this idea, the amplitude and phase tensors are flattened +and feed into two separate multi-layer perceptrons (MLP) to obtain +their features in the CSI latent space. We concatenated the 1D +features from both encoding branches, then the combined tensor is +fed to another MLP to perform feature fusion. +The next step is to transform the CSI latent space features to +feature maps in the spatial domain. As shown in Figure 4, the fused +1D feature is reshaped into a 24 × 24 2D feature map. Then, we +extract the spatial information by applying two convolution blocks +and obtain a more condensed map with the spatial dimension of 6×6. +Finally, four deconvolution layers are used to upsample the encoded +feature map in low dimensions to the size of 3 × 720 × 1280. We set +such an output tensor size to match the dimension commonly used +in RGB-image-input network. We now have a scene representation +in the image domain generated by WiFi signals. +3.3 +WiFi-DensePose RCNN +After we obtain the 3×720×1280 scene representation in the image +domain, we can utilize image-based methods to predict the UV +maps of human bodies. State-of-the-art pose estimation algorithms +are two-stage; first, they run an independent person detector to +estimate the bounding box and then conduct pose estimation from +person-wise image patches. However, as stated before, each element +in our CSI input tensors is a summary of the entire scene. It is not +possible to extract the signals corresponding to a single person +from a group of people in the scene. Therefore, we decide to adopt +a network structure similar to DensePose-RCNN [8], since it can +predict the dense correspondence of multiple humans in an end-to- +end fashion. +More specifically, in the WiFi-DensePose RCNN (Figure 5), we +extract the spatial features from the obtained 3 × 720 × 1280 image- +like feature map using the ResNet-FPN backbone [14]. Then, the +output will go through the region proposal network [20]. To bet- +ter exploit the complementary information of different sources, +the next part of our network contains two branches: DensePose +head and Keypoint head. Estimating keypoint locations is more +reliable than estimating dense correspondences, so we can train our +network to use keypoints to restrict DensePose predictions from +getting too far from the body joints of humans. The DensePose +head utilizes a Fully Convolutional Network (FCN) [16] to densely +predict human part labels and surface coordinates (UV coordinates) +within each part, while the keypoint head uses FCN to estimate the +keypoint heatmap. The results are combined and then fed into the +refinement unit of each branch, where each refinement unit con- +sists of two convolutional blocks followed by an FCN. The network +outputs a 17 × 56 × 56 keypoint mask and a 25 × 112 × 112 IUV map. +The process is demonstrated in Figure 5. It should be noted that the +modality translation network and the WiFi-DensePose RCNN are +trained together. +3.4 +Transfer Learning +Training the Modality Translation Network and WiFi-DensePose +RCNN network from a random initialization takes a lot of time +(roughly 80 hours). To improve the training efficiency, we conduct +transfer learning from an image-based DensPose network to our +WiFi-based network (See Figure 6 for details). +The idea is to supervise the training of the WiFi-based network +with the pre-trained image-based network. Directly initializing the +WiFi-based network with image-based network weights does not +work because the two networks get inputs from different domains +(image and channel state information). Instead, we first train an +image-based DensePose-RCNN model as a teacher network. Our +student network consists of the modality translation network and +the WiFi-DensePose RCNN. We fix the teacher network weights +and train the student network by feeding them with the synchro- +nized images and CSI tensors, respectively. We update the student +network such that its backbone (ResNet) features mimic that of +our teacher network. Our transfer learning goal is to minimize +the differences of multiple levels of feature maps generated by the +student model and those generated by the teacher model. There- +fore we calculate the mean squared error between feature maps. +The transfer learning loss from the teacher network to the student +network is: +𝐿𝑡𝑟 = 𝑀𝑆𝐸(𝑃2, 𝑃∗ +2)+𝑀𝑆𝐸(𝑃3, 𝑃∗ +3)+𝑀𝑆𝐸(𝑃4, 𝑃∗ +4)+𝑀𝑆𝐸(𝑃5, 𝑃∗ +5), (3) +where 𝑀𝑆𝐸(·) computes the mean squared error between two fea- +ture maps, {𝑃2, 𝑃3, 𝑃4, 𝑃5} is a set of feature maps produced by the +teacher network [14], and {𝑃∗ +2, 𝑃∗ +3, 𝑃∗ +4, 𝑃∗ +5} is the set of feature maps +produced by the student network [14]. +Benefiting from the additional supervision from the image-based +model, the student network gets higher performance and takes +fewer iterations to converge (Please see results in Table 5). +3.5 +Losses +The total loss of our approach is computed as: +𝐿 += +𝐿𝑐𝑙𝑠 + 𝐿𝑏𝑜𝑥 + 𝜆𝑑𝑝𝐿𝑑𝑝 + 𝜆𝑘𝑝𝐿𝑘𝑝 + 𝜆𝑡𝑟𝐿𝑡𝑟, +where 𝐿𝑐𝑙𝑠, 𝐿𝑏𝑜𝑥, 𝐿𝑑𝑝, 𝐿𝑘𝑝, 𝐿𝑡𝑟 are losses for the person classifica- +tion, bounding box regression, DensePose, keypoints, and transfer +learning respectively. The classification loss 𝐿𝑐𝑙𝑠 and the box regres- +sion loss 𝐿𝑏𝑜𝑥 are standard RCNN losses [9, 21]. The DensePose +loss 𝐿𝑑𝑝[8] consists of several sub-components: (1) Cross-entropy +loss for the coarse segmentation tasks. Each pixel is classified as +either belonging to the background or one of the 24 human body re- +gions. (2) Cross-entropy loss for body part classification and smooth +L1 loss for UV coordinate regression. These losses are used to de- +termine the exact coordinates of the pixels, i.e., 24 regressors are +created to break the full human into small parts and parameterize +each piece using a local two-dimensional UV coordinate system, +that identifies the position UV nodes on this surface part. + +Figure 5: WiFi-DensePose RCNN. The 3×720×1280 feature map from Figure 4 first goes through standard ResNet-FPN and ROI +pooling to extract person-wise features. The features are then processed by two heads:the Keypoint Head and the DensePose +Head. +Figure 6: Transfer learning from an image-based teacher network to our WiFi-based network. +We add 𝐿𝑘𝑝 to help the DensePose to balance between the torso +with more UV nodes and limbs with fewer UV nodes. Inspired by +Keypoint RCNN [9], we one-hot-encode each of the 17 ground truth +keypoints in one 56×56 heatmap, generating 17×56×56 keypoints +heatmaps and supervise the output with the Cross-Entropy Loss. To +closely regularize the Densepose regression, the keypoint heatmap +regressor takes the same input features used by the Denspose UV +maps. +4 +EXPERIMENTS +This section presents the experimental validation of our WiFi-based +DensePose. +4.1 +Dataset +We used the dataset 1 described in [31], which contains CSI samples +taken at 100Hz from receiver antennas and videos recorded at 20 +FPS. Time stamps are used to synchronize CSI and the video frames +such that 5 CSI samples correspond to 1 video frame. The dataset +1The identifiable information in this dataset has been removed for any privacy +concerns. +was gathered in sixteen spatial layouts: six captures in the lab +office and ten captures in the classroom. Each capture is around 13 +minutes with 1 to 5 subjects (8 subjects in total for the entire dataset) +performing daily activities under the layout described in Figure 2 +(a). The sixteen spatial layouts are different in their relative +locations/orientations of the WiFi-emitter antennas, person, +furniture, and WiFi-receiver antennas. +There are no manual annotations for the data set. We use the +MS-COCO-pre-trained dense model "R_101_FPN_s1x_legacy" 2 +and MS-COCO-pre-trained Keypoint R-CNN "R101-FPN" 3 to pro- +duce the pseudo ground truth. We denote the ground truth as +"R101-Pseudo-GT" (see an annotated example in Figure 7). The +R101-Pseudo-GT includes person bounding boxes, person-instance +segmentation masks, body-part UV maps, and person-wise key- +point coordinates. +2https://github.com/facebookresearch/detectron2/blob/main/projects/DensePose/ +doc/DENSEPOSE_IUV.md#ModelZoo +3https://github.com/facebookresearch/detectron2/blob/main/MODEL_ZOO.md# +coco-person-keypoint-detection-baselines-with-keypoint-r-cnn + +56 +56 +Keypoint +Head +1280 +17 +112 +720 +DensePose +ROI +Refinement +Head +Pooling +3 +112 +ResNet- +FPN +25Teacher Network +P +ResNet in image-based +P +Images +PA +Transfer Learning Loss +DensePose-RCNN +P +Compute the sum of +MSE for each level +P2 +Modality Translation +ResNet in WiFi- +P: +WiFi Signals +P4 +Network +DensePose RCNN +Ps +Student NetworkThroughout the section, we use R101-Puedo-GT to train our +WiFi-based DensePose model as well as finetuning the image-based +baseline "R_50_FPN_s1x_legacy". +Figure 7: Top two rows are the amplitude and phase of the +input WiFi signal. The bottom row shows R101-Psuedo-GT: +the ground truth dense pose and keypoints annotated by +running a pretrained image-based Densepose network on +the corresponding RGB image (see Section 4.1 for details). +4.2 +Training/Testing protocols and Metrics +We report results on two protocols: (1) Same layout: We train +on the training set in all 16 spatial layouts, and test on remaining +frames. Following [31], we randomly select 80% of the samples to +be our training set, and the rest to be our testing set. The training +and testing samples are different in the person’s location and pose, +but share the same person’s identities and background. This is a +reasonable assumption since the WiFi device is usually installed +in a fixed location. (2) Different layout: We train on 15 spatial +layouts and test on 1 unseen spatial layout. The unseen layout is in +the classroom scenarios. +We evaluate the performance of our algorithm in two aspects: +the ability to detect humans (bounding boxes) and accuracy of the +dense pose estimation. +To evaluate the performance of our models in detecting humans, +we calculate the standard average precision (AP) of person bounding +boxes at multiple IOU thresholds ranging from 0.5 to 0.95. +In addition, by MS-COCO [15] definition, we also compute AP-m +for median bodies that are enclosed in bounding boxes with sizes +between 32 × 32 and 96 × 96 pixels in a normalized 640 × 480 pixels +image space, and AP-l for large bodies that are enclosed in bounding +boxes larger than 96 × 96 pixels. +To measure the performance of DensePose detection, we follow +the original DensePose paper [8]. We first compute Geodesic Point +Similarity (GPS) as a matching score for dense correspondences: +𝐺𝑃𝑆𝑗 = +1 +|𝑃𝑗 | +∑︁ +𝑝 ∈𝑃𝑗 +exp( −𝑔(𝑖𝑝, ˆ𝑖𝑝)2 +2𝜅2 +), +(4) +where 𝑔 calculates the geodesic distance, 𝑃𝑗 denotes the ground +truth point annotations of person 𝑗, 𝑖𝑝 and ˆ𝑖𝑝 are the estimated and +ground truth vertex at point 𝑝 respectively, and 𝜅 is a normalizing +parameter (set to be 0.255 according to [8]). +One issue of GPS is that it does not penalize spurious predictions. +Therefore, estimations with all pixels classified as foreground are +favored. To alleviate this issue, masked geodesic point similarity +(GPSm) was introduced in [8], which incorporates both GPS and +segmentation masks. The formulation is as follows: +𝐺𝑃𝑆𝑚 = +√ +𝐺𝑃𝑆 · 𝐼, 𝐼 = 𝑀 ∩ ˆ𝑀 +𝑀 ∪ ˆ𝑀 +, +(5) +where 𝑀 and ˆ𝑀 are the predicted and ground truth foreground +segmentation masks. +Next, we can calculate DensePose average precision with GPS (de- +noted as dpAP· GPS) and GPSm (denoted as dpAP· GPSm) as thresh- +olds, following the same logic behind the calculation of bounding +box AP. +4.3 +Implementation Details +We implemented our approach in PyTorch. We set the training +batch size to 16 on a 4 GPU (Titan X) server. We empirically set +𝜆𝑑𝑝 = 0.6, 𝜆𝑘𝑝 = 0.3, 𝜆𝑡𝑟 = 0.1. We used a warmup multi-step +learning rate scheduler and set the initial learning rate as 1𝑒 − 5. +The learning rate increases to 1𝑒 − 3 during the first 2000 iterations, +then decreases to 1 +10 of its value every 48000 iterations. We trained +for 145, 000 iterations for our final model. +4.4 +WiFi-based DensePose under Same Layout +Under the Same Layout protocol, we compute the AP of human +bounding box detections as well as dpAP· GPS and dpAP· GPSm of +dense correspondence predictions. Results are presented in Table 1 +and Table 2, respectively. +Method +AP +AP@50 +AP@75 +AP-m +AP-l +WiFi +43.5 +87.2 +44.6 +38.1 +46.4 +Table 1: Average precision (AP) of WiFi-based DensePose un- +der the Same Layout protocol. All metrics are the higher the +better. +From Table 1, we can observe a high value (87.2) of AP@50, +indicating that our model can effectively detect the approximate +locations of human bounding boxes. The relatively low value (35.6) +for AP@75 suggests that the details of the human bodies are not +perfectly estimated. + +25.0 +22.5 +20.0 +plitude +17.5 +15.0 +12.5 +10.0 +7.5 +0 +5 +10 +15 +20 +25 +Subcarrier Index +3 +2 +1 +Phase +0 +-1 +2 +-3 +5 +10 +15 +20 +25 +0 +Subcarrier IndexMethod +dpAP · GPS +dpAP · GPS@50 +dpAP · GPS@75 +dpAP · GPSm +dpAP · GPSm@50 +dpAP · GPSm@75 +WiFi +45.3 +76.7 +47.7 +44.8 +73.6 +44.9 +Table 2: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based DensePose under the Same Layout protocol. +All metrics are the higher the better. +A similar pattern can be seen from the results of DensePose +estimations (see Table 2 for details). Experiments report high values +of dpAP · GPS@50 and dpAP · GPSm@50, but low values of dpAP · +GPS@75 and dpAP · GPSm@75. This demonstrates that our model +performs well at estimating the poses of human torsos, but still +struggles with detecting details like limbs. +4.5 +Comparison with Image-based DensePose +Method +AP +AP@50 +AP@75 +AP-m +AP-l +WiFi +43.5 +87.2 +44.6 +38.1 +46.4 +Image +84.7 +94.4 +77.1 +70.3 +83.8 +Table 3: Average precision (AP) of WiFi-based and image- +based DensePose under the Same Layout protocol. All met- +rics are the higher the better. +As discussed in Section 4.1, since there are no manual annota- +tions on the WiFi dataset, it is difficult to compare the performance +of WiFi-based DensePose with its Image-based counterpart. This is +a common drawback of many WiFi perception works including [31]. +Nevertheless, conducting such a comparison is still worthwhile +in assessing the current limit of WiFi-based perception. We tried an +image-based DensePose baseline "R_50_FPN_s1x_legacy" finetuned +on R101-Pseudo-GT under the Same Layout protocol. In addition, +as shown in Figure 9 and Figure 10, though certain defects still exist, +the estimations from our WiFi-based model are reasonably well +compared to the results produced by Image-based DensePose. +In the quantitative results in Table 3 and Table 4, the image-based +baseline produces very high APs due to the small difference between +its ResNet50 backbone and the Resnet101 backbone used to generate +R101-Pseudo-GT. This is to be expected. Our WiFi-based model +has much lower absolute metrics. However, it can be observed +from Table 3 that the difference between AP-m and AP-l values is +relatively small for the WiFi-based model. We believe this is because +individuals who are far away from cameras occupy less space in +the image, which leads to less information about these subjects. On +the contrary, WiFi signals incorporate all the information in the +entire scene, regardless of the subjects’ locations. +4.6 +Ablation Study +This section describes the ablation study to understand the effects +of phase information, keypoint supervision, and transfer learning +on estimating dense correspondences. Similar to section 4.4, the +models analyzed in this section are all trained under the same- +layout mentioned in section 4.2. +We start by training a baseline WiFi model, which does not in- +clude the phase encoder, the keypoint detection branch, or transfer +learning. The results are presented in the first row of both Table 5 +and Table 6 as a reference. +Addition of Phase information: We first examine whether +the phase information can enhance the baseline performance. As +shown in the second row of Table 5 and Table 6, the results for all +the metrics have slightly improved from the baseline. This proves +our hypothesis that the phase can reveal relevant information about +the dense human pose. +Addition of a keypoint detection branch: Having established +the advantage of incorporating phase information, we now evaluate +the effect of adding a keypoint branch to our model. The quantita- +tive results are summarized in the third row of Table 5 and Table 6. +Comparing with the numbers on the second row, we can observe +a slight increase in performance in terms of dpAP·GPS@50(from +77.4 to 78.8) and dpAP·GPSm@50 (from 75.7 to 76.8), and a more +noticeable improvement in terms of dpAP·GPS@75 (from 42.3 to +46.9) and dpAP·GPSm@75 (from 40.5 to 44.9). This indicates that +the keypoint branch provides effective references to dense pose +estimations, and our model becomes significantly better at detecting +subtle details (such as the limbs). +Effect of Transfer Learning: We aim to reduce the training +time for our model with the help of transfer learning. For each +model in Table 5, we continue training the model until there are +no significant changes in terms of performance. The last row of +Table 5 and Table 6 represents our final model with transfer learn- +ing. Though the final performance does not improve too much +compared to the model (with phase information and keypoints) +without transfer learning, it should be noted that the number of +training iterations decreases significantly from 186000 to 145000 +(this number includes the time to perform transfer learning as well +as training the main model). +4.7 +Performance in different layouts +All above results are obtained using the same layout for training and +testing. However, WiFi signals in different environments exhibit +significantly different propagation patterns. Therefore, it is still a +very challenging problem to deploy our model on data from an +untrained layout. +To test the robustness of our model, we conducted the previous +experiment under the different layout protocols, where there are +15 training layouts and 1 testing layout. The experimental results +are recorded in Table 7 and Table 8. +We can observe that our final model performs better than the +baseline model in the unseen domain, but the performance de- +creases significantly from that under the same layout protocol: the +AP performance drops from 43.5 to 27.3 and dpAP·GPS drops from +45.3 to 25.4. However, it should also be noted that the image-based +model suffers from the same domain generalization problem. We + +Method +dpAP · GPS +dpAP · GPS@50 +dpAP · GPS@75 +dpAP · GPSm +dpAP · GPSm@50 +dpAP · GPSm@75 +WiFi +45.3 +79.3 +47.7 +43.2 +77.4 +45.5 +Image +81.8 +93.7 +86.2 +84.0 +94.9 +86.8 +Table 4: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based and image-based DensePose under the Same +Layout protocol. All metrics are the higher the better. +Method +AP +AP@50 +AP@75 +AP-m +AP-l +Number of Trained Iterations +Amplitude-only Model +39.5 +85.4 +41.3 +34.4 +43.7 +174000 ++ Sanitized Phase Input +40.3 +85.9 +41.9 +34.6 +44.5 +180000 ++ Keypoint Supervision +42.9 +86.8 +44.1 +38.0 +45.8 +186000 ++ Transfer Learning +43.5 +87.2 +44.6 +38.1 +46.4 +145000 +Table 5: Ablation study of human detection under the Same-layout protocol. All metrics are the higher the better. +Method +dpAP · GPS +dpAP · GPS@50 +dpAP · GPS@75 +dpAP · GPSm +dpAP · GPSm@50 +dpAP · GPSm@75 +Amplitude-only Model +40.6 +76.6 +41.5 +39.7 +75.1 +40.3 ++ Sanitized Phase Input +41.2 +77.4 +42.3 +40.1 +75.7 +40.5 ++ Keypoint Supervision +44.6 +78.8 +46.9 +42.9 +76.8 +44.9 ++ Transfer Learning +45.3 +79.3 +47.7 +43.2 +77.4 +45.5 +Table 6: Ablation study of DensePose estimation under the Same-layout protocol. All metrics are the higher the better. +Method +AP +AP@50 +AP@75 +AP-m +AP-l +WiFi (base) +23.5 +48.1 +20.3 +19.4 +24.5 +WiFi (final) +27.3 +51.8 +24.2 +22.1 +28.6 +Image +60.6 +80.4 +52.1 +48.3 +65.8 +Table 7: Average precision (AP) of WiFi-based and image-based DensePose under the Different Layout protocol. All metrics +are the higher the better. +Method +dpAP · GPS +dpAP · GPS@50 +dpAP · GPS@75 +dpAP · GPSm +dpAP · GPSm@50 +dpAP · GPSm@75 +WiFi (base) +22.3 +47.3 +21.5 +20.9 +44.6 +21.8 +WiFi (final) +25.4 +50.2 +24.7 +23.2 +47.4 +26.5 +Image +60.2 +70.1 +62.3 +54.0 +72.7 +58.8 +Table 8: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based and image-based DensePose under the Differ- +ent Layout protocol. All metrics are the higher the better. +believe a more comprehensive dataset from a wide range of settings +can alleviate this issue. +4.8 +Failure cases +We observed two main types of failure cases. (1) When there are +body poses that rarely occurred in the training set, the WiFi-based +model is biased and is likely to produce wrong body parts (See exam- +ples (a-b) in Figure 8). (2) When there are three or more concurrent +subjects in one capture, it is more challenging for the WiFi-based +model to extract detailed information for each individual from the +amplitude and phase tensors of the entire capture. (See examples +(c-d) in Figure 8). We believe both of these issues can be resolved +by obtaining more comprehensive training data. +5 +CONCLUSION AND FUTURE WORK +In this paper, we demonstrated that it is possible to obtain dense +human body poses from WiFi signals by utilizing deep learning +architectures commonly used in computer vision. Instead of directly +training a randomly initialized WiFi-based model, we explored +rich supervision information to improve both the performance and +training efficiency, such as utilizing the CSI phase, adding keypoint +detection branch, and transfer learning from an image-based model. +The performance of our work is still limited by the public training +data in the field of WiFi-based perception, especially under different +layouts. In future work, we also plan to collect multi-layout data +and extend our work to predict 3D human body shapes from WiFi +signals. We believe that the advanced capability of dense perception + +(a) +(b) +(c) +(d) +Figure 8: Examples pf failure cases: (a-b) rare poses; (c-d) Three or more concurrent subjects. The first row is ground truth +dense pose estimation. The second row illustrates the predicted dense pose. +could empower the WiFi device as a privacy-friendly, illumination- +invariant, and cheap human sensor compared to RGB cameras and +Lidars. +REFERENCES +[1] Fadel Adib, Chen-Yu Hsu, Hongzi Mao, Dina Katabi, and Frédo Durand. 2015. +Capturing the Human Figure through a Wall. ACM Trans. Graph. 34, 6, Article +219 (oct 2015), 13 pages. https://doi.org/10.1145/2816795.2818072 +[2] Fadel Adib, Zach Kabelac, Dina Katabi, and Robert C. Miller. 2014. 3D Tracking via +Body Radio Reflections. In 11th USENIX Symposium on Networked Systems Design +and Implementation (NSDI 14). USENIX Association, Seattle, WA, 317–329. https: +//www.usenix.org/conference/nsdi14/technical-sessions/presentation/adib +[3] Thiemo Alldieck, Gerard Pons-Moll, Christian Theobalt, and Marcus Magnor. +2019. 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Learning Dense Correspondence via 3D-guided Cycle Consistency. +arXiv:1604.05383 [cs.CV] + diff --git a/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/load_file.txt b/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5a1d5d053c0dfd139fc10e83fb84d4a837b0b93 --- /dev/null +++ b/xNAyT4oBgHgl3EQfavfQ/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf,len=773 +page_content='DensePose From WiFi Jiaqi Geng jiaqigen@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='edu Carnegie Mellon University Pittsburgh, PA, USA Dong Huang donghuang@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='edu Carnegie Mellon University Pittsburgh, PA, USA Fernando De la Torre ftorre@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='edu Carnegie Mellon University Pittsburgh, PA, USA ABSTRACT Advances in computer vision and machine learning techniques have led to significant development in 2D and 3D human pose estimation from RGB cameras, LiDAR, and radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, human pose esti- mation from images is adversely affected by occlusion and lighting, which are common in many scenarios of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Radar and LiDAR technologies, on the other hand, need specialized hardware that is expensive and power-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Furthermore, placing these sensors in non-public areas raises significant privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To address these limitations, recent research has explored the use of WiFi antennas (1D sensors) for body segmentation and key-point body detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This paper further expands on the use of the WiFi signal in combination with deep learning architectures, commonly used in computer vision, to estimate dense human pose correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We developed a deep neural network that maps the phase and amplitude of WiFi signals to UV coordinates within 24 human regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The results of the study reveal that our model can estimate the dense pose of multiple subjects, with comparable performance to image-based approaches, by utilizing WiFi signals as the only input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This paves the way for low-cost, broadly accessible, and privacy-preserving algorithms for human sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Machine Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' • Hardware → Communica- tion hardware, interfaces and storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' KEYWORDS Pose Estimation, Dense Body Pose Estimation, WiFi Signals, Key- point Estimation, Body Segmentation, Object Detection, UV Coor- dinates, Phase and Amplitude, Phase Sanitization, Channel State Information, Domain Translation, Deep Neural Network, Mask- RCNN 1 INTRODUCTION Much progress has been made in human pose estimation using 2D [7, 8, 12, 22, 28, 33] and 3D [17, 32] sensors in the last few years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=', RGB sensors, LiDARs, radars), fueled by applications in autonomous driving and augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' These traditional sensors, however, are constrained by both technical and practical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' LiDAR and radar sensors are frequently seen as being out of reach for the average household or small business due to their high cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' For example, the medium price of one of the most common COTS LiDAR, Intel L515, is around 700 dollars, and the prices for ordinary radar detectors range from 200 dollars to 600 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In addition, these sensors are too power-consuming for daily and household use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' As for RGB cameras, narrow field of view and poor lighting conditions, such as glare and darkness, can have a severe impact on camera-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Occlusion is another obstacle that prevents the camera-based model from generating reasonable pose predictions in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This is especially worrisome for indoors scenarios, where furniture typically occludes people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' More importantly, privacy concerns prevent the use of these technologies in non-public places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' For instance, most people are uncomfortable with having cameras recording them in their homes, and in certain areas (such as the bathroom) it will not be feasible to install them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' These issues are particularly critical in healthcare applications, that are increasingly shifting from clinics to homes, where people are being monitored with the help of cameras and other sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' It is important to resolve the aforementioned prob- lems in order to better assist the aging population, which is the most susceptible (especially during COVID) and has a growing demand to keep them living independently at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We believe that WiFi signals can serve as a ubiquitous substitute for RGB images for human sensing in certain instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Illumination and occlusion have little effect on WiFi-based solutions used for interior monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In addition, they protect individuals’ privacy and the required equipment can be bought at a reasonable price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In fact, most households in developed countries already have WiFi at home, and this technology may be scaled to monitor the well-being of elder people or just identify suspicious behaviors at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The issue we are trying to solve is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 1 (first row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Given three WiFi transmitters and three aligned receivers, can we detect and recover dense human pose correspondence in clut- tered scenarios with multiple people (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 1 fourth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' It should be noted that many WiFi routers, such as TP-Link AC1750, come with 3 antennas, so our method only requires 2 of these routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Each of these router is around 30 dollars, which means our entire setup is still way cheaper than LiDAR and radar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Many factors make this a difficult task to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' First of all, WiFi-based perception[11, 30] is based on the Channel-state-information (CSI) that represents the ratio between the transmitted signal wave and the received signal wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The CSIs are complex decimal sequences that do not have spatial correspondence to spatial locations, such as the image pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Secondly, classic techniques rely on accurate measurement of time-of-fly and angle-of-arrival of the signal be- tween the transmitter and receiver [13, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' These techniques only locate the object’s center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' moreover, the localization accuracy is only around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 meters due to the random phase shift allowed by the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='11n/ac WiFi communication standard and potential interference with electronic devices under similar frequency range such as microwave oven and cellphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To address these issues, we derive inspiration from recent pro- posed deep learning architectures in computer vision, and propose a neural network architecture that can perform dense pose estima- tion from WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Fig 1 (bottom row) illustrates how our algorithm is able to estimate dense pose using only WiFi signal in scenarios with occlusion and multiple people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='00250v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='CV] 31 Dec 2022 Figure 1: The first row illustrates the hardware setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The second and third rows are the clips of amplitude and phase of the input WiFi signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The fourth row contains the dense pose estimation of our algorithm from only the WiFi signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 2 RELATED WORK This section briefly describes existing work on dense estimation from images and human sensing from WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our research aims to conduct dense pose estimation via WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In computer vision, the subject of dense pose estimation from pictures and video has received a lot of attention [6, 8, 18, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This task consists of finding the dense correspondence between image pixels and the dense vertices indexes of a 3D human body model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The pioneering work of Güler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' [8] mapped human images to dense correspondences of a human mesh model using deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' DensePose is based on instance segmentation architectures such as Mark-RCNN [9], and predicts body-wise UV maps for each pixel, where UV maps are flattened representations of 3d geometry, with coordinate points usually corresponding to the vertices of a 3d dimensional object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In this work, we borrow the same architecture as DensePose [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' however, our input will not be an image or video, but we use 1D WiFi signals to recover the dense correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Recently, there have been many extensions of DensePose pro- posed, especially in 3D human reconstruction with dense body parts [3, 35, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Shapovalov et al.’s [24] work focused on lifting dense pose surface maps to 3D human models without 3D supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Their network demonstrates that the dense correspondence alone (without using full 2D RGB images) contains sufficient information to generate posed 3D human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Compared to previous works on reconstructing 3D humans with sparse 2D keypoints, DensePose annotations are much denser and provide information about the 3D surface instead of 2D body joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' While there is a extensive literature on detection [19, 20], track- ing [4, 34], and dense pose estimation [8, 18] from images and videos, human pose estimation from WiFi or radar is a relatively unexplored problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' At this point, it is important to differentiate the current work on radar-based systems and WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The work of Adib et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' [2] proposed a Frequency Modulated Continuous Wave (FMCW) radar system (broad bandwidth from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='56GHz to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='25GHz) for indoor human localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' A limitation of this system is the specialized hardware for synchronizing the transmission, refrac- tion, and reflection to compute the Time-of-Flight (ToF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The system reached a resolution of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 cm on body localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In the following work [1], they improved the system by focusing on a moving per- son and generated a rough single-person outline with depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Recently, they applied deep learning approaches to do fine-grained human pose estimation using a similar system, named RF-Pose [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' These systems do not work under the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='11n/ac WiFi com- munication standard (40MHz bandwidth centered at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' They rely on additional high-frequency and high-bandwidth electromag- netic fields, which need specialized technology not available to the general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Recently, significant improvements have been made to radar-based human sensing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' mmMesh [36] generates 3ReceiverAntennas 6DBiRP-SMA 3TransmitterAntennas CameraforAnnotation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4GHz Copperdipoleantenna 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 inches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 inches 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 18 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 14 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 %0 Subcarrier Index 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 20 25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 20 25 Subcarrier Index 20 rrier Index Subcarrier Index Subcarrier Index3D human mesh from commercially portable millimeter-wave de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This system can accurately localize the vertices on the human mesh with an average error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='47 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, mmMesh does not work well with occlusions since high-frequency radio waves cannot penetrate objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Unlike the above radar systems, the WiFi-based solution [11, 30] used off-the-shelf WiFi adapters and 3dB omnidirectional antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The signal propagate as the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='11n/ac WiFi data packages transmitting between antennas, which does not introduce addi- tional interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, WiFi-based person localization using the traditional time-of-flight (ToF) method is limited by its wave- length and signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Most existing approaches only conduct center mass localization [5, 27] and single-person action classification [25, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Recently, Fei Wang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' [31] demonstrated that it is possible to detect 17 2D body joints and perform 2D se- mantic body segmentation mask using only WiFi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In this work, we go beyond [31] by estimating dense body pose, with much more accuracy than the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5m that the WiFi signal can pro- vide theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our dense posture outputs push above WiFi’s signal constraint in body localization, paving the road for complete dense 2D and possibly 3D human body perception through WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To achieve this, instead of directly training a randomly initialized WiFi-based model, we explored rich supervision information to improve both the performance and training efficiency, such as uti- lizing the CSI phase, adding keypoint detection branch, and transfer learning from the image-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3 METHODS Our approach produces UV coordinates of the human body sur- face from WiFi signals using three components: first, the raw CSI signals are cleaned by amplitude and phase sanitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Then, a two-branch encoder-decoder network performs domain translation from sanitized CSI samples to 2D feature maps that resemble im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The 2D features are then fed to a modified DensePose-RCNN architecture [8] to estimate the UV map, a representation of the dense correspondence between 2D and 3D humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To improve the training of our WiFi-input network, we conduct transfer learning, where we minimize the differences between the multi-level fea- ture maps produced by images and those produced by WiFi signals before training our main network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The raw CSI data are sampled in 100Hz as complex values over 30 subcarrier frequencies (linearly spaced within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4GHz±20MHz) transmitting among 3 emitter antennas and 3 reception antennas (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Each CSI sample contains a 3 × 3 real integer matrix and a 3 × 3 imaginary integer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The inputs of our network contained 5 consecutive CSI samples under 30 frequencies, which are organized in a 150×3×3 amplitude tensor and a 150×3×3 phase tensor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our network outputs include a 17 × 56 × 56 tensor of keypoint heatmaps (one 56 × 56 map for each of the 17 kepoints) and a 25 × 112 × 112 tensor of UV maps (one 112 × 112 map for each of the 24 body parts with one additional map for background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 Phase Sanitization The raw CSI samples are noisy with random phase drift and flip (see Figure 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Most WiFi-based solutions disregard the phase of CSI (a) Layout of WiFi devices and human bodies (b) The 3 × 3 dimensions of the CSI tensor Figure 2: CSI samples from Wifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (a) the layout of WiFi de- vices and human bodies, and (b) the 3 × 3 tensor dimen- sion corresponds to the 3 × 3 transmitter-receiver antenna pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' For instance, 𝐸1 denotes the first emitter and 𝑅1 de- notes the first receiver, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' By incorporating the 5 consecu- tive complex-valued CSI samples (100 samples/second) un- der 30 subcarrier frequencies, the two input tensors to our network are a 150 × 3 × 3 amplitude tensor and a 150 × 3 × 3 phase tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' signals and rely only on their amplitude (see Figure 3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' As shown in our experimental validation, discarding the phase information have a negative impact on the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In this section, we perform sanitization to obtain stable phase values to enable full use of the CSI information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In raw CSI samples (5 consecutive samples visualized in Fig- ure 3(a-b)), the amplitude (𝐴) and phase (Φ) of each complex ele- ment 𝑧 = 𝑎+𝑏𝑖 are computed using the formulation 𝐴 = √︁ (𝑎2 + 𝑏2) and Φ = 𝑎𝑟𝑐𝑡𝑎𝑛(𝑏/𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Note that the range of the arctan function is from −𝜋 to 𝜋 and the phase values outside this range get wrapped, leading to a discontinuity in phase values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our first sanitization step is to unwrap the phase following [10]: Δ𝜙𝑖,𝑗 = Φ𝑖,𝑗+1 − Φ𝑖,𝑗 if Δ𝜙𝑖,𝑗 > 𝜋, Φ𝑖,𝑗+1 = Φ𝑖,𝑗 + Δ𝜙𝑖,𝑗 − 2𝜋 if Δ𝜙𝑖,𝑗 < −𝜋, Φ𝑖,𝑗+1 = Φ𝑖,𝑗 + Δ𝜙𝑖,𝑗 + 2𝜋, (1) where 𝑖 denotes the index of the measurements in the five consecu- tive samples, and 𝑗 denotes the index of the subcarriers(frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Following unwrapping, each of the flipping phase curves in Fig- ure 3(b) are restored to continuous curves in Figure 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Observe that among the 5 phase curves captured in 5 consecutive samples in Figure 3(c), there are random jiterings that break the temporal order among the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To keep the temporal order of signals, previous work [23] mentioned linear fitting as a popular approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, directly applying linear fitting to Figure 3(c) further amplified the jitering instead of fixing it (see the failed results in Figure 3(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' From Figure 3(c), we use median and uniform filters to eliminate outliers in both the time and frequency domain which leads to Figure 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' we obtain the fully sanitized phase values by applying the linear fitting method following the equations below: 𝛼1 = Φ𝐹 − Φ1 2𝜋𝐹 𝛼0 = 1 𝐹 ∑︁ 1≤𝑓 ≤𝐹 𝜙𝑓 ˆ𝜙𝑓 = 𝜙𝑓 − (𝛼1𝑓 + 𝑎0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (2) R1 R2 R3 E1 E2 E3(a) Original CSI Amplitude (b) Original CSI Phase (c) Phase after unwrapping (d) Phase after unwrapping + linear fitting (e) Phase after unwrapping + filtering (f) Phase after unwrapping + filtering + linear fitting Figure 3: Sanitization steps of CSI sequences described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In each subfigure, we plot five consecutive samples (five colored curves) each containing CSI data of 30 IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='11n/ac sub-Carrier frequencies (horizontal axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Figure 4: Modality Translation Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Two encoders extract the features from the amplitude and phase in the CSI domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Then the features are fused and reshaped before going through an encoder-decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The output is a 3 × 720 × 1280 feature map in the image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' where 𝐹 denotes the largest subcarrier index (30 in our case) and ˆ𝜙𝑓 is the sanitized phase values at subcarrier 𝑓 (the 𝑓 th frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In Figure 3(f), the final phase curves are temporally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 Modality Translation Network In order to estimate the UV maps in the spatial domain from the 1D CSI signals, we first transform the network inputs from the CSI domain to the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This is done with the Modality Translation Network (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We first extract the CSI latent space features using two encoders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' one for the amplitude tensor Encoder Encoder- Decoder Flatten 1280 Phase Feature Fusion 24 Reshape 24 720 Flatten 3 Amp Encoder30 25 Amplitude 20 15 10 5 0 5 10 15 20 25 30 Subcarrier Indexm 2 1 Phase 0 1 2 3 0 5 10 15 20 25 Subcarrier Index0 5 hase 10 Ph 15 20 0 5 10 15 20 25 Subcarrier Index4 2 Phase 1 0 1 2 0 5 10 15 20 25 Subcarrier Index0 5 Phase 10 15 20 0 5 10 15 20 25 Subcarrier Index1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 0 5 10 15 20 25 Subcarrier Indexand the other for the phase tensor, where both tensors have the size of 150 × 3 × 3 (5 consecutive samples, 30 frequencies, 3 emitters and 3 receivers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Previous work on human sensing with WiFi [30] stated that Convolutional Neural Network (CNN) can be used to extract spatial features from the last two dimensions (the 3 × 3 transmitting sensor pairs) of the input tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We, on the other hand, believe that locations in the 3×3 feature map do not correlate with the locations in the 2D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' More specifically, as depicted in Figure 2(b), the element that is colored in blue represents a 1D summary of the entire scene captured by emitter 1 and receiver 3 (E1 R3), instead of local spatial information of the top right corner of the 2D scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Therefore, we consider that each of the 1350 elements (in both tensors) captures a unique 1D summary of the entire scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Following this idea, the amplitude and phase tensors are flattened and feed into two separate multi-layer perceptrons (MLP) to obtain their features in the CSI latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We concatenated the 1D features from both encoding branches, then the combined tensor is fed to another MLP to perform feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The next step is to transform the CSI latent space features to feature maps in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' As shown in Figure 4, the fused 1D feature is reshaped into a 24 × 24 2D feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Then, we extract the spatial information by applying two convolution blocks and obtain a more condensed map with the spatial dimension of 6×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Finally, four deconvolution layers are used to upsample the encoded feature map in low dimensions to the size of 3 × 720 × 1280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We set such an output tensor size to match the dimension commonly used in RGB-image-input network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We now have a scene representation in the image domain generated by WiFi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 WiFi-DensePose RCNN After we obtain the 3×720×1280 scene representation in the image domain, we can utilize image-based methods to predict the UV maps of human bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' State-of-the-art pose estimation algorithms are two-stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' first, they run an independent person detector to estimate the bounding box and then conduct pose estimation from person-wise image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, as stated before, each element in our CSI input tensors is a summary of the entire scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' It is not possible to extract the signals corresponding to a single person from a group of people in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Therefore, we decide to adopt a network structure similar to DensePose-RCNN [8], since it can predict the dense correspondence of multiple humans in an end-to- end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' More specifically, in the WiFi-DensePose RCNN (Figure 5), we extract the spatial features from the obtained 3 × 720 × 1280 image- like feature map using the ResNet-FPN backbone [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Then, the output will go through the region proposal network [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To bet- ter exploit the complementary information of different sources, the next part of our network contains two branches: DensePose head and Keypoint head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Estimating keypoint locations is more reliable than estimating dense correspondences, so we can train our network to use keypoints to restrict DensePose predictions from getting too far from the body joints of humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The DensePose head utilizes a Fully Convolutional Network (FCN) [16] to densely predict human part labels and surface coordinates (UV coordinates) within each part, while the keypoint head uses FCN to estimate the keypoint heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The results are combined and then fed into the refinement unit of each branch, where each refinement unit con- sists of two convolutional blocks followed by an FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The network outputs a 17 × 56 × 56 keypoint mask and a 25 × 112 × 112 IUV map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The process is demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' It should be noted that the modality translation network and the WiFi-DensePose RCNN are trained together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 Transfer Learning Training the Modality Translation Network and WiFi-DensePose RCNN network from a random initialization takes a lot of time (roughly 80 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To improve the training efficiency, we conduct transfer learning from an image-based DensPose network to our WiFi-based network (See Figure 6 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The idea is to supervise the training of the WiFi-based network with the pre-trained image-based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Directly initializing the WiFi-based network with image-based network weights does not work because the two networks get inputs from different domains (image and channel state information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Instead, we first train an image-based DensePose-RCNN model as a teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our student network consists of the modality translation network and the WiFi-DensePose RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We fix the teacher network weights and train the student network by feeding them with the synchro- nized images and CSI tensors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We update the student network such that its backbone (ResNet) features mimic that of our teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our transfer learning goal is to minimize the differences of multiple levels of feature maps generated by the student model and those generated by the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' There- fore we calculate the mean squared error between feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The transfer learning loss from the teacher network to the student network is: 𝐿𝑡𝑟 = 𝑀𝑆𝐸(𝑃2, 𝑃∗ 2)+𝑀𝑆𝐸(𝑃3, 𝑃∗ 3)+𝑀𝑆𝐸(𝑃4, 𝑃∗ 4)+𝑀𝑆𝐸(𝑃5, 𝑃∗ 5), (3) where 𝑀𝑆𝐸(·) computes the mean squared error between two fea- ture maps, {𝑃2, 𝑃3, 𝑃4, 𝑃5} is a set of feature maps produced by the teacher network [14], and {𝑃∗ 2, 𝑃∗ 3, 𝑃∗ 4, 𝑃∗ 5} is the set of feature maps produced by the student network [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Benefiting from the additional supervision from the image-based model, the student network gets higher performance and takes fewer iterations to converge (Please see results in Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 Losses The total loss of our approach is computed as: 𝐿 = 𝐿𝑐𝑙𝑠 + 𝐿𝑏𝑜𝑥 + 𝜆𝑑𝑝𝐿𝑑𝑝 + 𝜆𝑘𝑝𝐿𝑘𝑝 + 𝜆𝑡𝑟𝐿𝑡𝑟, where 𝐿𝑐𝑙𝑠, 𝐿𝑏𝑜𝑥, 𝐿𝑑𝑝, 𝐿𝑘𝑝, 𝐿𝑡𝑟 are losses for the person classifica- tion, bounding box regression, DensePose, keypoints, and transfer learning respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The classification loss 𝐿𝑐𝑙𝑠 and the box regres- sion loss 𝐿𝑏𝑜𝑥 are standard RCNN losses [9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The DensePose loss 𝐿𝑑𝑝[8] consists of several sub-components: (1) Cross-entropy loss for the coarse segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Each pixel is classified as either belonging to the background or one of the 24 human body re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (2) Cross-entropy loss for body part classification and smooth L1 loss for UV coordinate regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' These losses are used to de- termine the exact coordinates of the pixels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=', 24 regressors are created to break the full human into small parts and parameterize each piece using a local two-dimensional UV coordinate system, that identifies the position UV nodes on this surface part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Figure 5: WiFi-DensePose RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The 3×720×1280 feature map from Figure 4 first goes through standard ResNet-FPN and ROI pooling to extract person-wise features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The features are then processed by two heads:the Keypoint Head and the DensePose Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Figure 6: Transfer learning from an image-based teacher network to our WiFi-based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We add 𝐿𝑘𝑝 to help the DensePose to balance between the torso with more UV nodes and limbs with fewer UV nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Inspired by Keypoint RCNN [9], we one-hot-encode each of the 17 ground truth keypoints in one 56×56 heatmap, generating 17×56×56 keypoints heatmaps and supervise the output with the Cross-Entropy Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To closely regularize the Densepose regression, the keypoint heatmap regressor takes the same input features used by the Denspose UV maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4 EXPERIMENTS This section presents the experimental validation of our WiFi-based DensePose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 Dataset We used the dataset 1 described in [31], which contains CSI samples taken at 100Hz from receiver antennas and videos recorded at 20 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Time stamps are used to synchronize CSI and the video frames such that 5 CSI samples correspond to 1 video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The dataset 1The identifiable information in this dataset has been removed for any privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' was gathered in sixteen spatial layouts: six captures in the lab office and ten captures in the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Each capture is around 13 minutes with 1 to 5 subjects (8 subjects in total for the entire dataset) performing daily activities under the layout described in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The sixteen spatial layouts are different in their relative locations/orientations of the WiFi-emitter antennas, person, furniture, and WiFi-receiver antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' There are no manual annotations for the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We use the MS-COCO-pre-trained dense model "R_101_FPN_s1x_legacy" 2 and MS-COCO-pre-trained Keypoint R-CNN "R101-FPN" 3 to pro- duce the pseudo ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We denote the ground truth as "R101-Pseudo-GT" (see an annotated example in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The R101-Pseudo-GT includes person bounding boxes, person-instance segmentation masks, body-part UV maps, and person-wise key- point coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='com/facebookresearch/detectron2/blob/main/projects/DensePose/ doc/DENSEPOSE_IUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='md#ModelZoo 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='com/facebookresearch/detectron2/blob/main/MODEL_ZOO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='md# coco-person-keypoint-detection-baselines-with-keypoint-r-cnn 56 56 Keypoint Head 1280 17 112 720 DensePose ROI Refinement Head Pooling 3 112 ResNet- FPN 25Teacher Network P ResNet in image-based P Images PA Transfer Learning Loss DensePose-RCNN P Compute the sum of MSE for each level P2 Modality Translation ResNet in WiFi- P: WiFi Signals P4 Network DensePose RCNN Ps Student NetworkThroughout the section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' we use R101-Puedo-GT to train our WiFi-based DensePose model as well as finetuning the image-based baseline "R_50_FPN_s1x_legacy".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Figure 7: Top two rows are the amplitude and phase of the input WiFi signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The bottom row shows R101-Psuedo-GT: the ground truth dense pose and keypoints annotated by running a pretrained image-based Densepose network on the corresponding RGB image (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 Training/Testing protocols and Metrics We report results on two protocols: (1) Same layout: We train on the training set in all 16 spatial layouts, and test on remaining frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Following [31], we randomly select 80% of the samples to be our training set, and the rest to be our testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The training and testing samples are different in the person’s location and pose, but share the same person’s identities and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This is a reasonable assumption since the WiFi device is usually installed in a fixed location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (2) Different layout: We train on 15 spatial layouts and test on 1 unseen spatial layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The unseen layout is in the classroom scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We evaluate the performance of our algorithm in two aspects: the ability to detect humans (bounding boxes) and accuracy of the dense pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To evaluate the performance of our models in detecting humans, we calculate the standard average precision (AP) of person bounding boxes at multiple IOU thresholds ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In addition, by MS-COCO [15] definition, we also compute AP-m for median bodies that are enclosed in bounding boxes with sizes between 32 × 32 and 96 × 96 pixels in a normalized 640 × 480 pixels image space, and AP-l for large bodies that are enclosed in bounding boxes larger than 96 × 96 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To measure the performance of DensePose detection, we follow the original DensePose paper [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We first compute Geodesic Point Similarity (GPS) as a matching score for dense correspondences: 𝐺𝑃𝑆𝑗 = 1 |𝑃𝑗 | ∑︁ 𝑝 ∈𝑃𝑗 exp( −𝑔(𝑖𝑝, ˆ𝑖𝑝)2 2𝜅2 ), (4) where 𝑔 calculates the geodesic distance, 𝑃𝑗 denotes the ground truth point annotations of person 𝑗, 𝑖𝑝 and ˆ𝑖𝑝 are the estimated and ground truth vertex at point 𝑝 respectively, and 𝜅 is a normalizing parameter (set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='255 according to [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' One issue of GPS is that it does not penalize spurious predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Therefore, estimations with all pixels classified as foreground are favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To alleviate this issue, masked geodesic point similarity (GPSm) was introduced in [8], which incorporates both GPS and segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The formulation is as follows: 𝐺𝑃𝑆𝑚 = √ 𝐺𝑃𝑆 · 𝐼, 𝐼 = 𝑀 ∩ ˆ𝑀 𝑀 ∪ ˆ𝑀 , (5) where 𝑀 and ˆ𝑀 are the predicted and ground truth foreground segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Next, we can calculate DensePose average precision with GPS (de- noted as dpAP· GPS) and GPSm (denoted as dpAP· GPSm) as thresh- olds, following the same logic behind the calculation of bounding box AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 Implementation Details We implemented our approach in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We set the training batch size to 16 on a 4 GPU (Titan X) server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We empirically set 𝜆𝑑𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6, 𝜆𝑘𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3, 𝜆𝑡𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We used a warmup multi-step learning rate scheduler and set the initial learning rate as 1𝑒 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The learning rate increases to 1𝑒 − 3 during the first 2000 iterations, then decreases to 1 10 of its value every 48000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We trained for 145, 000 iterations for our final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 WiFi-based DensePose under Same Layout Under the Same Layout protocol, we compute the AP of human bounding box detections as well as dpAP· GPS and dpAP· GPSm of dense correspondence predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Results are presented in Table 1 and Table 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Method AP AP@50 AP@75 AP-m AP-l WiFi 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 Table 1: Average precision (AP) of WiFi-based DensePose un- der the Same Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' From Table 1, we can observe a high value (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2) of AP@50, indicating that our model can effectively detect the approximate locations of human bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The relatively low value (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6) for AP@75 suggests that the details of the human bodies are not perfectly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 plitude 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 0 5 10 15 20 25 Subcarrier Index 3 2 1 Phase 0 1 2 3 5 10 15 20 25 0 Subcarrier IndexMethod dpAP · GPS dpAP · GPS@50 dpAP · GPS@75 dpAP · GPSm dpAP · GPSm@50 dpAP · GPSm@75 WiFi 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 Table 2: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based DensePose under the Same Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' A similar pattern can be seen from the results of DensePose estimations (see Table 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Experiments report high values of dpAP · GPS@50 and dpAP · GPSm@50, but low values of dpAP · GPS@75 and dpAP · GPSm@75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This demonstrates that our model performs well at estimating the poses of human torsos, but still struggles with detecting details like limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 Comparison with Image-based DensePose Method AP AP@50 AP@75 AP-m AP-l WiFi 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 Image 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 Table 3: Average precision (AP) of WiFi-based and image- based DensePose under the Same Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All met- rics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1, since there are no manual annota- tions on the WiFi dataset, it is difficult to compare the performance of WiFi-based DensePose with its Image-based counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This is a common drawback of many WiFi perception works including [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Nevertheless, conducting such a comparison is still worthwhile in assessing the current limit of WiFi-based perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We tried an image-based DensePose baseline "R_50_FPN_s1x_legacy" finetuned on R101-Pseudo-GT under the Same Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In addition, as shown in Figure 9 and Figure 10, though certain defects still exist, the estimations from our WiFi-based model are reasonably well compared to the results produced by Image-based DensePose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In the quantitative results in Table 3 and Table 4, the image-based baseline produces very high APs due to the small difference between its ResNet50 backbone and the Resnet101 backbone used to generate R101-Pseudo-GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Our WiFi-based model has much lower absolute metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, it can be observed from Table 3 that the difference between AP-m and AP-l values is relatively small for the WiFi-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We believe this is because individuals who are far away from cameras occupy less space in the image, which leads to less information about these subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' On the contrary, WiFi signals incorporate all the information in the entire scene, regardless of the subjects’ locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 Ablation Study This section describes the ablation study to understand the effects of phase information, keypoint supervision, and transfer learning on estimating dense correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Similar to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4, the models analyzed in this section are all trained under the same- layout mentioned in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We start by training a baseline WiFi model, which does not in- clude the phase encoder, the keypoint detection branch, or transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The results are presented in the first row of both Table 5 and Table 6 as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Addition of Phase information: We first examine whether the phase information can enhance the baseline performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' As shown in the second row of Table 5 and Table 6, the results for all the metrics have slightly improved from the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This proves our hypothesis that the phase can reveal relevant information about the dense human pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Addition of a keypoint detection branch: Having established the advantage of incorporating phase information, we now evaluate the effect of adding a keypoint branch to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The quantita- tive results are summarized in the third row of Table 5 and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Comparing with the numbers on the second row, we can observe a slight increase in performance in terms of dpAP·GPS@50(from 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8) and dpAP·GPSm@50 (from 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 to 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8), and a more noticeable improvement in terms of dpAP·GPS@75 (from 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 to 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9) and dpAP·GPSm@75 (from 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 to 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' This indicates that the keypoint branch provides effective references to dense pose estimations, and our model becomes significantly better at detecting subtle details (such as the limbs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Effect of Transfer Learning: We aim to reduce the training time for our model with the help of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' For each model in Table 5, we continue training the model until there are no significant changes in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The last row of Table 5 and Table 6 represents our final model with transfer learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Though the final performance does not improve too much compared to the model (with phase information and keypoints) without transfer learning, it should be noted that the number of training iterations decreases significantly from 186000 to 145000 (this number includes the time to perform transfer learning as well as training the main model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 Performance in different layouts All above results are obtained using the same layout for training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, WiFi signals in different environments exhibit significantly different propagation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Therefore, it is still a very challenging problem to deploy our model on data from an untrained layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' To test the robustness of our model, we conducted the previous experiment under the different layout protocols, where there are 15 training layouts and 1 testing layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The experimental results are recorded in Table 7 and Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We can observe that our final model performs better than the baseline model in the unseen domain, but the performance de- creases significantly from that under the same layout protocol: the AP performance drops from 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 and dpAP·GPS drops from 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' However, it should also be noted that the image-based model suffers from the same domain generalization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We Method dpAP · GPS dpAP · GPS@50 dpAP · GPS@75 dpAP · GPSm dpAP · GPSm@50 dpAP · GPSm@75 WiFi 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 Image 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 Table 4: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based and image-based DensePose under the Same Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Method AP AP@50 AP@75 AP-m AP-l Number of Trained Iterations Amplitude-only Model 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 174000 + Sanitized Phase Input 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 180000 + Keypoint Supervision 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 186000 + Transfer Learning 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 145000 Table 5: Ablation study of human detection under the Same-layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Method dpAP · GPS dpAP · GPS@50 dpAP · GPS@75 dpAP · GPSm dpAP · GPSm@50 dpAP · GPSm@75 Amplitude-only Model 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 + Sanitized Phase Input 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 + Keypoint Supervision 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 + Transfer Learning 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 Table 6: Ablation study of DensePose estimation under the Same-layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Method AP AP@50 AP@75 AP-m AP-l WiFi (base) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 WiFi (final) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 Image 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 Table 7: Average precision (AP) of WiFi-based and image-based DensePose under the Different Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Method dpAP · GPS dpAP · GPS@50 dpAP · GPS@75 dpAP · GPSm dpAP · GPSm@50 dpAP · GPSm@75 WiFi (base) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 WiFi (final) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='5 Image 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 Table 8: DensePose Average precision (dpAP · GPS, dpAP · GPSm) of WiFi-based and image-based DensePose under the Differ- ent Layout protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' All metrics are the higher the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' believe a more comprehensive dataset from a wide range of settings can alleviate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content='8 Failure cases We observed two main types of failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (1) When there are body poses that rarely occurred in the training set, the WiFi-based model is biased and is likely to produce wrong body parts (See exam- ples (a-b) in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (2) When there are three or more concurrent subjects in one capture, it is more challenging for the WiFi-based model to extract detailed information for each individual from the amplitude and phase tensors of the entire capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (See examples (c-d) in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We believe both of these issues can be resolved by obtaining more comprehensive training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK In this paper, we demonstrated that it is possible to obtain dense human body poses from WiFi signals by utilizing deep learning architectures commonly used in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Instead of directly training a randomly initialized WiFi-based model, we explored rich supervision information to improve both the performance and training efficiency, such as utilizing the CSI phase, adding keypoint detection branch, and transfer learning from an image-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The performance of our work is still limited by the public training data in the field of WiFi-based perception, especially under different layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' In future work, we also plan to collect multi-layout data and extend our work to predict 3D human body shapes from WiFi signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' We believe that the advanced capability of dense perception (a) (b) (c) (d) Figure 8: Examples pf failure cases: (a-b) rare poses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' (c-d) Three or more concurrent subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The first row is ground truth dense pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' The second row illustrates the predicted dense pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' could empower the WiFi device as a privacy-friendly, illumination- invariant, and cheap human sensor compared to RGB cameras and Lidars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' REFERENCES [1] Fadel Adib, Chen-Yu Hsu, Hongzi Mao, Dina Katabi, and Frédo Durand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNAyT4oBgHgl3EQfavfQ/content/2301.00250v1.pdf'} +page_content=' Capturing the Human Figure through a Wall.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..f5ded034ac17252e0e556c080791d7701e667c16 --- /dev/null +++ b/xtAyT4oBgHgl3EQfa_c9/content/tmp_files/2301.00252v1.pdf.txt @@ -0,0 +1,1704 @@ +A Comparative Study of Image Disguising Methods +for Confidential Outsourced Learning +Sagar Sharma +Bytedance +Seattle, WA +sagar.sharma@bytedance.com +Yuechun Gu, Keke Chen +Trustworthy and Intelligent Computing Lab +Marquette University, Milwaukee WI +{ethan.gu, keke.chen}@marquette.edu +Abstract +Large training data and expensive model tweaking are +standard features of deep learning for images. As a result, +data owners often utilize cloud resources to develop large- +scale complex models, which raises privacy concerns. Existing +solutions are either too expensive to be practical or do not +sufficiently protect the confidentiality of data and models. +In this paper, we study and compare novel image disguising +mechanisms, DisguisedNets and InstaHide, aiming to achieve +a better trade-off among the level of protection for outsourced +DNN model training, the expenses, and the utility of data. +DisguisedNets are novel combinations of image blocktization, +block-level random permutation, and two block-level secure +transformations: random multidimensional projection (RMT) +and AES pixel-level encryption (AES).InstaHide is an image +mixup and random pixel flipping technique [16]. We have +analyzed and evaluated them under a multi-level threat model. +RMT provides a better security guarantee than InstaHide, +under the Level-1 adversarial knowledge with well-preserved +model quality. In contrast, AES provides a security guarantee +under the Level-2 adversarial knowledge, but it may affect +model quality more. The unique features of image disguising +also help us to protect models from model-targeted attacks. We +have done an extensive experimental evaluation to understand +how these methods work in different settings for different +datasets. +I. INTRODUCTION +Deep Neural Networks (DNN) have shown impressive +performance across diverse domains such as image classifi- +cation, natural language processing, speech recognition, and +recommendation systems. However, DNN training is resource- +intensive and time-consuming, requiring large training data, +careful model architecture selection, and exhaustive model +parameter tweaking. As a result, data owners or model de- +velopers often utilize multiple cloud GPUs or online model +training services, such as Google Colab, to lower their costs. +Despite its popularity, outsourcing DNN learning to the +cloud raises privacy and security concerns about the sensi- +tive training data and trained models [31], [5]. On the one +hand, cloud users cannot verifiably prevent the cloud provider +from getting access to their data. In practice, using public +clouds often means fully trusting your cloud provider. On +the other hand, public cloud providers are not immune to +security attacks, which may lead to data breaches through +insider [4], [5] and external attacks [25], [37]. Additionally, +membership inference attacks [34], model inversion attacks +[10], and adversarial example exploration [3], [28] can be +applied to models directly to explore the training examples in +DNN learning. Therefore, data and models in training, testing, +and transferring between the cloud and the client are seriously +threatened. +There have been a few efforts trying to address this critical +issue. However, all of them are not satisfactory. +• Encrypted data and models. The first approach is to train +encrypted DNN models over encrypted data. However, +due to the large training data and expensive training pro- +cess in deep learning, cryptographic model training ap- +proaches are too expensive to be practical. A recent study +on training small-scale neural networks +[26] (e.g., just +two layers with a maximum of 128 neurons per layer) has +shown astonishingly high communication, computation, +and storage costs. As a result, cryptographic approaches +are often limited to training small models [26], [30] or +securely applying trained DNNs for prediction [40], [29], +[16]. +• Federated learning. Another possible solution is to par- +tition the dataset and the learning task into sensitive and +non-sensitive partitions and use cloud-client federated +learning [19]. The non-sensitive portion, assuming it is +much larger than the sensitive one, is exported to and +processed by the cloud. Correspondingly, the learning +process is partitioned and distributed between the cloud +and the client, ensuring that the intermediate information +exchanged between the two parties does not breach +privacy. Collaborative deep learning framework enhanced +with differential privacy [32], [1] may be tweaked into +such a partition-based setting. However, reported attacks +[14] allow an adversarial collaborator (e.g., a compro- +mised cloud in the cloud-client scenario) to generate +images resembling the sensitive classes owned by the +victim parties (the trusted data owner). +• Trusted +execution +environments. +Hardware-assisted +trusted execution environments (TEEs), such as Intel +SGX, can also be applied to deep learning in the cloud. +The idea is to create a secure enclave in the specific +memory area (enclave page cache (EPC)) so that no +arXiv:2301.00252v1 [cs.CR] 31 Dec 2022 + +other process/thread can access the content in the +enclave. Memory pages are also automatically encrypted +when they are swapped to the disk. However, recent +studies on side-channel attacks [8] make this approach +challenging to develop and deploy. Attackers can peek +or infer the plaintext content inside the secure enclave +via side channels, such as page fault interrupts and cache +loading. +Furthermore, to work with GPUs, costly cryptographic +approaches have been applied to achieve partial data +confidentiality in transferring data between CPU and +GPU [35], [27], which does not meet the performance +requirement for training large DNNs or with large training +data. GPU manufacturers may develop TEEs for GPU 1 +However, it’s to be tested to determine how secure it is +in terms of side-channel attacks. +Researchers have also explored the application of differ- +ential privacy (DP) [6] in distributed (federated) learning +scenarios [32] or a trusted central training server [1]. However, +DP works for the setting of sharing data and models without +breaching individual training examples’ privacy. It does not +meet the need for data and model confidentiality. +Scope and contributions. This study compares two dif- +ferent image disguising approaches: InstaHide [16] and our +recently developed DisguisedNets. These image disguising +mechanisms protect the training data and also possibly the +learned models by casting training data into a confidential +transformed space where powerful DNN models can still +learn features and patterns distinguishing image classes and +leverage the power of GPUs in the cloud. The intuition +is twofold. (1) Apply appropriate transformations and data +protection mechanisms so that the disguised images cannot be +effectively reconstructed and re-link to the original images. (2) +Meanwhile, powerful deep learning techniques can still pick +up the unique topological and geometric features preserved +in the transformed space to distinguish the originally defined +classes of images in the transformed space. By doing so, the +tie between the original training data and the learned model +in the transformed space is broken, which also disables any +model-based exploration [3], [28], [34], [10]. In the end, we +can approximately preserve the image distinguishability for the +target classification task while minimizing the recoverability +of individual images. +There are several unique contributions. +1) We have designed two image disguising mechanisms: +AES-based (AES) and random-projection-based (RMT) +for image-based DNN learning to preserve training data +and model confidentiality in outsourced training. The +goal is to study and achieve a good balance between the +utility of disguised images and the level of confidential- +ity protection. +2) We have carefully analyzed the potential attacks under +the outsourced deep learning settings and the resilience +of disguising mechanisms to attacks on data and model +1Nvidia has announced a GPU TEE in their Hopper architecture. +confidentiality. So that users can choose the correspond- +ing method under their preferred threat model. +3) We have conducted extensive experimental evaluations +on public datasets to show the trade-offs of different dis- +guising schemes and related parameter settings between +data utility and their resilience to attacks. We also show +how the disguising methods work to protect models from +model-targeted attacks and +II. RELATED WORK +Sensitive deep learning assets may include training data, +models, and online testing data. Protection methods may target +the model training phase or the application (i.e., inference) +phase when both phases can be exported to the public cloud. +However, typically, the computational complexity and the +demand on resources of model training is far more than +model application. Thus, many studies have focused on the +application phase, e.g., a cloud-based model inference service +hosting the trained model and making predictions for a user- +provided image [12], [29], [17]. +In contrast, due to the high computational complexity of +training algorithms and the large size of training data, there +is no practical cryptographic approach, e.g., homomorphic +encryption or secure multi-party computing based approaches +for protecting model training. A few recent studies in this +direction have shown prohibitively high costs even for a +small neural network model [26], [30]. Trusted execution +environments (TEE) with masked GPU operations are applied +to speed up training [27], [35]. However, no TEE-based deep +learning method has addressed severe side-channel attacks [8]. +Researchers have looked at protection methods for image +training data in the outsourcing context. Noise addition [7], +image blurring [24], and morphing [23] are weak as the visible +features of the images are still perceivable and understand- +able. Such transformations do not defeat simple visual re- +identification. Figure 1 shows how easy it is to visually re- +identify the content in the original images by observing the +transformed ones with the mentioned techniques. +A recent method InstaHide [16] applies the idea of mixing +up images [41] in a training set with public images with linear +combination to obfuscate the content, along with randomized +signs of pixel values for further protection. However, it is +vulnerable to image reconstruction attacks [2] that need only +to know the disguised images and public image sets (i.e., the +Level-1 adversarial knowledge, as we will discuss). +Thus, on the one end, existing disguising mechanisms are +too weak to protect almost nothing. On the other end, if +an encryption mechanism, e.g., homomorphic encryption, or +a complex cryptographic protocol, is applied, such linking +or reconstruction would be impossible. However, the current +cryptographic schemes incur extremely high costs in almost +all aspects of computation, communication, and storage. Thus, +they are impractical for resource-intensive tasks like training +a DNN model. Hardware-assisted approaches, such as TEEs, +are still under investigation to ensure the expected security +properties. Along with all these possible approaches, we aim + +TABLE I +RELATED WORK ON TRAINING PHASE PROTECTION. +Sample Related +Work +Method +Weaknesses +Strengths +Mohassel et al. +[26] +Train DNN models over masked or +encrypted data with cryptographic +protocols. +Involve high communication, computation, and +storage costs; Require extensive re-design and +custom implementation of the DNN architec- +ture; Involve iterative interactions between the +data owner and cloud provider. +Provide semantic security. Model +quality is fully preserved. +Tramer +et +al. +[35] and Ng et +al. [27] +Use TEEs for confidential CPU op- +erations and masked data for con- +fidential GPU operations +side channel attacks are an unaddressed con- +cern +More efficient than cryptographic +protocols, but GPU operations on +masked data are still expensive +Abadi et al. [1] +and Shokri et +al. [33] +Apply differential privacy to ran- +domize intermediate gradients +Aim to share trained models, and thus does +not preserve model confidentiality; result in +significant drop in model quality; vulnerable +to generative adversarial network attacks. +Preserve individuals’ privacy. +Fan [7] +Train shallow neural network lo- +cally and outsources intermediate +representation to the cloud for +deeper training. +The intermediate representation of images re- +veals the visual characteristics of the related +images, vulnerable to visual re-identification +attacks. +NA +Li et al [24] +Applies differential privacy to hide +sensitive pixels in images +Does not hide the global visual characteristic +content of images, vulnerable to visual re- +identification attacks +NA +Zhang +et +al. +[41] and Huang +et al. [16] +Mix-up images from the training +set and public domain with random +selection and weighing to hide the +content of the sensitive image. +Vulnerable to ciphertext-only (Level 1 adver- +sarial knowledge) image re-construction at- +tacks; May expose trained model to wide va- +riety of model-based and membership attacks +for the inside-dataset setting. +Fast training; No changes to the +training architecture. +original +DP ε = 0.5 +noisy +original +(a) +original +15.8 dB +16.9 dB +blurred +original +(b) +original +morphed +(c) +Fig. 1. +(a) Differentially private noise addition to images +[7]; (b) The +reconstructed blurred images in PrivyNet [24]; (c) The morphed images [23] +to explore and develop new image-disguising methods to +achieve good balances among costs, data utility, and security +guarantees. Table I summarizes the current work in protecting +data and models in the training phase. +III. THREAT MODELING +We are concerned with the confidentiality of the sensitive +training image data and the DNN models in the outsourced +training phase. Here, we make some relevant security assump- +tions for our disguising mechanisms. 1) We consider the cloud +provider to be an honest-but-curious adversary, which implies +that a curious provider will still honestly deliver desired results +to the data owner. However, it may keep a copy of the data +and programs it can observe. 2) The adversary can observe +the training data, the training process, and the trained models, +including the structure of the DNN architecture and parameter +settings for training. Thus, they can probe the observed items +with methods such as image reconstruction, re-identification, +and membership attacks. (3) We do not address evasive attacks +and poisoning attacks [3], [28], where adversaries will tamper +with the training data, which can be guarded with training data +integrity checking. (5) The client infrastructure and commu- +nication channels are secure. +Assets to Be Protected. Under a certain protection mecha- +nism, we generalize that the training data D is transformed +to fkey(D), and the model M += M(D) is changed to +gkey(D). The attacker might want to know whether an image +is likely used to train a model, e.g., the membership inference +attack [34], observe training images that may contain sensitive +objects, or steal a proprietary training dataset. The attack might +also target the model if the model is exposed, e.g., using +model-inversion attacks [9] to explore the private information +of training data. +Adversarial Prior Knowledge. The adversary may have + +Created by UNREGISTERED +http./tantamorh.com +FantaMorphtwo levels of prior knowledge. For each level, we may design +a disguising technique. +• Level-1: They may know what the model is used for, e.g., +the background application, the distribution of the data, +e.g., face images, and the type of disguising technique +used, but do not know the disguising parameter setting +for a specific dataset that serves as the secret key to the +protocol. +• Level-2: In addition to Level-1 knowledge, they may try +to obtain pairs of images and their disguised versions via +other attacking channels (not including the ones they are +targeting). They hope to use these known pairs to explore +various image reconstruction attacks. +Potential Attacks. Recent studies have shown that attackers +can explore training/testing examples and models, for example, +to find adversarial examples misleading the prediction of deep +neural networks [3], [28]. Such attacks depend on adversaries’ +clear understanding of the original image data and the ability +to access the developed models freely. Outsourced learning +without protection makes these attacks easier to deploy. +With a protection mechanism on data and models, we +consider a fundamental attack: training image re-identification +that aims at linking the protected images to identifiable original +images. We introduce a model-based re-identification test – +DNN examiner, which uses a model trained on the original +data to tell whether a protected image is re-identifiable. Note +that some related methods [7], [24] are not resilient to human +visual re-identification, which does not protect confidentiality. +Since the image disguising mechanisms break the link between +the original training data and the learned models (in the +transformed space), the existing model-oriented attacks do not +work anymore without successfully breaking the disguising +mechanism. Attackers thus depend on reconstruction attacks: +reverse the disguising mechanism to approximately reconstruct +the original images and then try to re-identify the reconstructed +images. We use the DNN examiner approach to evaluate how +successful a reconstruction attack works in our experiments. +IV. DISGUISEDNETS – A NOVEL IMAGE DISGUISING +MECHANISM FOR OUTSOURCED DEEP LEARNING +In the following, we will introduce an image disguising +framework that incorporates pixel-block partitioning, random +block permutation, and block-wise transformations of images +along with noise additions. The premise is that after the +dramatic transformation, it is difficult to link the disguised +images to the original images, while, unlike pure encryption +schemes, it still preserves some essential patterns for distin- +guishing between classes of images that allow DNN learning +methods to capture. This amalgam of multiple transformations +provides a sufficiently large parameter space so that the attacks +are computationally intractable (Section V) under the Level-1 +prior knowledge. +Figure +2 depicts the DisguisedNets framework. A data +owner disguises her private images before outsourcing them +to the cloud for DNN learning. She can either fully outsource +the entire image datasets and the learning procedure to the +cloud or selectively retain sensitive images in the cloud-client +partitioning setting. She transforms all of her images using +one secure transformation key secret to her. Note that this +transformation should be at a reasonable cost, practical for a +client’s infrastructure to process. +Cloud Provider +DNN +Model +GPU Processing +Data Owner +Train +Disguised test images {TK(Xnew)} +Predictions {y’new} +Disguised Images +{TK(Xi),yi} +Fig. 2. Image disguising framework for DNN learning. +Specifically, assume the data owner owns a set of images +for training, notated as pairs D = {(Xi, yi)}, where Xi is +the image pixel matrix (l × m and 3 × l × m for grayscale +and RGB images respectively) and yi the corresponding label. +We formally define a disguising mechanism as follows. Let +the disguising mechanism be a transformation TK, where K +is the secret key that depends on the selected perturbation +techniques. By applying image disguising, the training data +is transformed to {(T(Xi), ci)} with ci mapped to 0, 1, . . . +randomly representing the classes yi. The original model is a +function M(D), which is learned with a DNN learning method +G: G(D) → M(D). The image disguising mechanism enables +the same learning method G to be applied to the transformed +data directly without any modification: G(DT ) → MT (DT ). +For any new data Xnew, the model application (or inference) +is defined as MT (T(Xnew)), i.e., the new data transformed +with the same key. To make such a transformation method +practical for modeling, i.e., a model trained with transformed +data still working satisfactorily, a user may expect the error +of modeling is not far away from the original model’s. Thus, +a utility-preserving mechanism should have +|Err(MT ) − Err(M)| < δ +where δ is the level of model quality degradation acceptable +to the user. While for a specific DNN modeling method and a +specific dataset, it’s difficult to theoretically justify what this +gap will be, one can always directly evaluate the model quality +to check whether it is acceptable for the application. We have +empirically evaluated the δ levels for different mechanisms, +datasets, and a few popular DNN modeling architectures in +experiments. +A. Pixel-Block Partitioning and Block-based Random Permu- +tation +In this section, we present one way of image transformation: +image block permutation, that will be combined with other +mechanisms later. +An image Xl×m is first partitioned into t blocks of uniform +size r × s. If we label the blocks sequentially as v =< +1, 2, 3, 4, . . . , t >, a pseudorandom permutation of the image, +Tπ(X), shuffles the blocks and reassemble the corresponding + +E(A) +E(A;) +Data Owner +On-demand +Processing +Models +E(Ak) +Cluster +Model +Cloud +consumers +Data contributorsimage accordingly. Block-based permutation preserves the in- +block information and the relative positions of related blocks. +Thus, we understand it preserves a great amount of information +for effective modeling. However, while the permutation may +break the global patterns of the images and achieve good +visual privacy already, the between-block characteristics such +as boundaries, color, content shape, and texture of the origi- +nal neighboring blocks may provide clues for adversaries to +recover the original image – imagine the jigsaw puzzle! For +large t, such attacks can be time-consuming due to the vague +similarity between block boundaries. However, with the prior +knowledge: a pair of original image and its block-permuted +image, it’s not difficult to solve such a jigsaw puzzle. Thus, +we use this as an auxiliary step enhancing other steps in the +disguising framework. +B. Pixel-Block Transformations +Next, we establish pixel-block-level protection mechanisms +that aim to preserve the data utility for DNN modeling +and further increase the resilience to attacks. We consider +two candidate mechanisms: random projection and encryption +schemes, and discuss their characteristics. Specifically, when +an image is partitioned into t pixel blocks for random permu- +tation, we get a list of t parameters {Ki, i = 1 . . . t}, one for +the pixel-block at the same position across the whole dataset. +We name the specific position of the pixel block in the image +the pixel-block position. The list of parameters acts as a secret +key and will be shared, together with the permutation key, +by each image in the dataset. The purpose of this setting is to +maximize the preservation of distinguishable patterns between +image classes – i.e., a pair of similar image patterns (blocks) +can still be transformed to another pair of (likely) similar ones +after applying the disguising mechanism. +Randomized Multidimensional Transformation (RMT). +For an image represented as a pixel matrix X, a general linear +transformation can be defined as G(X) = R(X + ∆), where +Rm×m is a random orthogonal matrix generated following the +Haar distribution [11], or a random invertible matrix, e.g., a +random projection matrix [39], and ∆ is an optional noise +matrix. We call this method the randomized multidimensional +transformation. When an image is partitioned into t blocks +for random permutation, we prepare a list of random matrices +{Ri, i = 1..t}, one for each image-block position and share +this list for each image. Such transformation is known to +preserve (or approximately preserve by random projection) the +Euclidean distance between columns of the matrix X. For real +application, we may arrange the pixel blocks accordingly to +form the column of X. For example, a 4x4 pixel matrix can +be partitioned into 4 2x2 block to preserve the smaller block- +level similarity with RMT. Figure 3 shows the effects of RMT +on MNIST and CIFAR-10 datasets. +AES Block Transformation (AES). The existing AES +encryption schemes typically use 128-bit encryption keys, +which encode every 16-byte data block sequentially. If we use +AES for pixel-block encryption, assuming each pixel is stored +in one byte, 16 original pixels are mapped to 16 encrypted +Algorithm 1 DN RMT (X, t, Key) +Require: X: image of size l × m; t: number of blocks; Key += {permutation key, transformation matrices, noise level +∈ [0, N]} +1: r, s ← compute image block size with l × m and t; +2: Partition image Xl×m into blocks X1, X2, . . . , Xt; +3: Shuffle the image blocks pseudorandomly with permuta- +tion key +4: for each block i, i = 1 . . . t do +5: +∆i ← Generate random matrix with elements from +the uniform distribution in [0,N]; +6: +use the transformation matrix at the position i: Ri; +7: +Yi ← Ri(Xi + ∆i); +8: end for +9: Re-assemble {Yi} to make the transformed image Y and +return Y ; +8 x 8 blocks +2 x 2 blocks +32 x 32 image +28 x 28 image +T{RMT, π, N}(X) +T{RMT, π, N}(X) +2x2 blocks, N=0 2x2 blocks, N=100 +2x2 blocks, N=0 +2x2 blocks, N=25 +Fig. 3. Block-wise RMT+Noise on MNIST and CIFAR-10 images. +bytes (pixels), and a whole pixel block is encoded to 16-byte +units. Putting all encrypted pixel blocks together, we get a +disguised image. For clear presentation, when we talk about +AES encryption block, i.e., 16 bytes for a 128-bit encryption +key, we use the 16-byte “encryption unit”, which are different +from “pixel blocks” we have been using previously in our +image disguising framework. Figure 4 shows some example +AES transformations on images. +AES - ECB +AES - CBC +original +AES - ECB +AES - CBC +original +Fig. 4. Pixel-block based AES encryption of MNIST and CIFAR-10 images. +We consider two AES modes in our design. (1) We observe +that with the AES Cipher Block Chaining (CBC) mode, any +pixel-level change in the pixel block between two images + +0 +5 +10 +15 +20 +25 +70 +5 +10 +15 +20 +25original +Noise Addition ++ Block-wise +Encryption + +Permutation +Encrypted +with padding +Scaled up, +Encrypted, & +Scaled-down +Fig. 5. AES-ECB encryption of MNIST image with different strategy. +will result in different encoding results for most 16-byte AES +blocks in this pixel block position, making it not ideal for +our purpose. (2) Then, we turn to the AES Electronic Code +Book (ECB) mode that can be considered as a fixed mapping +function between 16-byte original data to 16-byte encrypted +data. Different from CBC, the neighboring 16-byte blocks do +not affect the encoding of the current block. This matches our +requirement of data utility preservation, e.g., to preserve the +block-level distinguishable patterns after the transformation. +To preserve more information, based on the intuition of +smaller blocks preserving more inter-pixel-block information, +we can also use unit sizes smaller than the regular size, which +is 16 pixels for 128-bit ECB. The method is to scale up the +image first, e.g., from 32x32 to 256x256 (where each pixel is +duplicated eight times), then encrypt it by 16-pixel units, and +finally scale down to the size 32x32. Please refer to Figure +5 for the detailed example. It’s equivalent to encrypt 2-pixel +units in the original 32x32 images. We found by reducing the +block size, the model quality can be improved with the cost of +lower attack resilience to the Level-2-knowledge-based attack. +Algorithm 2 DN AES (X, k, t, Key) +Require: X: image of size l × m; k: scale-up factor; t: +number of blocks; Key = {permutation key, AES keys, +p = probability of salt-pepper noise } +Ensure: the selection of k and t results in image blocks that +can be further partitioned to 4x4 pixel patches; +1: Xlk×mk ← scaled up the image; +2: r, s ← compute image block size with lk × mk and t; +3: Partition image Xl×m into blocks X1, X2, . . . , Xt; +4: Shuffle the image blocks pseudorandomly with permuta- +tion key; +5: for each block i, i = 1 . . . t do +6: +Yi ← for each pixel in block Xi, with the probability +p, it’s randomly turned to white or block pixel (salt-pepper +noise); +7: +E(Yi) ← with the AES CBC mode, every 16-byte +segment (4x4 pixel patch) is encypted to 16 bytes of AES +digest with AES key i; +8: end for +9: re-assemble image blocks E(Yi) and return E(Y ) +C. Complexity Analysis +The additional costs of the disguising methods consist of the +encoding cost and the possible additional learning cost, i.e., it +may take more rounds to converge. We leave the second part +to the experimental evaluation and analyze the encoding cost +here. +For an image partitioned into t blocks with each block +l × m, the RMT transformation involves t matrix-matrix +multiplications and matrix additions. As the numbers t, m, and +l are all small, the cost of RMT per image is low: O(tlm2). +For an image of l×m with a scale-up constant of s, the AES- +128 encryption cost is l × m × s/16 times of AES encryption. +Our experimental evaluation shows that per image cost is less +than 10 ms and can be comfortably done by any PC or mobile +phone. +D. Model Protection via Image Disguising +Note that the models trained with disguised data work +only on disguised data. We show this property also protects +models from existing model-targeted attacks. So far, we have +seen model-inversion attacks [9], [42], membership-inference +attacks [34], [15], and model-extraction attacks [36], [18]. +Model-extraction attacks assume the attacker can freely +access the model, e.g., via a cloud-based prediction API. With +such a service, the attacker can try various images to collect +their outputs and then use the input-output pairs to reconstruct +the model. Our threat model assumes the attacker can copy or +save the trained model for analysis. Thus, the attacker does not +need to perform model-extraction attacks. As the models only +work on disguised test images, without the secret disguising +key, they are useless to the attacker. +Membership-inference attack aims to estimate the possi- +bility of a target example belonging to the training data of +a model. To perform such an attack, the attacker must first +apply the disguising method (with the secret key) to the +target example so that the model can be used. This step +effectively blocks the attack or at least significantly increases +the difficulty. To successfully conduct the MIA attack on the +disguised model, the attacker may need to manipulate an +authorized user to transform the example and intercept the +transformed one, which corresponds to the mentioned Level- +2 knowledge. Thus, the disguising mechanism establishes an +effective defensive line. +Model-inversion attack uses a learning procedure, e.g., a +GAN method [42], to progressively adjust randomly generated +or seed images from similar domains towards most likely +training examples. When applied to the models trained on +disguised data, the model-inversion attack recovers only the +disguised training data, not the original data. Again, the +disguising mechanism builds a defensive line on this attack. +We will show how the RMT mechanism works against model- +inversion attacks in experiments. +V. ATTACK ANALYSIS +This section aims to analyze the possible threats to the +proposed disguising mechanisms and clarify the applica- + +0 +5 +10 +15 +20 +25 +30 +OP +5 +10 +15 +20 +上 +25 +300 +5 +10 +15 +20 +25 +30 +OF +5 +10 +15 +20 +25 +300 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +303ble settings. With Level-1 adversarial knowledge, Disguised- +Nets mechanisms provide strong confidentiality protection, as +shown in the discussion of “brute-force attacks”. In contrast, +other related methods are still struggling with visual re- +identification by human eyes [24], [7] or disguised-image- +based reconstruction attacks [2]. We also analyze more so- +phisticated reconstruction attacks that depend on Level-2 ad- +versarial knowledge. +A. Level-1 Adversarial Knowledge and Attacks +Recall that Level-1 knowledge includes knowing the dis- +guised images and possibly the model domain, i.e., the types +of images and the background application. It is clear that +with only Level-1 knowledge, the brute-force attack on AES +schemes is not possible, and thus we focus on the scheme +using multidimensional projection. +Visual Re-identification. The first simple attack is to visu- +ally identify images by human attackers. We have shown that +simple methods like noise addition, morphing, and shallow- +network-based transformation are not resilient to this attack. +However, many other attacks may use re-identification as the +last step, i.e., reconstruction attacks. It’s inefficient for human +evaluators to check each image to determine the protection +level of an image disguising mechanism. Thus, we propose +the DNN examiner approach for evaluation purposes: let a +DNN trained on the original datasets to perform the visual +re-identification task for human evaluators. We will use DNN +examiners in experiments. +Brute-Force. The brute-force attack method for image re- +construction is to enumerate each possible parameter setting of +the disguising mechanism and then check the recovered result +with re-identification. As AES encryption is already resilient +to the brute-force attack, we examine the RMT method only. +Let’s start with a block-level transformation for any image +block i with RMT. With X′ +i = XiRi, the adversary knows +only X′ +i. In the brute force attack, the number of possible Xi +is determined by the number of possible Ri matrices. We show +that the number of possible Ri (even limited to orthogonal +ones) can be exponentially large for given parameters. +Proposition 1. For values encoded in h-bit finite field, there +are O(2hm) candidate orthogonal matrices Rm×m. +The proof is based on the theory of orthogonal matrix +group [13], the detail of which is skipped here. With a typical +setting in our experiments, e.g., h = 8 and m = 28 for the +MNIST dataset, the overall complexity is O(2224), which is +more than sufficient to protect from computationally-bounded +attackers. Combined with the random permutation of blocks, +the attack complexity is even higher. Thus, a brute-force attack +is generally impractical for the proposed methods. +Clustering Attack. Carlini et al. [2] utilized a clustering +method to attack InstaHide [16] disguised images. InstaHide +uses the random mix-up method to generate disguised images. +Depending on the random weight distribution, some disguised +images might be dominated by the same image, which likely +forms a cluster of images that can be used to de-mask and +de-noise. As InstaHide disguised images are essentially linear +combinations of plaintext images, the attack result can be +visually re-identified. +Important questions are whether our disguising methods +can generate images with clustering structures and whether +such clusters can be used to break our disguising methods. +To answer these questions, we visualize the disguised training +data with t-SNE [38] to understand the existence of clustering +structure in the Euclidean-distance space. Figure 6 shows +that RMT might preserve the clustering structures for some +datasets: for simpler datasets like MNIST and FASHION, the +clustering structure is well preserved, while others do not. +In contrast, AES does not preserve any clustering structure, +as shown in Figure 7. While AES not preserving clustering +structures to leave less information to attackers, it also affects +data utility and leads to lower-quality models, as we will show +in experiments. +(a) RMT on MNNST +(b) RMT on CIFAR10 +Fig. 6. t-SNE visualization of RMT disguised datasets (4x4 blocks). Colors +represent different labels. A dense area covered with one color means that the +clustering structure matches the label distribution well for the specific subset. +Next, can such preserved clustering structures be used for +attacks? An attack on InstaHide [2] has used image clusters to +de-noise and de-mask, as InstaHide uses the random weights +mix-up mechanism. However, unlike InstaHide, clustering +structures of RMT-disguised images do not help attackers +identify original images. However, it may help attackers infer +additional information with other domain knowledge. For +example, if the original training samples’ distribution (and + +100 +label +1 +75 +2 +3 +4 +5 +50 +9 +25 +0 +-25 +-50 +-75 +80 +-60 +-40 +20 +0 +20 +40 +60 +80 +Xlabel +80 +0 +2 +3 +4 +60 +5 +6 +7 +8 +9 +40 +20 +0 +-20 +-40 +20 +0 +20 +40 +60 +X(a) AES on MNIST +(b) AES on CIFAR10 +Fig. 7. t-SNE visualization of AES disguised datasets (4x4 blocks) +thus the clustering structure) is also known, it may allow the +attacker to identify the mapping between a specific cluster of +original images and a cluster of disguised images. As distances +between samples are not preserved, it’s still difficult to figure +out the sample-to-sample mapping. +B. Level-2 Adversarial Knowledge and Attacks +We move one more step further to study the more challeng- +ing issue: what if a powerful adversary can obtain additional +knowledge: pairs of original images and their transformed +ones? This assumption corresponds to the chosen-plaintext +attack in cryptographic analysis [20]. This study helps us +understand when we should not use a proposed disguising +method. Below, we focus on the codebook attack on the AES- +ECB-based disguising mechanism and the regression-based +attack on the RMT mechanism. +Codebook Attack. The assumption is that the adversary is +knowledgeable of the encryption procedure described previ- +ously but does not have the encryption/decryption key. Since +the AES ECB method is deterministic, the basic attack is to +build a mapping (i.e., the codebook) between the plaintext unit +(e.g., the 16-byte pixel block) and its encrypted counterpart +(e.g., the 16-byte AES cipher block). By processing the +known image pairs, the adversary constructs a codebook as +a dictionary mapping 16-byte pixel blocks to encrypted 16- +byte blocks. Since different images, especially those in the +same class, might share some 16-byte pixel blocks, some 16- +byte encrypted blocks in the targeted images are likely already +in the codebook, which will be used to recover the original +blocks. For encrypted pixel blocks not present in the codebook, +the adversary may use a fixed pattern, e.g., all zero values or +most likely values to pad. By repeating this procedure for +each 16-byte block, the adversary can recover some parts of +the image, which can be further re-identified via human eyes +or models at the adversary’s hands. +Possible mitigation methods. Let the hit rate be defined +as the probability that an encrypted pixel block can find a +match in the codebook. This attack can become less effective +if we add salt-and-pepper noises to the original images before +encoding. This step will reduce the hit rate significantly and +make the mapping non-unique: the same 16-byte pixel block +can be mapped to different ciphertexts. We will evaluate the +success rate of this attack in experiments, using the accuracy +that the DNN examiner trained with the original image space +correctly classifies the reconstructed images. +Projection Matrix Estimation Attack. Note that noise +addition can easily defend the RMT method from the code- +book attack, which is already a part of the RMT method. +However, if the adversary has obtained enough original and +transformed image pairs, there is a possibility that the trans- +formation matrix might be estimated with linear regression. +Specifically, a noise-added block-wise transformation, e.g., +Yi = Ri(Xi + ∆i), where ∆i is a random noise matrix, re- +generated for each image block Xi, and drawn uniformly at +random from [0, N] where N is the tunable noise level. With +enough known pairs of (Xi, Yi), the regression method can be +applied to estimate Ri. Generally, the more known pairs, the +more precise the estimation can be. However, it’s unclear how +the noise level affects the effectiveness of estimation and how +we can achieve a good balance between data utility and attack +resilience. We will examine the regression-based attacks in the +experiments. +Note that the recently proposed InstaHide [16] method also +somewhat matches this definition of image disguising. It also +requires learning from the disguised examples {(T(Xi), ci)}. +However, the learned model MT is still applicable to the +original test data, i.e., the application phase uses MT (Xnew). +They also show that the performance of MT (Xnew) is very +close to M(Xnews), which implies MT +≈ M. leads to +serious problems, such as the impossibility result and a +clustering-based attack, as Carlini et al. [2] show. In contrast, +our proposed methods require strictly MT (T(Xnew) in the +application phase, which eliminates the possible information +leakage targeting the models and the clustering of disguised +training images. +VI. EXPERIMENTS +The experiments have three goals. (1) The proposed Dis- +guisedNets mechanisms involve parameter settings, which may +affect data utility. (2) While the proposed methods are resilient +to attacks under Level-1 knowledge, we need to understand +the intrinsic trade-offs between data utility and the methods + +100 +label +0 +75 +2 +3 +4 +5 +50 +6 +8 +25 +9 +-25 +50 +75 +100 +100 +75 +50 +25 +0 +25 +50 +75 +100 +X20 +label +0 +1 +2 +3 +4 +10 +5 +6 +7 +8 +9 +10 +. +20 +-30 +30 +20 +10 +0 +10 +x0 +10 +20 +30 +40 +50 +0 +1 +2 +3 +Epochs +Loss +Baseline +RMT +AES +(a) Convergence on MNIST +0 +10 +20 +30 +40 +50 +0 +1 +2 +3 +Epochs +Loss +Baseline +RMT +AES +(b) Convergence on CIFAR10 +Fig. 8. Convergence speed on disguised images. Baseline: models trained on +original datasets. +enhancing the resilience to Level-2 attacks. (3) As we have +discussed, the proposed methods have unique benefits in +defending model-based attacks, which we will demonstrate in +experiments. +Datasets. We use four prevalent DNN benchmarking +datasets: MNIST, FASHION, CIFAR10, and LFW [22] for ex- +periments. MNIST (handwritten digits) and FASHION (fash- +ion items) are gray-scale 28×28-pixel images with ten classes. +CIFAR10 has 60 thousand 32 × 32 color images distributed +into ten classes. LFW is a labeled face database. It is relatively +small, with only a few thousand samples. We used five folds +of random sampling to estimate the standard deviation of +modeling results, which are also used for later experiments. +Table II summarizes the datasets, the techniques used to +train the base models, and their baseline model accuracy on +the original image data. All the models are implemented with +PyTorch. +TABLE II +DATASETS AND BASELINE ACCURACY. TR: TRAINING, TE: TESTING +Datasets +Records +ImageSize +Network +BaselineAccuracy +MNIST +(60K Tr,10K Te) +{28 × 28} +AlexNet +96.7 ± 0.2% +FASHION +(60K Tr,10K Te) +{28 × 28} +AlexNet +88.7 ± 0.3% +CIFAR-10 +(50K Tr,10K Te) +{32 × 32} +ResNet-18 +93.4 ± 0.2% +LFW +(1164 Tr, 292 Te) +{60 × 48} +ResNet-18 +94.3 ± 2.0% +A. Parameter Settings for Level-1 Attacks +Since all the proposed methods are resilient to Level-1 +attacks, we focus on the utility preservation aspect in this +section. +Costs. The disguised images are used directly with the +existing DNN training algorithms without any modification to +the algorithm or data. We have briefly analyzed the per image +disguising cost in Section IV-C, which can be comfortably +handled by a mobile phone. Another question is whether the +disguised images will extend the training time. Fig 8 shows the +evaluation of convergence speed on MNIST and CIFAR10 for +the three methods: the baseline, RMT, and AES. The baseline +refers to the models reported in Table II. Both RMT and AES +run with the basic setting of 4x4 blocks. All of the methods +converge with 50 epochs, but AES appears more unstable on +CIFAR10. +RMT Mechanism. We look at the effects of block size and +noise levels on models trained on images transformed with +RMT methods. For easier presentation, we convert block size +into the number of blocks: 1 block on the x-axis means the +image is not split into blocks; while 196 blocks means 196 +2x2 blocks for 32x32 images (CIFAR10) or padded 28x28 +images (MNIST and FASHION), and 196 4x3 blocks for +padded 60x48 images (LFW). Thus, a smaller block size +results in a larger number of blocks after partitioning, as the +image size is fixed. If more than one block is generated in +partitioning, we also apply a secret block-wise permutation. +Figure 9 (a) shows that the model quality is slightly decreased +with smaller block sizes (more blocks per image). Overall, +the model quality is well preserved, only 2-3% worse than +the baseline. It’s also understandable that the simpler images, +MNIST and FASHION, are more resilient to noise addition +and more sophisticated ones are sensitive to noise as shown +in Figure 9 (b). +1 +4 +16 +49 +64 +196 +50% +60% +70% +80% +90% +100% +Block Counts +Avg. Model Quality +MNIST +FASHION +CIFAR-10 +LFW +(a) Effect of Block Size. +0 25 50 +100 +200 +50% +60% +70% +80% +90% +100% +Noise Levels +Avg. Model Quality +MNIST +FASHION +CIFAR-10 +LFW +(b) Effect of Noise Levels +Fig. 9. Effects of block size and varying noise levels on model quality for +RMT-disguised images. +AES Mechanism. We have done experiments to understand +the effect of block-size setting for the AES ECB based +block protection. We use “pixel blocks” for partitioning and +permutation, and “units” for AES encryption units. A pixel +block typically contains more than one unit. Recall that AES +uses 16 bytes as the encryption unit if 128-bit encryption is +used. Our partitioning schemes follow this restriction of unit +size to make sure that each block has integer times of 16 +bytes. Figure 10 shows different block size settings from 1 +block (e.g., 32x32 per block for 32x32 images) to 64 blocks +(e.g., 4x4 pixels per block for 32x32 images). +We tested two schemes: no scaling vs scaling. The no- +scaling scheme uses the block size ≥ 16 bytes, while scaling +can use even smaller block sizes. Specifically, when we use +a block size ¡ 16, e.g., 2x2 blocks, the scaling up factors +are determined for the x and y axes, corresponding, e.g., the +scaling factor for x-axis is 2 and also 2 for y-axis for 2x2 +blocks, so that we can partition the scaled image with 4x4 +blocks. Figure 10 (a) shows that the model quality can be +affected by the no-scaling scheme. For some datasets, e.g., +CIFAR10 and LFW, the model quality is too low to be used. +Figure 10 (b) shows that the model quality is boosted to +the level comparable to the RMT’s results for MNIST and +FASHION, while the other two still stay at unusable levels. +The possible reason is that the colored (multi-channel) images +contain more noisy image blocks, which changes significantly +after the AES transformation. In summary, different from the + +RMT scheme, the AES scheme may only work for some +datasets. +64 +16 +4 +1 +0% +20% +40% +60% +80% +100% +Block Counts +Avg. Model Quality +MNIST +FASHION +CIFAR-10 +LFW +(a) no scaling +14 +16 +64 +0% +20% +40% +60% +80% +100% +Block Counts +Avg. Model Quality +MNIST +FASHION +CIFAR-10 +LFW +(b) scaling up, then scaling down +Fig. 10. Effect of block size on model quality for AES-ECB-disguised images +B. Resilience on Level-2 Attacks: AES Scheme +With the known additional knowledge, i.e., pairs of original +and disguised images, the disguising mechanisms might be +under the reconstruction attack, and attackers can visually +check the reconstructed images to re-identify the features of +original images. +To effectively evaluate the re-identification step, we use +a DNN trained on the original image data to simulate the +attacker in the visual re-identification process. The intuition is +that if any features in the disguised (or reconstructed) images +can be detected visually by the adversary, it can be used to link +the disguised (or reconstructed) images to the original images. +Such linking is often probabilistic, and we can use the linking +success rate (i.e., the accuracy of prediction) to gauge the +threat level of attacks. As DNNs perform comparably well as +human experts do in the image-based classification tasks [21], +we believe such “DNN examiners” can satisfactorily simulate +the attacker. +We train DNN examiners with the original training data +using the same DNN architectures detailed in Table II. We then +apply the DNN examiners to see whether the reconstructed +images can be correctly classified to their original labels. +To minimize the impact of DNN architecture and different +baseline accuracy, we define the attack success rate as +accuracy of DNN examiner on disguised/reconstructed images +accuracy of DNN examiner on original images +× 100. +1) Resilience to Codebook Attack for the AES ECB method: +Assume the attacker knows m pairs of original images and +their ECB encrypted ones, and also other information such as +their pixel-block sizes. The codebook attack uses the known +pairs to construct a mapping between the known plaintext +16-byte pixels (or a reduced number of pixels if the scaling +up/down method is used to preserve more utility) and the +corresponding encrypted 16-byte pixels. The attacker might be +able to use the codebook to partially recover the original pixel +blocks of a disguised image (with random pixel patches for +unrecognized blocks). We use the DNN examiner to examine +the quality of reconstructed images. +As MNIST and Fashion perform reasonably well with the +AES scheme (Figure 10 compared to the other two, we pick +only the MNIST data for clear presentation – the Fashion data +has a similar pattern. Figure 11 compares the attack results on +16-pixel encryption units (subfigure (a)) and 2-pixel encryption +units with scaling (subfigure (b)). The attacker’s known pairs +are selected randomly from the training data, while the targeted +images are selected from the testing data. 16-pixel encryption +unit gives a one-to-one mapping between the original pixel +units and the encrypted ones. We observed hit rates are quite +low (lower than 10%), but success rates are increasing steadily +due to the increased codebook size. Overall, attackers will +need a large number of pairs to achieve a good success rate. +2-pixel encryption unit may create a one-to-many mapping +between original pixel units and the encrypted ones, due to the +scale up/down processes. We used the Python library function +for image scaling. With the scaling process, we observed that +hit rates initially increase to around 10% and then drop to 2- +3%. However, the success rate quickly reaches the plateau – +around 50% with only 20 image pairs. Therefore, no-scaling +method is more resilient to attacks – both the hit rates and +success rates grow slowly and knowing the whole training data +does not help improve the success rates much. In contrast, the +scaling method can help gain better model quality. However, it +might be vulnerable to Level-2 attacks. There seems an abrupt +trade-off the user may have to make. +Aiming at achieving a better balance of utility and attack +resilience for the setting of the 2-pixel encryption unit, we +found that it’s possible to defend from the codebook attack by +adding “salt-and-pepper” noises to the original images. The +AES encrypted pixel block changes dramatically when any +of the original pixel changes, which helps reduce the attack +success rate. Figure 12 shows by adding a small amount of +noise, e.g., 2-3%, the attack success rate drops by 10%, while +the model quality is not significant damaged. Certainly, the +level of noise should be carefully chosen to avoid destroying +the data utility: an increase of noise intensity to 4% will +dramatically degrade the model quality as Figure 12 (b) shows. +1 +5000 20000 40000 60000 +0% +10% +20% +30% +40% +50% +Known pair counts +Hit Rate +Attack Success Rate +(a) 16-pixel encoding unit +1 +10 +20 +30 +0% +20% +40% +60% +80% +100% +Known pair counts +Hit Rate +Attack Success Rate +(b) 2-pixel encoding unit (scaling up +to 16 pixels, then scaling down) +Fig. 11. Codebook attack on MNIST dataset with varying number of known +pairs. +C. Resilience on Level-2 Attacks: RMT Scheme +We study how known pairs can be effectively used to +attack the RMT method. Again, we assume a stronger attack +scenario: the attacker already knows the pixel-block size +and the permutation pattern. By known only one pair of + +0.5 +1 +2 +3 +4 +0% +20% +40% +60% +80% +100% +Noise Level +Attack Success Rate +Model Quality +(a) 16-pixel encoding unit +0.5 +1 +2 +3 +4 +0% +20% +40% +60% +80% +100% +Noise Level +Attack Success Rate +Model Quality +(b) 2-pixel encoding unit (scaling up +and then down) +Fig. 12. Protecting AES-based disguising with noise addition (MNIST data) +1 +4 +16 +49 +64 196 256 +0% +10% +20% +30% +40% +Block Counts +Attack Success Rate +MNIST +FASHION +CIFAR-10 +LFW +(a) direct re-identification attack is +not effective +1 +10 +20 +30 +0% +20% +40% +60% +80% +100% +120% +Known pair counts +Avg. Attack Success Rate +MNIST +Fashion +CIFAR-10 +LFW +(b) Regression attacks on the noise- +added RMT disguising method can be +effective with enough known pairs. +Images with block-size 7 × 7 and +noise level u = 100 +Fig. 13. Attacks on RMT-disguised images. +images, RMT without noise addition can be easily broken +– the block-wise transformation parameters {Ri, i = 1..m} +can be straightforwardly recovered. Different from the “salt- +and-pepper” noise for selected pixels in the enhanced AES +scheme, we generate a noise value for each pixel and add +it to the original pixel value before applying the projection, +i.e., Yi = (Xi + ∆i)Ri, where the noise ∆i is drawn from +the uniform distribution U(0, u). With noise addition, the +known attack method is to use linear regression to estimate +the parameters {Ri}, the accuracy of which is affected by the +noise intensity (i.e., the variance of noise) and the number of +available pairs. +Figure 13 (a) shows that direct re-identification (with Level- +1 attack) is generally not effective at all. However, Figure 13 +(b) shows that the regression attack is surprisingly effective +on all datasets. With a small number of known image pairs, +the attack can achieve surprisingly high success rates. Thus, +it’s not safe to use the RMT scheme when Level-2 attack +knowledge is possibly available. +D. Use Image Disguising to Protect Models +Exposing models may have high risks, as shown in +model-inversion attacks, membership-inference attacks, and +model-extraction attacks. This experiment shows that image- +disguising methods can work effectively against such model- +targeted attacks. We take model-inversion (MI) attacks, for +example, which try to recover training data from the exposed +model. +MNIST +FASHION +LFW +CIFAR-10 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Attack Success Rate +Original +RMT +Fig. 14. RMT protects models from model-inversion attacks. Original: the MI +attack applied to the original non-protected model to recover images. RMT: +the MI attack applied to the model trained on RMT-disguised training data. +TABLE III +BEST RESULT UNDER LEVEL-1 ASSUMPTION +Datasets +No Disguise +RMT Disguise +AES Disguise +MNIST +96.7 ±0.2% +96.6 ± 0.4 % +91.6 ± 1.1% +FASHION +88.7 ±0.3% +85.1 ± 0.6 % +68.9±1.4% +CIFAR-10 +93.4 ±0.2% +89.3%±0.1% +11.3 ± 1.7 % +LFW +94.3 ±2.0 +92.6±2.3% +17.2±0.4% +The experiment exposes the models trained on RMT dis- +guised images for a recent model-inversion (MI) attack [42] +that has shown good performance in recovering training data. +Specifically, we used a 4x4 block without noise addition for +RMT to generate disguised data and models. We then apply +the MI attack to generate 2000 images for each dataset (200 +for each class). To compare the performance of the MI attack, +we use the DNN examiners trained on the original datasets to +recognize the recovered images. Figure 14 shows that models +trained on RMT-disguised data are very resilient to the MI +attack. Indeed, the MI attack recovers the RMT-disguised +training data, which are different from the original images +and thus still unrecognizable. The results are consistently +worse than the DNN examiners applied to the RMT-disguised +training data directly (Figure 13 a). +E. Discussion +Based on the experimental results, we have the following +observations. +• With Level-1 adversarial knowledge, the RMT mecha- +nism preserves good data utility for most datasets. In +contrast, the AES scheme only keeps data utility for some +datasets. Table III summarizes the best result under the +Level-1 adversarial knowledge assumption. +• With Level-2 adversaries, the RMT mechanism should +not be used as the attack success rate will be high. The +AES scheme with a small encryption unit and small +(e.g., 2%) noise addition is resilient to the codebook +attack and still preserves model quality for some datasets. +Table IV summarizes the best results for AES. As the +AES scheme does not work on CIFAR10 and LFW, so +far, we haven’t discovered satisfactory utility-preserving +disguising methods against Level-2 adversaries. + +TABLE IV +AES BEST RESULT UNDER LEVEL-2 ASSUMPTION: ENCRYPTION UNIT +2X1 (WITH SCALING), NOISE LEVEL 2%. +Datasets +No Disguise +Model Accuracy +Attack Success Rate +MNIST +96.7 ±0.2% +90.14 ±1.1% +30.76±0.87% +FASHION +88.7 ±0.3% +73.08± 0.86% +23.51± 0.27% +• Finally, if only Level-1 adversaries are expected, RMT +can also be used to effectively protect from model- +targeted attacks, as the models trained on RMT disguised +data can only be applied to disguised data. +VII. CONCLUSION +Outsourcing large image datasets to the cloud for deep +learning has been an economical and popular option, but it +also raises concerns about data and model confidentiality. The +existing solutions are either too expensive to be practical, +vulnerable to different model-based adversarial attacks, or +ineffective in protecting the image content. By focusing on +the training image reconstruction and re-identification attacks, +we propose image disguising mechanisms that efficiently +thwart the attacks and preserve model quality. The combi- +nation of random image-block permutation and block-wise +AES encryption or multidimensional transformation (RMT) +does not require any changes to the existing DNN modeling +architectures. Experimental results show that the RMT method +can preserve the model quality and provide sufficient attack +resilience under Level-1 adversarial knowledge – adversaries +knowing only the disguised images and the domain informa- +tion. The AES method improves the attack resilience against +Level-2 adversaries who manage to obtain pairs of original +images and disguised ones. 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In CVPR, 2020. + diff --git a/xtAyT4oBgHgl3EQfa_c9/content/tmp_files/load_file.txt b/xtAyT4oBgHgl3EQfa_c9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dd63b6303e9eb3c91ae8c738adf15c5840c9d39 --- /dev/null +++ b/xtAyT4oBgHgl3EQfa_c9/content/tmp_files/load_file.txt @@ -0,0 +1,983 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf,len=982 +page_content='A Comparative Study of Image Disguising Methods for Confidential Outsourced Learning Sagar Sharma Bytedance Seattle, WA sagar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='sharma@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='com Yuechun Gu, Keke Chen Trustworthy and Intelligent Computing Lab Marquette University, Milwaukee WI {ethan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='gu, keke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='chen}@marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='edu Abstract Large training data and expensive model tweaking are standard features of deep learning for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As a result, data owners often utilize cloud resources to develop large- scale complex models, which raises privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Existing solutions are either too expensive to be practical or do not sufficiently protect the confidentiality of data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In this paper, we study and compare novel image disguising mechanisms, DisguisedNets and InstaHide, aiming to achieve a better trade-off among the level of protection for outsourced DNN model training, the expenses, and the utility of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' DisguisedNets are novel combinations of image blocktization, block-level random permutation, and two block-level secure transformations: random multidimensional projection (RMT) and AES pixel-level encryption (AES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='InstaHide is an image mixup and random pixel flipping technique [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have analyzed and evaluated them under a multi-level threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' RMT provides a better security guarantee than InstaHide, under the Level-1 adversarial knowledge with well-preserved model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, AES provides a security guarantee under the Level-2 adversarial knowledge, but it may affect model quality more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The unique features of image disguising also help us to protect models from model-targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have done an extensive experimental evaluation to understand how these methods work in different settings for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' INTRODUCTION Deep Neural Networks (DNN) have shown impressive performance across diverse domains such as image classifi- cation, natural language processing, speech recognition, and recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, DNN training is resource- intensive and time-consuming, requiring large training data, careful model architecture selection, and exhaustive model parameter tweaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As a result, data owners or model de- velopers often utilize multiple cloud GPUs or online model training services, such as Google Colab, to lower their costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Despite its popularity, outsourcing DNN learning to the cloud raises privacy and security concerns about the sensi- tive training data and trained models [31], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' On the one hand, cloud users cannot verifiably prevent the cloud provider from getting access to their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In practice, using public clouds often means fully trusting your cloud provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' On the other hand, public cloud providers are not immune to security attacks, which may lead to data breaches through insider [4], [5] and external attacks [25], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Additionally, membership inference attacks [34], model inversion attacks [10], and adversarial example exploration [3], [28] can be applied to models directly to explore the training examples in DNN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Therefore, data and models in training, testing, and transferring between the cloud and the client are seriously threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' There have been a few efforts trying to address this critical issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, all of them are not satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Encrypted data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The first approach is to train encrypted DNN models over encrypted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, due to the large training data and expensive training pro- cess in deep learning, cryptographic model training ap- proaches are too expensive to be practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A recent study on training small-scale neural networks [26] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', just two layers with a maximum of 128 neurons per layer) has shown astonishingly high communication, computation, and storage costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As a result, cryptographic approaches are often limited to training small models [26], [30] or securely applying trained DNNs for prediction [40], [29], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Another possible solution is to par- tition the dataset and the learning task into sensitive and non-sensitive partitions and use cloud-client federated learning [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The non-sensitive portion, assuming it is much larger than the sensitive one, is exported to and processed by the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Correspondingly, the learning process is partitioned and distributed between the cloud and the client, ensuring that the intermediate information exchanged between the two parties does not breach privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Collaborative deep learning framework enhanced with differential privacy [32], [1] may be tweaked into such a partition-based setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, reported attacks [14] allow an adversarial collaborator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a compro- mised cloud in the cloud-client scenario) to generate images resembling the sensitive classes owned by the victim parties (the trusted data owner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Trusted execution environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Hardware-assisted trusted execution environments (TEEs), such as Intel SGX, can also be applied to deep learning in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The idea is to create a secure enclave in the specific memory area (enclave page cache (EPC)) so that no arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='00252v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='CR] 31 Dec 2022 other process/thread can access the content in the enclave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Memory pages are also automatically encrypted when they are swapped to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, recent studies on side-channel attacks [8] make this approach challenging to develop and deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Attackers can peek or infer the plaintext content inside the secure enclave via side channels, such as page fault interrupts and cache loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Furthermore, to work with GPUs, costly cryptographic approaches have been applied to achieve partial data confidentiality in transferring data between CPU and GPU [35], [27], which does not meet the performance requirement for training large DNNs or with large training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' GPU manufacturers may develop TEEs for GPU 1 However, it’s to be tested to determine how secure it is in terms of side-channel attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Researchers have also explored the application of differ- ential privacy (DP) [6] in distributed (federated) learning scenarios [32] or a trusted central training server [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, DP works for the setting of sharing data and models without breaching individual training examples’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It does not meet the need for data and model confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Scope and contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This study compares two dif- ferent image disguising approaches: InstaHide [16] and our recently developed DisguisedNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' These image disguising mechanisms protect the training data and also possibly the learned models by casting training data into a confidential transformed space where powerful DNN models can still learn features and patterns distinguishing image classes and leverage the power of GPUs in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The intuition is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (1) Apply appropriate transformations and data protection mechanisms so that the disguised images cannot be effectively reconstructed and re-link to the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (2) Meanwhile, powerful deep learning techniques can still pick up the unique topological and geometric features preserved in the transformed space to distinguish the originally defined classes of images in the transformed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By doing so, the tie between the original training data and the learned model in the transformed space is broken, which also disables any model-based exploration [3], [28], [34], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In the end, we can approximately preserve the image distinguishability for the target classification task while minimizing the recoverability of individual images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' There are several unique contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1) We have designed two image disguising mechanisms: AES-based (AES) and random-projection-based (RMT) for image-based DNN learning to preserve training data and model confidentiality in outsourced training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The goal is to study and achieve a good balance between the utility of disguised images and the level of confidential- ity protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2) We have carefully analyzed the potential attacks under the outsourced deep learning settings and the resilience of disguising mechanisms to attacks on data and model 1Nvidia has announced a GPU TEE in their Hopper architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' So that users can choose the correspond- ing method under their preferred threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 3) We have conducted extensive experimental evaluations on public datasets to show the trade-offs of different dis- guising schemes and related parameter settings between data utility and their resilience to attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We also show how the disguising methods work to protect models from model-targeted attacks and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' RELATED WORK Sensitive deep learning assets may include training data, models, and online testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Protection methods may target the model training phase or the application (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', inference) phase when both phases can be exported to the public cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, typically, the computational complexity and the demand on resources of model training is far more than model application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, many studies have focused on the application phase, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a cloud-based model inference service hosting the trained model and making predictions for a user- provided image [12], [29], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, due to the high computational complexity of training algorithms and the large size of training data, there is no practical cryptographic approach, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', homomorphic encryption or secure multi-party computing based approaches for protecting model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A few recent studies in this direction have shown prohibitively high costs even for a small neural network model [26], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Trusted execution environments (TEE) with masked GPU operations are applied to speed up training [27], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, no TEE-based deep learning method has addressed severe side-channel attacks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Researchers have looked at protection methods for image training data in the outsourcing context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Noise addition [7], image blurring [24], and morphing [23] are weak as the visible features of the images are still perceivable and understand- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Such transformations do not defeat simple visual re- identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 1 shows how easy it is to visually re- identify the content in the original images by observing the transformed ones with the mentioned techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A recent method InstaHide [16] applies the idea of mixing up images [41] in a training set with public images with linear combination to obfuscate the content, along with randomized signs of pixel values for further protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, it is vulnerable to image reconstruction attacks [2] that need only to know the disguised images and public image sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the Level-1 adversarial knowledge, as we will discuss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, on the one end, existing disguising mechanisms are too weak to protect almost nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' On the other end, if an encryption mechanism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', homomorphic encryption, or a complex cryptographic protocol, is applied, such linking or reconstruction would be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, the current cryptographic schemes incur extremely high costs in almost all aspects of computation, communication, and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, they are impractical for resource-intensive tasks like training a DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Hardware-assisted approaches, such as TEEs, are still under investigation to ensure the expected security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Along with all these possible approaches, we aim TABLE I RELATED WORK ON TRAINING PHASE PROTECTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Sample Related Work Method Weaknesses Strengths Mohassel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [26] Train DNN models over masked or encrypted data with cryptographic protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Involve high communication, computation, and storage costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Require extensive re-design and custom implementation of the DNN architec- ture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Involve iterative interactions between the data owner and cloud provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Provide semantic security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model quality is fully preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Tramer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [35] and Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [27] Use TEEs for confidential CPU op- erations and masked data for con- fidential GPU operations side channel attacks are an unaddressed con- cern More efficient than cryptographic protocols, but GPU operations on masked data are still expensive Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [1] and Shokri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [33] Apply differential privacy to ran- domize intermediate gradients Aim to share trained models, and thus does not preserve model confidentiality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' result in significant drop in model quality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' vulnerable to generative adversarial network attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Preserve individuals’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Fan [7] Train shallow neural network lo- cally and outsources intermediate representation to the cloud for deeper training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The intermediate representation of images re- veals the visual characteristics of the related images, vulnerable to visual re-identification attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' NA Li et al [24] Applies differential privacy to hide sensitive pixels in images Does not hide the global visual characteristic content of images, vulnerable to visual re- identification attacks NA Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [41] and Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [16] Mix-up images from the training set and public domain with random selection and weighing to hide the content of the sensitive image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Vulnerable to ciphertext-only (Level 1 adver- sarial knowledge) image re-construction at- tacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' May expose trained model to wide va- riety of model-based and membership attacks for the inside-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Fast training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' No changes to the training architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' original DP ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='5 noisy original (a) original 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='8 dB 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='9 dB blurred original (b) original morphed (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (a) Differentially private noise addition to images [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (b) The reconstructed blurred images in PrivyNet [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (c) The morphed images [23] to explore and develop new image-disguising methods to achieve good balances among costs, data utility, and security guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Table I summarizes the current work in protecting data and models in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' THREAT MODELING We are concerned with the confidentiality of the sensitive training image data and the DNN models in the outsourced training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Here, we make some relevant security assump- tions for our disguising mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1) We consider the cloud provider to be an honest-but-curious adversary, which implies that a curious provider will still honestly deliver desired results to the data owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, it may keep a copy of the data and programs it can observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2) The adversary can observe the training data, the training process, and the trained models, including the structure of the DNN architecture and parameter settings for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, they can probe the observed items with methods such as image reconstruction, re-identification, and membership attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (3) We do not address evasive attacks and poisoning attacks [3], [28], where adversaries will tamper with the training data, which can be guarded with training data integrity checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (5) The client infrastructure and commu- nication channels are secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Assets to Be Protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Under a certain protection mecha- nism, we generalize that the training data D is transformed to fkey(D), and the model M = M(D) is changed to gkey(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The attacker might want to know whether an image is likely used to train a model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the membership inference attack [34], observe training images that may contain sensitive objects, or steal a proprietary training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The attack might also target the model if the model is exposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', using model-inversion attacks [9] to explore the private information of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Adversarial Prior Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The adversary may have Created by UNREGISTERED http.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='/tantamorh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='com FantaMorphtwo levels of prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For each level, we may design a disguising technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Level-1: They may know what the model is used for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the background application, the distribution of the data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', face images, and the type of disguising technique used, but do not know the disguising parameter setting for a specific dataset that serves as the secret key to the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Level-2: In addition to Level-1 knowledge, they may try to obtain pairs of images and their disguised versions via other attacking channels (not including the ones they are targeting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' They hope to use these known pairs to explore various image reconstruction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Potential Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Recent studies have shown that attackers can explore training/testing examples and models, for example, to find adversarial examples misleading the prediction of deep neural networks [3], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Such attacks depend on adversaries’ clear understanding of the original image data and the ability to access the developed models freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Outsourced learning without protection makes these attacks easier to deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With a protection mechanism on data and models, we consider a fundamental attack: training image re-identification that aims at linking the protected images to identifiable original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We introduce a model-based re-identification test – DNN examiner, which uses a model trained on the original data to tell whether a protected image is re-identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Note that some related methods [7], [24] are not resilient to human visual re-identification, which does not protect confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Since the image disguising mechanisms break the link between the original training data and the learned models (in the transformed space), the existing model-oriented attacks do not work anymore without successfully breaking the disguising mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Attackers thus depend on reconstruction attacks: reverse the disguising mechanism to approximately reconstruct the original images and then try to re-identify the reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We use the DNN examiner approach to evaluate how successful a reconstruction attack works in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' DISGUISEDNETS – A NOVEL IMAGE DISGUISING MECHANISM FOR OUTSOURCED DEEP LEARNING In the following, we will introduce an image disguising framework that incorporates pixel-block partitioning, random block permutation, and block-wise transformations of images along with noise additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The premise is that after the dramatic transformation, it is difficult to link the disguised images to the original images, while, unlike pure encryption schemes, it still preserves some essential patterns for distin- guishing between classes of images that allow DNN learning methods to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This amalgam of multiple transformations provides a sufficiently large parameter space so that the attacks are computationally intractable (Section V) under the Level-1 prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 2 depicts the DisguisedNets framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A data owner disguises her private images before outsourcing them to the cloud for DNN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' She can either fully outsource the entire image datasets and the learning procedure to the cloud or selectively retain sensitive images in the cloud-client partitioning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' She transforms all of her images using one secure transformation key secret to her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Note that this transformation should be at a reasonable cost, practical for a client’s infrastructure to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Cloud Provider DNN Model GPU Processing Data Owner Train Disguised test images {TK(Xnew)} Predictions {y’new} Disguised Images {TK(Xi),yi} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Image disguising framework for DNN learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Specifically, assume the data owner owns a set of images for training, notated as pairs D = {(Xi, yi)}, where Xi is the image pixel matrix (l × m and 3 × l × m for grayscale and RGB images respectively) and yi the corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We formally define a disguising mechanism as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Let the disguising mechanism be a transformation TK, where K is the secret key that depends on the selected perturbation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By applying image disguising, the training data is transformed to {(T(Xi), ci)} with ci mapped to 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' randomly representing the classes yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The original model is a function M(D), which is learned with a DNN learning method G: G(D) → M(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The image disguising mechanism enables the same learning method G to be applied to the transformed data directly without any modification: G(DT ) → MT (DT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For any new data Xnew, the model application (or inference) is defined as MT (T(Xnew)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the new data transformed with the same key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To make such a transformation method practical for modeling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a model trained with transformed data still working satisfactorily, a user may expect the error of modeling is not far away from the original model’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, a utility-preserving mechanism should have |Err(MT ) − Err(M)| < δ where δ is the level of model quality degradation acceptable to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' While for a specific DNN modeling method and a specific dataset, it’s difficult to theoretically justify what this gap will be, one can always directly evaluate the model quality to check whether it is acceptable for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have empirically evaluated the δ levels for different mechanisms, datasets, and a few popular DNN modeling architectures in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Pixel-Block Partitioning and Block-based Random Permu- tation In this section, we present one way of image transformation: image block permutation, that will be combined with other mechanisms later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' An image Xl×m is first partitioned into t blocks of uniform size r × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' If we label the blocks sequentially as v =< 1, 2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' , t >, a pseudorandom permutation of the image, Tπ(X), shuffles the blocks and reassemble the corresponding E(A) E(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=') Data Owner On-demand Processing Models E(Ak) Cluster Model Cloud consumers Data contributorsimage accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Block-based permutation preserves the in- block information and the relative positions of related blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, we understand it preserves a great amount of information for effective modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, while the permutation may break the global patterns of the images and achieve good visual privacy already, the between-block characteristics such as boundaries, color, content shape, and texture of the origi- nal neighboring blocks may provide clues for adversaries to recover the original image – imagine the jigsaw puzzle!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For large t, such attacks can be time-consuming due to the vague similarity between block boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, with the prior knowledge: a pair of original image and its block-permuted image, it’s not difficult to solve such a jigsaw puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, we use this as an auxiliary step enhancing other steps in the disguising framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Pixel-Block Transformations Next, we establish pixel-block-level protection mechanisms that aim to preserve the data utility for DNN modeling and further increase the resilience to attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We consider two candidate mechanisms: random projection and encryption schemes, and discuss their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Specifically, when an image is partitioned into t pixel blocks for random permu- tation, we get a list of t parameters {Ki, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t}, one for the pixel-block at the same position across the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We name the specific position of the pixel block in the image the pixel-block position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The list of parameters acts as a secret key and will be shared, together with the permutation key, by each image in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The purpose of this setting is to maximize the preservation of distinguishable patterns between image classes – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a pair of similar image patterns (blocks) can still be transformed to another pair of (likely) similar ones after applying the disguising mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Randomized Multidimensional Transformation (RMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For an image represented as a pixel matrix X, a general linear transformation can be defined as G(X) = R(X + ∆), where Rm×m is a random orthogonal matrix generated following the Haar distribution [11], or a random invertible matrix, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a random projection matrix [39], and ∆ is an optional noise matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We call this method the randomized multidimensional transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' When an image is partitioned into t blocks for random permutation, we prepare a list of random matrices {Ri, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='.t}, one for each image-block position and share this list for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Such transformation is known to preserve (or approximately preserve by random projection) the Euclidean distance between columns of the matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For real application, we may arrange the pixel blocks accordingly to form the column of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For example, a 4x4 pixel matrix can be partitioned into 4 2x2 block to preserve the smaller block- level similarity with RMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 3 shows the effects of RMT on MNIST and CIFAR-10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' AES Block Transformation (AES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The existing AES encryption schemes typically use 128-bit encryption keys, which encode every 16-byte data block sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' If we use AES for pixel-block encryption, assuming each pixel is stored in one byte, 16 original pixels are mapped to 16 encrypted Algorithm 1 DN RMT (X, t, Key) Require: X: image of size l × m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t: number of blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Key = {permutation key, transformation matrices, noise level ∈ [0, N]} 1: r, s ← compute image block size with l × m and t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2: Partition image Xl×m into blocks X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' , Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 3: Shuffle the image blocks pseudorandomly with permuta- tion key 4: for each block i, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t do 5: ∆i ← Generate random matrix with elements from the uniform distribution in [0,N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 6: use the transformation matrix at the position i: Ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 7: Yi ← Ri(Xi + ∆i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 8: end for 9: Re-assemble {Yi} to make the transformed image Y and return Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 8 x 8 blocks 2 x 2 blocks 32 x 32 image 28 x 28 image T{RMT, π, N}(X) T{RMT, π, N}(X) 2x2 blocks, N=0 2x2 blocks, N=100 2x2 blocks, N=0 2x2 blocks, N=25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Block-wise RMT+Noise on MNIST and CIFAR-10 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' bytes (pixels), and a whole pixel block is encoded to 16-byte units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Putting all encrypted pixel blocks together, we get a disguised image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For clear presentation, when we talk about AES encryption block, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 16 bytes for a 128-bit encryption key, we use the 16-byte “encryption unit”, which are different from “pixel blocks” we have been using previously in our image disguising framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 4 shows some example AES transformations on images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' AES - ECB AES - CBC original AES - ECB AES - CBC original Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Pixel-block based AES encryption of MNIST and CIFAR-10 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We consider two AES modes in our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (1) We observe that with the AES Cipher Block Chaining (CBC) mode, any pixel-level change in the pixel block between two images 0 5 10 15 20 25 70 5 10 15 20 25original Noise Addition + Block-wise Encryption + Permutation Encrypted with padding Scaled up, Encrypted, & Scaled-down Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' AES-ECB encryption of MNIST image with different strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' will result in different encoding results for most 16-byte AES blocks in this pixel block position, making it not ideal for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (2) Then, we turn to the AES Electronic Code Book (ECB) mode that can be considered as a fixed mapping function between 16-byte original data to 16-byte encrypted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Different from CBC, the neighboring 16-byte blocks do not affect the encoding of the current block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This matches our requirement of data utility preservation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', to preserve the block-level distinguishable patterns after the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To preserve more information, based on the intuition of smaller blocks preserving more inter-pixel-block information, we can also use unit sizes smaller than the regular size, which is 16 pixels for 128-bit ECB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The method is to scale up the image first, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', from 32x32 to 256x256 (where each pixel is duplicated eight times), then encrypt it by 16-pixel units, and finally scale down to the size 32x32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Please refer to Figure 5 for the detailed example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It’s equivalent to encrypt 2-pixel units in the original 32x32 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We found by reducing the block size, the model quality can be improved with the cost of lower attack resilience to the Level-2-knowledge-based attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Algorithm 2 DN AES (X, k, t, Key) Require: X: image of size l × m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' k: scale-up factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t: number of blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Key = {permutation key, AES keys, p = probability of salt-pepper noise } Ensure: the selection of k and t results in image blocks that can be further partitioned to 4x4 pixel patches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1: Xlk×mk ← scaled up the image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2: r, s ← compute image block size with lk × mk and t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 3: Partition image Xl×m into blocks X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' , Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 4: Shuffle the image blocks pseudorandomly with permuta- tion key;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 5: for each block i, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t do 6: Yi ← for each pixel in block Xi, with the probability p, it’s randomly turned to white or block pixel (salt-pepper noise);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 7: E(Yi) ← with the AES CBC mode, every 16-byte segment (4x4 pixel patch) is encypted to 16 bytes of AES digest with AES key i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 8: end for 9: re-assemble image blocks E(Yi) and return E(Y ) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Complexity Analysis The additional costs of the disguising methods consist of the encoding cost and the possible additional learning cost, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', it may take more rounds to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We leave the second part to the experimental evaluation and analyze the encoding cost here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For an image partitioned into t blocks with each block l × m, the RMT transformation involves t matrix-matrix multiplications and matrix additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As the numbers t, m, and l are all small, the cost of RMT per image is low: O(tlm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For an image of l×m with a scale-up constant of s, the AES- 128 encryption cost is l × m × s/16 times of AES encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Our experimental evaluation shows that per image cost is less than 10 ms and can be comfortably done by any PC or mobile phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model Protection via Image Disguising Note that the models trained with disguised data work only on disguised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We show this property also protects models from existing model-targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' So far, we have seen model-inversion attacks [9], [42], membership-inference attacks [34], [15], and model-extraction attacks [36], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model-extraction attacks assume the attacker can freely access the model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', via a cloud-based prediction API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With such a service, the attacker can try various images to collect their outputs and then use the input-output pairs to reconstruct the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Our threat model assumes the attacker can copy or save the trained model for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, the attacker does not need to perform model-extraction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As the models only work on disguised test images, without the secret disguising key, they are useless to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Membership-inference attack aims to estimate the possi- bility of a target example belonging to the training data of a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To perform such an attack, the attacker must first apply the disguising method (with the secret key) to the target example so that the model can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This step effectively blocks the attack or at least significantly increases the difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To successfully conduct the MIA attack on the disguised model, the attacker may need to manipulate an authorized user to transform the example and intercept the transformed one, which corresponds to the mentioned Level- 2 knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, the disguising mechanism establishes an effective defensive line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model-inversion attack uses a learning procedure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', a GAN method [42], to progressively adjust randomly generated or seed images from similar domains towards most likely training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' When applied to the models trained on disguised data, the model-inversion attack recovers only the disguised training data, not the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Again, the disguising mechanism builds a defensive line on this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We will show how the RMT mechanism works against model- inversion attacks in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' ATTACK ANALYSIS This section aims to analyze the possible threats to the proposed disguising mechanisms and clarify the applica- 0 5 10 15 20 25 30 OP 5 10 15 20 上 25 300 5 10 15 20 25 30 OF 5 10 15 20 25 300 5 10 15 20 25 30 0 5 10 15 20 25 303ble settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With Level-1 adversarial knowledge, Disguised- Nets mechanisms provide strong confidentiality protection, as shown in the discussion of “brute-force attacks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, other related methods are still struggling with visual re- identification by human eyes [24], [7] or disguised-image- based reconstruction attacks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We also analyze more so- phisticated reconstruction attacks that depend on Level-2 ad- versarial knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Level-1 Adversarial Knowledge and Attacks Recall that Level-1 knowledge includes knowing the dis- guised images and possibly the model domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the types of images and the background application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It is clear that with only Level-1 knowledge, the brute-force attack on AES schemes is not possible, and thus we focus on the scheme using multidimensional projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Visual Re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The first simple attack is to visu- ally identify images by human attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have shown that simple methods like noise addition, morphing, and shallow- network-based transformation are not resilient to this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, many other attacks may use re-identification as the last step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', reconstruction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It’s inefficient for human evaluators to check each image to determine the protection level of an image disguising mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, we propose the DNN examiner approach for evaluation purposes: let a DNN trained on the original datasets to perform the visual re-identification task for human evaluators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We will use DNN examiners in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Brute-Force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The brute-force attack method for image re- construction is to enumerate each possible parameter setting of the disguising mechanism and then check the recovered result with re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As AES encryption is already resilient to the brute-force attack, we examine the RMT method only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Let’s start with a block-level transformation for any image block i with RMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With X′ i = XiRi, the adversary knows only X′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In the brute force attack, the number of possible Xi is determined by the number of possible Ri matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We show that the number of possible Ri (even limited to orthogonal ones) can be exponentially large for given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For values encoded in h-bit finite field, there are O(2hm) candidate orthogonal matrices Rm×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The proof is based on the theory of orthogonal matrix group [13], the detail of which is skipped here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With a typical setting in our experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', h = 8 and m = 28 for the MNIST dataset, the overall complexity is O(2224), which is more than sufficient to protect from computationally-bounded attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Combined with the random permutation of blocks, the attack complexity is even higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, a brute-force attack is generally impractical for the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Clustering Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Carlini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [2] utilized a clustering method to attack InstaHide [16] disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' InstaHide uses the random mix-up method to generate disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Depending on the random weight distribution, some disguised images might be dominated by the same image, which likely forms a cluster of images that can be used to de-mask and de-noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As InstaHide disguised images are essentially linear combinations of plaintext images, the attack result can be visually re-identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Important questions are whether our disguising methods can generate images with clustering structures and whether such clusters can be used to break our disguising methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To answer these questions, we visualize the disguised training data with t-SNE [38] to understand the existence of clustering structure in the Euclidean-distance space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 6 shows that RMT might preserve the clustering structures for some datasets: for simpler datasets like MNIST and FASHION, the clustering structure is well preserved, while others do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, AES does not preserve any clustering structure, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' While AES not preserving clustering structures to leave less information to attackers, it also affects data utility and leads to lower-quality models, as we will show in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (a) RMT on MNNST (b) RMT on CIFAR10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t-SNE visualization of RMT disguised datasets (4x4 blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Colors represent different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A dense area covered with one color means that the clustering structure matches the label distribution well for the specific subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Next, can such preserved clustering structures be used for attacks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' An attack on InstaHide [2] has used image clusters to de-noise and de-mask, as InstaHide uses the random weights mix-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, unlike InstaHide, clustering structures of RMT-disguised images do not help attackers identify original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, it may help attackers infer additional information with other domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For example, if the original training samples’ distribution (and 100 label 1 75 2 3 4 5 50 9 25 0 25 50 75 80 60 40 20 0 20 40 60 80 Xlabel 80 0 2 3 4 60 5 6 7 8 9 40 20 0 20 40 20 0 20 40 60 X(a) AES on MNIST (b) AES on CIFAR10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' t-SNE visualization of AES disguised datasets (4x4 blocks) thus the clustering structure) is also known, it may allow the attacker to identify the mapping between a specific cluster of original images and a cluster of disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As distances between samples are not preserved, it’s still difficult to figure out the sample-to-sample mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Level-2 Adversarial Knowledge and Attacks We move one more step further to study the more challeng- ing issue: what if a powerful adversary can obtain additional knowledge: pairs of original images and their transformed ones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This assumption corresponds to the chosen-plaintext attack in cryptographic analysis [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This study helps us understand when we should not use a proposed disguising method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Below, we focus on the codebook attack on the AES- ECB-based disguising mechanism and the regression-based attack on the RMT mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Codebook Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The assumption is that the adversary is knowledgeable of the encryption procedure described previ- ously but does not have the encryption/decryption key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Since the AES ECB method is deterministic, the basic attack is to build a mapping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the codebook) between the plaintext unit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the 16-byte pixel block) and its encrypted counterpart (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the 16-byte AES cipher block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By processing the known image pairs, the adversary constructs a codebook as a dictionary mapping 16-byte pixel blocks to encrypted 16- byte blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Since different images, especially those in the same class, might share some 16-byte pixel blocks, some 16- byte encrypted blocks in the targeted images are likely already in the codebook, which will be used to recover the original blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For encrypted pixel blocks not present in the codebook, the adversary may use a fixed pattern, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', all zero values or most likely values to pad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By repeating this procedure for each 16-byte block, the adversary can recover some parts of the image, which can be further re-identified via human eyes or models at the adversary’s hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Possible mitigation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Let the hit rate be defined as the probability that an encrypted pixel block can find a match in the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This attack can become less effective if we add salt-and-pepper noises to the original images before encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This step will reduce the hit rate significantly and make the mapping non-unique: the same 16-byte pixel block can be mapped to different ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We will evaluate the success rate of this attack in experiments, using the accuracy that the DNN examiner trained with the original image space correctly classifies the reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Projection Matrix Estimation Attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Note that noise addition can easily defend the RMT method from the code- book attack, which is already a part of the RMT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, if the adversary has obtained enough original and transformed image pairs, there is a possibility that the trans- formation matrix might be estimated with linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Specifically, a noise-added block-wise transformation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', Yi = Ri(Xi + ∆i), where ∆i is a random noise matrix, re- generated for each image block Xi, and drawn uniformly at random from [0, N] where N is the tunable noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With enough known pairs of (Xi, Yi), the regression method can be applied to estimate Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Generally, the more known pairs, the more precise the estimation can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, it’s unclear how the noise level affects the effectiveness of estimation and how we can achieve a good balance between data utility and attack resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We will examine the regression-based attacks in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Note that the recently proposed InstaHide [16] method also somewhat matches this definition of image disguising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It also requires learning from the disguised examples {(T(Xi), ci)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, the learned model MT is still applicable to the original test data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the application phase uses MT (Xnew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' They also show that the performance of MT (Xnew) is very close to M(Xnews), which implies MT ≈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' leads to serious problems, such as the impossibility result and a clustering-based attack, as Carlini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' [2] show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, our proposed methods require strictly MT (T(Xnew) in the application phase, which eliminates the possible information leakage targeting the models and the clustering of disguised training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' EXPERIMENTS The experiments have three goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (1) The proposed Dis- guisedNets mechanisms involve parameter settings, which may affect data utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (2) While the proposed methods are resilient to attacks under Level-1 knowledge, we need to understand the intrinsic trade-offs between data utility and the methods 100 label 0 75 2 3 4 5 50 6 8 25 9 25 50 75 100 100 75 50 25 0 25 50 75 100 X20 label 0 1 2 3 4 10 5 6 7 8 9 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 20 30 30 20 10 0 10 x0 10 20 30 40 50 0 1 2 3 Epochs Loss Baseline RMT AES (a) Convergence on MNIST 0 10 20 30 40 50 0 1 2 3 Epochs Loss Baseline RMT AES (b) Convergence on CIFAR10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Convergence speed on disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Baseline: models trained on original datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' enhancing the resilience to Level-2 attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' (3) As we have discussed, the proposed methods have unique benefits in defending model-based attacks, which we will demonstrate in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We use four prevalent DNN benchmarking datasets: MNIST, FASHION, CIFAR10, and LFW [22] for ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' MNIST (handwritten digits) and FASHION (fash- ion items) are gray-scale 28×28-pixel images with ten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' CIFAR10 has 60 thousand 32 × 32 color images distributed into ten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' LFW is a labeled face database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It is relatively small, with only a few thousand samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We used five folds of random sampling to estimate the standard deviation of modeling results, which are also used for later experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Table II summarizes the datasets, the techniques used to train the base models, and their baseline model accuracy on the original image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' All the models are implemented with PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' TABLE II DATASETS AND BASELINE ACCURACY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' TR: TRAINING, TE: TESTING Datasets Records ImageSize Network BaselineAccuracy MNIST (60K Tr,10K Te) {28 × 28} AlexNet 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2% FASHION (60K Tr,10K Te) {28 × 28} AlexNet 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3% CIFAR-10 (50K Tr,10K Te) {32 × 32} ResNet-18 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2% LFW (1164 Tr, 292 Te) {60 × 48} ResNet-18 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='0% A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Parameter Settings for Level-1 Attacks Since all the proposed methods are resilient to Level-1 attacks, we focus on the utility preservation aspect in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The disguised images are used directly with the existing DNN training algorithms without any modification to the algorithm or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have briefly analyzed the per image disguising cost in Section IV-C, which can be comfortably handled by a mobile phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Another question is whether the disguised images will extend the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Fig 8 shows the evaluation of convergence speed on MNIST and CIFAR10 for the three methods: the baseline, RMT, and AES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The baseline refers to the models reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Both RMT and AES run with the basic setting of 4x4 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' All of the methods converge with 50 epochs, but AES appears more unstable on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' RMT Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We look at the effects of block size and noise levels on models trained on images transformed with RMT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For easier presentation, we convert block size into the number of blocks: 1 block on the x-axis means the image is not split into blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' while 196 blocks means 196 2x2 blocks for 32x32 images (CIFAR10) or padded 28x28 images (MNIST and FASHION), and 196 4x3 blocks for padded 60x48 images (LFW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, a smaller block size results in a larger number of blocks after partitioning, as the image size is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' If more than one block is generated in partitioning, we also apply a secret block-wise permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 9 (a) shows that the model quality is slightly decreased with smaller block sizes (more blocks per image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Overall, the model quality is well preserved, only 2-3% worse than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' It’s also understandable that the simpler images, MNIST and FASHION, are more resilient to noise addition and more sophisticated ones are sensitive to noise as shown in Figure 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1 4 16 49 64 196 50% 60% 70% 80% 90% 100% Block Counts Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model Quality MNIST FASHION CIFAR-10 LFW (a) Effect of Block Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 0 25 50 100 200 50% 60% 70% 80% 90% 100% Noise Levels Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model Quality MNIST FASHION CIFAR-10 LFW (b) Effect of Noise Levels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Effects of block size and varying noise levels on model quality for RMT-disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' AES Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We have done experiments to understand the effect of block-size setting for the AES ECB based block protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We use “pixel blocks” for partitioning and permutation, and “units” for AES encryption units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' A pixel block typically contains more than one unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Recall that AES uses 16 bytes as the encryption unit if 128-bit encryption is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Our partitioning schemes follow this restriction of unit size to make sure that each block has integer times of 16 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 10 shows different block size settings from 1 block (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 32x32 per block for 32x32 images) to 64 blocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 4x4 pixels per block for 32x32 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We tested two schemes: no scaling vs scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The no- scaling scheme uses the block size ≥ 16 bytes, while scaling can use even smaller block sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Specifically, when we use a block size ¡ 16, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 2x2 blocks, the scaling up factors are determined for the x and y axes, corresponding, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the scaling factor for x-axis is 2 and also 2 for y-axis for 2x2 blocks, so that we can partition the scaled image with 4x4 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 10 (a) shows that the model quality can be affected by the no-scaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' For some datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', CIFAR10 and LFW, the model quality is too low to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 10 (b) shows that the model quality is boosted to the level comparable to the RMT’s results for MNIST and FASHION, while the other two still stay at unusable levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The possible reason is that the colored (multi-channel) images contain more noisy image blocks, which changes significantly after the AES transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In summary, different from the RMT scheme, the AES scheme may only work for some datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 64 16 4 1 0% 20% 40% 60% 80% 100% Block Counts Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model Quality MNIST FASHION CIFAR-10 LFW (a) no scaling 14 16 64 0% 20% 40% 60% 80% 100% Block Counts Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Model Quality MNIST FASHION CIFAR-10 LFW (b) scaling up, then scaling down Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Effect of block size on model quality for AES-ECB-disguised images B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Resilience on Level-2 Attacks: AES Scheme With the known additional knowledge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', pairs of original and disguised images, the disguising mechanisms might be under the reconstruction attack, and attackers can visually check the reconstructed images to re-identify the features of original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To effectively evaluate the re-identification step, we use a DNN trained on the original image data to simulate the attacker in the visual re-identification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The intuition is that if any features in the disguised (or reconstructed) images can be detected visually by the adversary, it can be used to link the disguised (or reconstructed) images to the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Such linking is often probabilistic, and we can use the linking success rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the accuracy of prediction) to gauge the threat level of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As DNNs perform comparably well as human experts do in the image-based classification tasks [21], we believe such “DNN examiners” can satisfactorily simulate the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We train DNN examiners with the original training data using the same DNN architectures detailed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We then apply the DNN examiners to see whether the reconstructed images can be correctly classified to their original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To minimize the impact of DNN architecture and different baseline accuracy, we define the attack success rate as accuracy of DNN examiner on disguised/reconstructed images accuracy of DNN examiner on original images × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1) Resilience to Codebook Attack for the AES ECB method: Assume the attacker knows m pairs of original images and their ECB encrypted ones, and also other information such as their pixel-block sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The codebook attack uses the known pairs to construct a mapping between the known plaintext 16-byte pixels (or a reduced number of pixels if the scaling up/down method is used to preserve more utility) and the corresponding encrypted 16-byte pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The attacker might be able to use the codebook to partially recover the original pixel blocks of a disguised image (with random pixel patches for unrecognized blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We use the DNN examiner to examine the quality of reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As MNIST and Fashion perform reasonably well with the AES scheme (Figure 10 compared to the other two, we pick only the MNIST data for clear presentation – the Fashion data has a similar pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 11 compares the attack results on 16-pixel encryption units (subfigure (a)) and 2-pixel encryption units with scaling (subfigure (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The attacker’s known pairs are selected randomly from the training data, while the targeted images are selected from the testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 16-pixel encryption unit gives a one-to-one mapping between the original pixel units and the encrypted ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We observed hit rates are quite low (lower than 10%), but success rates are increasing steadily due to the increased codebook size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Overall, attackers will need a large number of pairs to achieve a good success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 2-pixel encryption unit may create a one-to-many mapping between original pixel units and the encrypted ones, due to the scale up/down processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We used the Python library function for image scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With the scaling process, we observed that hit rates initially increase to around 10% and then drop to 2- 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, the success rate quickly reaches the plateau – around 50% with only 20 image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Therefore, no-scaling method is more resilient to attacks – both the hit rates and success rates grow slowly and knowing the whole training data does not help improve the success rates much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, the scaling method can help gain better model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, it might be vulnerable to Level-2 attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' There seems an abrupt trade-off the user may have to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Aiming at achieving a better balance of utility and attack resilience for the setting of the 2-pixel encryption unit, we found that it’s possible to defend from the codebook attack by adding “salt-and-pepper” noises to the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The AES encrypted pixel block changes dramatically when any of the original pixel changes, which helps reduce the attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 12 shows by adding a small amount of noise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 2-3%, the attack success rate drops by 10%, while the model quality is not significant damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Certainly, the level of noise should be carefully chosen to avoid destroying the data utility: an increase of noise intensity to 4% will dramatically degrade the model quality as Figure 12 (b) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 1 5000 20000 40000 60000 0% 10% 20% 30% 40% 50% Known pair counts Hit Rate Attack Success Rate (a) 16-pixel encoding unit 1 10 20 30 0% 20% 40% 60% 80% 100% Known pair counts Hit Rate Attack Success Rate (b) 2-pixel encoding unit (scaling up to 16 pixels, then scaling down) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Codebook attack on MNIST dataset with varying number of known pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Resilience on Level-2 Attacks: RMT Scheme We study how known pairs can be effectively used to attack the RMT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Again, we assume a stronger attack scenario: the attacker already knows the pixel-block size and the permutation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By known only one pair of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='5 1 2 3 4 0% 20% 40% 60% 80% 100% Noise Level Attack Success Rate Model Quality (a) 16-pixel encoding unit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='5 1 2 3 4 0% 20% 40% 60% 80% 100% Noise Level Attack Success Rate Model Quality (b) 2-pixel encoding unit (scaling up and then down) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Protecting AES-based disguising with noise addition (MNIST data) 1 4 16 49 64 196 256 0% 10% 20% 30% 40% Block Counts Attack Success Rate MNIST FASHION CIFAR-10 LFW (a) direct re-identification attack is not effective 1 10 20 30 0% 20% 40% 60% 80% 100% 120% Known pair counts Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Attack Success Rate MNIST Fashion CIFAR-10 LFW (b) Regression attacks on the noise- added RMT disguising method can be effective with enough known pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Images with block-size 7 × 7 and noise level u = 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Attacks on RMT-disguised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' images, RMT without noise addition can be easily broken – the block-wise transformation parameters {Ri, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='.m} can be straightforwardly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Different from the “salt- and-pepper” noise for selected pixels in the enhanced AES scheme, we generate a noise value for each pixel and add it to the original pixel value before applying the projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', Yi = (Xi + ∆i)Ri, where the noise ∆i is drawn from the uniform distribution U(0, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With noise addition, the known attack method is to use linear regression to estimate the parameters {Ri}, the accuracy of which is affected by the noise intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', the variance of noise) and the number of available pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 13 (a) shows that direct re-identification (with Level- 1 attack) is generally not effective at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, Figure 13 (b) shows that the regression attack is surprisingly effective on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With a small number of known image pairs, the attack can achieve surprisingly high success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Thus, it’s not safe to use the RMT scheme when Level-2 attack knowledge is possibly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Use Image Disguising to Protect Models Exposing models may have high risks, as shown in model-inversion attacks, membership-inference attacks, and model-extraction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' This experiment shows that image- disguising methods can work effectively against such model- targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We take model-inversion (MI) attacks, for example, which try to recover training data from the exposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' MNIST FASHION LFW CIFAR-10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 Attack Success Rate Original RMT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' RMT protects models from model-inversion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Original: the MI attack applied to the original non-protected model to recover images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' RMT: the MI attack applied to the model trained on RMT-disguised training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' TABLE III BEST RESULT UNDER LEVEL-1 ASSUMPTION Datasets No Disguise RMT Disguise AES Disguise MNIST 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4 % 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='1% FASHION 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='6 % 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4% CIFAR-10 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='1% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 % LFW 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='4% The experiment exposes the models trained on RMT dis- guised images for a recent model-inversion (MI) attack [42] that has shown good performance in recovering training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Specifically, we used a 4x4 block without noise addition for RMT to generate disguised data and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We then apply the MI attack to generate 2000 images for each dataset (200 for each class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' To compare the performance of the MI attack, we use the DNN examiners trained on the original datasets to recognize the recovered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Figure 14 shows that models trained on RMT-disguised data are very resilient to the MI attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Indeed, the MI attack recovers the RMT-disguised training data, which are different from the original images and thus still unrecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The results are consistently worse than the DNN examiners applied to the RMT-disguised training data directly (Figure 13 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Discussion Based on the experimental results, we have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With Level-1 adversarial knowledge, the RMT mecha- nism preserves good data utility for most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In contrast, the AES scheme only keeps data utility for some datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Table III summarizes the best result under the Level-1 adversarial knowledge assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' With Level-2 adversaries, the RMT mechanism should not be used as the attack success rate will be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The AES scheme with a small encryption unit and small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=', 2%) noise addition is resilient to the codebook attack and still preserves model quality for some datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Table IV summarizes the best results for AES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' As the AES scheme does not work on CIFAR10 and LFW, so far, we haven’t discovered satisfactory utility-preserving disguising methods against Level-2 adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' TABLE IV AES BEST RESULT UNDER LEVEL-2 ASSUMPTION: ENCRYPTION UNIT 2X1 (WITH SCALING), NOISE LEVEL 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Datasets No Disguise Model Accuracy Attack Success Rate MNIST 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='2% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='14 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='1% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='87% FASHION 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='7 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='3% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='08± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='86% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='51± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content='27% Finally, if only Level-1 adversaries are expected, RMT can also be used to effectively protect from model- targeted attacks, as the models trained on RMT disguised data can only be applied to disguised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' CONCLUSION Outsourcing large image datasets to the cloud for deep learning has been an economical and popular option, but it also raises concerns about data and model confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The existing solutions are either too expensive to be practical, vulnerable to different model-based adversarial attacks, or ineffective in protecting the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' By focusing on the training image reconstruction and re-identification attacks, we propose image disguising mechanisms that efficiently thwart the attacks and preserve model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The combi- nation of random image-block permutation and block-wise AES encryption or multidimensional transformation (RMT) does not require any changes to the existing DNN modeling architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' Experimental results show that the RMT method can preserve the model quality and provide sufficient attack resilience under Level-1 adversarial knowledge – adversaries knowing only the disguised images and the domain informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The AES method improves the attack resilience against Level-2 adversaries who manage to obtain pairs of original images and disguised ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' However, the AEs method may seriously damage some datasets’ utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We also show that the disguising methods can protect the trained models from model-targeted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' The future work will be focused on new image disguising mechanisms that can more efficiently preserve utility with stronger security guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' We will also extend the related research to non-image data.' metadata={'source': 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model-inversion attacks against deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} +page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtAyT4oBgHgl3EQfa_c9/content/2301.00252v1.pdf'} diff --git a/z9E0T4oBgHgl3EQfdACa/content/tmp_files/2301.02371v1.pdf.txt b/z9E0T4oBgHgl3EQfdACa/content/tmp_files/2301.02371v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee73bfe5446d0ec4aa4f7cb45cad4f44b354ef5d --- /dev/null +++ b/z9E0T4oBgHgl3EQfdACa/content/tmp_files/2301.02371v1.pdf.txt @@ -0,0 +1,2231 @@ +Anchor3DLane: Learning to Regress 3D Anchors for Monocular 3D Lane +Detection +Shaofei Huang1,2 +Zhenwei Shen3* +Zehao Huang3 +Zihan Ding4 +Jiao Dai1,2 +Jizhong Han1,2 +Naiyan Wang3 +Si Liu4 +1 Institute of Information Engineering, Chinese Academy of Sciences +2 School of Cyber Security, University of Chinese Academy of Sciences +3 TuSimple +4 Institute of Artificial Intelligence, Beihang University +{nowherespyfly, zehaohuang18, zihanding819, winsty}@gmail.com +shenzhenwei@outlook.com +{hanjizhong, daijiao}@iie.ac.cn +liusi@buaa.edu.cn +Abstract +Monocular 3D lane detection is a challenging task due +to its lack of depth information. A popular solution to 3D +lane detection is to first transform the front-viewed (FV) im- +ages or features into the bird-eye-view (BEV) space with +inverse perspective mapping (IPM) and detect lanes from +BEV features. However, the reliance of IPM on flat ground +assumption and loss of context information makes it inac- +curate to restore 3D information from BEV representations. +An attempt has been made to get rid of BEV and predict 3D +lanes from FV representations directly, while it still under- +performs other BEV-based methods given its lack of struc- +tured representation for 3D lanes. In this paper, we define +3D lane anchors in the 3D space and propose a BEV-free +method named Anchor3DLane to predict 3D lanes directly +from FV representations. 3D lane anchors are projected +to the FV features to extract their features which contain +both good structural and context information to make ac- +curate predictions. +We further extend Anchor3DLane to +the multi-frame setting to incorporate temporal informa- +tion for performance improvement. In addition, we also +develop a global optimization method that makes use of +the equal-width property between lanes to reduce the lat- +eral error of predictions. Extensive experiments on three +popular 3D lane detection benchmarks show that our An- +chor3DLane outperforms previous BEV-based methods and +achieves state-of-the-art performances. +1. Introduction +Monocular 3D lane detection, which aims at estimating +the 3D coordinates of lane lines from a frontal-viewed im- +*Work done while at TuSimple +Front-Viewed +Image/Feature +IPM +BEV Lane +Detection +2D Lane +Detection +Depth +Estimation +Projecting +Dense Depth Map +Segmentation Mask +Front-Viewed +Image/Feature +Bird-Eye-Viewed +Image/Feature +Lane Heights +Front-Viewed +Image/Feature +Anchor +Projection +3D Anchors +Feature +Sampling +Iterative +Regression +BEV Lanes +3D Lanes +3D Lanes +3D Lanes +[ "$%& ] +[ "$%& ] +[ "$%& ] +3D Lane +Detection +(a) +(b) +(c) +Figure 1. (a) BEV-based methods, which perform lane detection +in the warped BEV images or features. (b) Non-BEV method, +which projects 2D lane predictions back to 3D space with esti- +mated depth. (c) Our Anchor3DLane projects 3D anchors into FV +features to sample features for 3D prediction directly. +age, is one of the essential modules in autonomous driv- +ing systems. Accurate and robust perception of 3D lanes +is not only critical for stable lane keeping, but also serves +as an important component for downstream tasks like high- +definition map construction [21, 25], and trajectory plan- +ning [1, 43]. However, due to the lack of depth informa- +tion, estimating lanes in 3D space directly from 2D image +domain still remains very challenging. +arXiv:2301.02371v1 [cs.CV] 6 Jan 2023 + +045 +0.4 +0.35 +0.3 +0.25 +0.2 +100 +80 +0.15 +60 +0.1 +40 +0.05 +2:0 +0 +lane] +[ane2 +lane3 +lane42.0 +1.5 +1.0 +0.5 +0.0 +0.5 +-1.0 +1.5 +2.0 +0 +20 +5 +40 +10 +60 +15 +80 +20 +100 +252.0 +15 +10 +0.5 +0.0 +-0.5 +o +0.5 +2.0 +0 +5 +20 +10. +40 +25 +60 +80 +20. +GOr +25A straightforward way to tackle the above challenges +is to detect lanes from the bird-eye-viewed (BEV) space. +As illustrated in Figure 1(a), a common practice of BEV- +based methods [5, 7, 8, 20] is to warp images or features +from frontal-viewed (FV) space to BEV with inverse per- +spective mapping (IPM), thereby transforming the 3D lane +detection task into 2D lane detection task in BEV. To project +the detected BEV lanes back into 3D space, coordinates of +the lane points are then combined with their corresponding +height values which are estimated by a height estimation +head. Though proven effective, their limitations are still ob- +vious: (1) IPM relies on a strict assumption of flat ground, +which does not hold true for uphill or downhill cases. (2) +Since IPM warps the images on the basis of ground, some +useful height information as well as the context information +above the road surface are lost inevitably. For example, ob- +jects like vehicles on the road are severely distorted after +warping. Therefore, information lost brought by IPM hin- +ders the accurate restoration of 3D information from BEV +representations. +Given the above limitations of BEV, some works tried +to predict 3D lanes from FV directly. As illustrated in Fig- +ure 1(b), SALAD [39] decomposes 3D lane detection task +into 2D lane segmentation and dense depth estimation. The +segmented 2D lanes are projected into 3D space with cam- +era intrinsic parameters and the estimated depth informa- +tion. Even though getting rid of the flat ground assumption, +SALAD lacks structured representations of 3D lanes. As a +result, it is unnatural to extend it to more complex 3D lane +settings like multi-view or multi-frame. +Moreover, their +performance is still far behind the state-of-the-art methods +due to the unstructured representation. +In this paper, we propose a novel BEV-free method +named Anchor3DLane to predict 3D lanes directly from +FV concisely and effectively. +As shown in Figure 1(c), +our Anchor3DLane defines lane anchors as rays in the 3D +space with given pitches and yaws. +Afterward, we first +project them to corresponding 2D points in FV space us- +ing camera parameters, and then obtain their features by +bilinear sampling. A simple classification head and a re- +gression head are adopted to generate classification proba- +bilities and 3D offsets from anchors respectively to make +final predictions. Unlike the information loss in IPM, sam- +pling from original FV features retains richer context infor- +mation around lanes, which helps estimate 3D information +more accurately. Moreover, our 3D lane anchors can be it- +eratively refined to sample more accurate features to better +capture complex variations of 3D lanes. Furthermore, An- +chor3DLane can be easily extended to the multi-frame set- +ting by projecting 3D anchors to adjacent frames with the +assistance of camera poses between frames, which further +improves performances over single-frame prediction. +In addition, we also utilize global constraints to refine the +challenging distant parts due to low resolution. The moti- +vation is based on an intuitive insight that lanes in the same +image appear to be parallel in most cases except for the fork +lanes, i.e., distances between different point pairs on each +lane pair are nearly consistent. By applying a global equal- +width optimization to non-fork lane pairs, we adjust 3D lane +predictions to make the width of lane pairs consistent from +close to far. The lateral error of distant parts of lane lines +can be further reduced through the above adjustment. +Our contributions are summarized as follows: +• We propose a novel Anchor3DLane framework that di- +rectly defines anchors in 3D space and regresses 3D +lanes directly from FV without introducing BEV. An +extension to the multi-frame setting of Anchor3DLane +is also proposed to leverage the well-aligned temporal +information for further performance improvement. +• We develop a global optimization method to utilize the +equal-width properties of lanes for refinement. +• Without bells and whistles, our Anchor3DLane out- +performs previous BEV-based methods and achieves +state-of-the-art performances on three popular 3D lane +detection benchmarks. +2. Related Works +2.1. 2D Lane Detection +2D lane detection [12, 22, 24, 32, 40] aims at obtaining +the accurate shape and locations of 2D lanes in the images. +Earlier works [2, 10, 13, 36, 42] mainly focus on extract- +ing low-level handcrafted features, such as edge and color +information. However, these approaches often have com- +plex feature extraction and post-processing designs and are +less robust under changing scenarios. With the development +of deep learning, CNN-based methods have been explored +recently and achieve notable performance. Segmentation- +based methods [11, 23, 24, 26] formulate 2D lane detection +task as a per-pixel classification problem and typically focus +on how to explore more effective and semantically informa- +tive features. To make predictions more sparse and flexible, +keypoint-based methods [15,27,35,38] model lane lines as +sets of ordered keypoints and associate keypoints belonged +to the same lane together by postprocessing. Apart from the +above methods, anchor-based methods [17, 19, 31, 41] are +also popular in 2D lane detection task due to their concise- +ness and effectiveness. LineCNN [17] first defines straight +rays emitted from the image boundary to fit the shape of +2D lane lines. Non-Maximum Suppression (NMS) is then +applied to the 2D lanes to keep only lanes with higher confi- +dence. LaneATT [31] develops anchor-based feature pool- +ing to extract features for the 2D anchors. CLRNet [41] +learns to refine the initial anchors iteratively through the +feature pyramid. + +2.2. 3D Lane Detection +Since projecting 2D lanes back into 3D space suffers +from inaccuracy as well as less robustness, 3D lane detec- +tion task is proposed to predict lanes in 3D space end to +end. Some works utilize multiple sensors, such as stereo +cameras [4] and Lidar-camera [3] to restore 3D informa- +tion. However, the collection and annotation cost of multi- +sensor data is expensive, restricting the practical applica- +tion of these methods. Therefore, monocular camera image +based 3D lane detection [6–8,20,39] attracts more attention. +Due to the good geometric properties of lanes in the +perspective of BEV, 3DLaneNet [7] utilizes IPM to trans- +form features from FV into BEV and then regresses the an- +chor offsets of lanes in the BEV space. CLGo [20] trans- +forms raw images into BEV images with the estimated +camera pitches and heights and fits the lane lines by pre- +dicting polynomial parameters. Since IPM relies heavily +on the flat ground assumption, lanes represented in BEV +space may be misaligned with 3D space in rough ground +cases. To deal with the above issue, Gen-LaneNet [8] fur- +ther makes a distinction between the virtual top view gen- +erated by IPM and the true top view in 3D space for better +space alignment. Persformer [5] utilizes deformable atten- +tion to generate BEV features more adaptively and robustly. +SALAD [39] tries to get rid of BEV by decomposing 3D +lane detection into 2D lane segmentation and dense depth +estimation tasks. Different from the above methods, our +Anchor3DLane defines anchors in the 3D space to explicitly +model 3D lanes and bridge the gap between FV space and +3D space. The projection and sampling operations ensure +the accuracy of anchor feature extraction, enabling effec- +tively predicting 3D lanes directly from FV representations +without introducing BEV. +3. Method +The overall architecture of our Anchor3DLane is il- +lustrated in Figure 3. +Given a front-viewed image I ∈ +RH×W ×3 as input, where H and W denote the height and +width of the input image, a CNN backbone (e.g., ResNet- +18 [9]) is adopted to extract 2D visual features represented +in FV space. +To enlarge the receptive field of the net- +work, we further insert a single Transformer layer [34] af- +ter the backbone to obtain the enhanced 2D feature map +F ∈ RHf ×Wf ×C, where Hf, Wf, and C represent the +height, width and channel number of feature map respec- +tively. 3D anchors are then projected to this feature map +F with the assistance of camera parameters, and the cor- +responding anchor features are sampled using bilinear in- +terpolation. Afterward, we apply a classification head and a +regression head to the sampled anchor features to make pre- +dictions, with each head composed of several lightweight +fully connected layers. Furthermore, the predictions can be +𝑿𝒈 +𝒀𝒈 +𝒁𝒈 +𝑶𝒈 +𝜽 +𝝓 +𝑶𝒄 +𝒁𝒄 +𝒀𝒄 +𝑿𝒄 +𝒒𝟒 +𝒑𝟑 +𝒒𝟐 +𝒒𝟏 +𝒒'𝟒 +𝒒'𝟑 +𝒒'𝟐 +𝒒'𝟏 +𝒙𝒔 +Road +3D Anchor +Front-viewed +Image/Feature +Anchor +Projection +Camera +𝒑𝟏 +𝒒𝟑 +𝒑𝟐 +𝒑𝟒 +3D Lane +Figure 2. Illustration of 3D anchor and 3D lane in the ground +coordinate system. +regarded as new 3D anchors for iterative regression. +3.1. 3D Lane Representation +We first revisit the representation of 3D lanes in this sec- +tion. As shown in Figure 2, two different coordinate sys- +tems are involved in our paper, including the camera coordi- +nate system and the ground coordinate system. The camera +coordinate directly corresponds with the FV image and is +a right-handed coordinate system defined by origin Oc and +Xc, Yc, Zc axes, with Oc located at the center of the cam- +era and Zc pointing forward vertical to the camera plane. +3D lanes are commonly annotated in the ground coordinate +system, of which the origin Og is set right below Oc, x- +axis Xg points positive to the right, y-axis Yg points posi- +tive forwards and z-axis Zg points positive upwards. A 3D +lane is described by 3D points with N uniformly sampled +y-coordinates y = {yk}N +k=1. Thus, we denote the i-th 3D +lane as Gi = {pk +i }N +k=1 and its k-th point is represented as +pk +i = (xk +i , yk, zk +i , visk +i ), where the first 3 elements denote +the location of pk +i in the ground coordinate system and the +last one denotes the visibility of pk +i . It is worth noting that +we elaborate our method based on the ground coordinate +system following the common practices adopted in previ- +ous works [7, 8]. However, our Anchor3DLane is able to +work in an arbitrary 3D coordinate system as long as cam- +era calibration is available. +3.2. Anchor3DLane +3.2.1 +Representation of 3D Lane Anchors +Our 3D lane anchors are defined in the same coordinate sys- +tem as 3D lanes, i.e., ground coordinate, for ease of posi- +tion regression. As illustrated in Figure 2, a 3D anchor is +a ray starting from (xs, 0, 0) with pitch θ and yaw φ. Sim- +ilar to 3D lanes, we also sample N points for each anchor + +Transformer +Layer +3D Anchors +Projecting +[+, 7%⟶'] +… +Anchor Features +Iterative +Regression +Camera Parameters +Backbone +Classification +Head +Regression +Head +4 ∈ ℝ'!×)!×* +3D Proposals +Figure 3. The overall architecture of Anchor3DLane. Given a front-viewed input image, a CNN backbone and a Transformer layer are +adopted to first extract visual feature F. 3D anchors are then projected to sample their features from F given camera parameters. Afterward, +a classification head and a regression head are applied to make the final predictions. The lane predictions can also serve as new 3D anchors +for iterative regression. +by the same y-coordinates and represent the j-th 3D an- +chor by Aj = {qk +j }N +k=1, and its k-th point is denoted by +qk +j = (xk +j , yk, zk +j ). Different from previous works [5, 7] +that define anchors in the BEV plane, our 3D anchors have +pitches to the ground and could fit the lane shape better. +3.2.2 +Anchor Projection and Feature Sampling +To obtain features of 3D anchors, we first project them +into the plane of FV feature F using camera parameters as +shown in Figure 2. Given an anchor Aj, we take its k-th +point qk +j as an example to explain the projection operation +and omit the subscript j for simplicity as follows: +� +� +˜uk +˜vk +dk +� +� = KTg→c +� +��� +xk +yk +zk +1 +� +��� , +(1) +uk = Wf/W · ˜uk +dk , +(2) +vk = Hf/H · ˜vk +dk , +(3) +where K ∈ R3×3 denotes camera intrinsic parameters, +Tg→c ∈ R3×4 denotes the transform matrix from ground +coordinate to camera coordinate, and dk denotes the depth +of qk to the camera plane. Through the above formula- +tions, qk is projected to position (uk, vk) in FV feature F. +Finally, the feature of anchor Aj is obtained through bilin- +ear interpolation within the neighborhood of the projected +points and is represented as {F(uk,vk)}N +k=1. +3.2.3 +3D Lane Prediction +We concatenate features of points belonging to the same +anchor as its feature representation. Then we apply a clas- +sification head and a regression head to the anchor fea- +tures for predicting classification probabilities cj ∈ RL, +anchor points offsets (∆xj +∈ +RN, ∆zj +∈ +RN) += +{(∆xk +j , ∆zk +j )}N +k=1 and visibility of each point visj ∈ RN +respectively, with j ∈ [1, M]. L and M denote the num- +bers of lane types and 3D anchors respectively. +In this +way, 3D lane proposals are generated as {Pj = (cj, xj + +∆xj, y, zj + ∆zj, visj)}M +j=1. Furthermore, these 3D lane +proposals can also serve as new anchors in the following it- +erative regression steps as illustrated in Figure 3. Through +iterative regression, proposals can be refined progressively +to better fit the lane shape. +During training, we associate n nearest anchors to each +ground-truth lane and the rest are defined as negative sam- +ples. Distance metric between ground-truth Gi and anchor +Aj is calculated as follows: +D(Gi, Aj) = +�N +k=1 visk +i · +� +(xk +i − xk +j )2 + (zk +i − zk +j )2 +�N +k=1 visk +i +. +(4) +This metric is also used in Non-Maximum Suppression +(NMS) during inference to keep a reasonable number of +proposals except that distances are calculated between visi- +ble parts of two proposals. +We adopt focal loss [18] for training classification to bal- +ance the positive and negative proposals as follows: +Lcls = − +M +� +j=1 +L +� +l=1 +αl(1 − cl +j)γ log cl +j, +(5) +where αl and γ are the hyperparamters for focal loss. The +regression loss is only calculated between the positive pro- + +07 +0.5 +0.0 +2.5. +5.0 +..0 +112.5 +50 +.5 +281 +4812.0 +245 +0.5 +0.0 +015 +-.5 +7.0 +0 +0 +5 +2461 +461 +6.0 +80] +20posals and their assigned ground-truth lanes following [8]: +Lreg = +Mp +� +i=1 +N +� +k=1 +(∥ ˆ +vis +k +i · (xk +i + ∆xk +i − ˆxk +i )∥1 ++ +Mp +� +i=1 +N +� +k=1 +∥ ˆ +vis +k +i · (zk +i + ∆zk +i − ˆzk +i )∥1) ++ +Mp +� +i=1 +N +� +k=1 +∥ ˆ +vis +k +i − visk +i ∥1. +(6) +Mp represents the total number of positive proposals. Here +we use ˆxk +i , ˆzk +i and ˆ +vis +k +i to denote the x, z coordinates and +visibility of the ground-truth lane points. +The total loss function of our Anchor3DLane is a com- +bination of the above two losses with corresponding coeffi- +cients: +L = λclsLcls + λregLreg. +(7) +3.3. Temporal Context Modeling +Thanks to the design of 3D anchors, our Anchor3DLane +can be easily extended to multi-frame 3D lane detection. +Given a 3D point (xt, yt, zt) in the t-th frame’s ground co- +ordinate system, we transform it to the t′-th frame’s ground +coordinate system with the following formulation: +� +� +xt′ +yt′ +zt′ +� +� = Tg(t)→g(t′) +� +��� +xt +yt +zt +1 +� +��� , +(8) +where Tg(t)→g(t′) ∈ R3×4 denotes the transformation ma- +trix from t-th frame to t′-th frame. Together with Equa- +tion 1, anchors defined in the current frame can be projected +to previous frames for sampling their features. For each an- +chor, we take its points from the current frame as query and +points from previous frames as key and value to conduct +cross-frame attention for feature aggregation. By integrat- +ing the well-aligned anchor features from multiple frames, +temporal context is incorporated into our Anchor3DLane to +enlarge its perception range. +3.4. Optimization with Equal-Width Constraint +In most cases, lanes in 3D space are nearly parallel with +each other, which is helpful in generating robust 3D estima- +tions from monocular 2D images. In this work, we lever- +age this geometry property of 3D lanes and formulate it as +an equal-width constraint to adjust the x-coordinates of lane +predictions. Given two lane predictions Pj = {pk +j }N +k=1 and +Pj′ = {pk +j′}N +k=1, width between Pj and Pj′ at point pair +pk +j and pk +j′ is calculated as: +wk +j,j′ = | cos θk +j (xk +j + ˜∆xk +j − xk +j′ − ˜∆xk +j′)|, +(9) +where ˜∆xk +j and ˜∆xk +j′ denote the adjustment to xk +j and xk +j′ +to be optimized respectively and θk +j denotes the normal di- +rection of the adjusted lane at pk +j . The objective function of +equal-width constraint is as follows: +min +{ ˜∆xj}j∈[1,Q] +1 +Q(Q − 1) +Q +� +j=1 +Q +� +j′=1,j′̸=j +L(wj,j′) ++ α 1 +Q +Q +� +j=1 +∥ ˜∆xj∥2, +(10) +where +L(wj,j′) = +N +� +k=1 +|wk +j,j′ − 1 +N +N +� +k′=1 +wk′ +j,j′|. +(11) +We use Q to denote the number of lane predictions after +NMS. L(wj,j′) restricts the width between Pj and Pj′ to +be consistent and the second item serves as a regularization +to avoid the adjusted results being too far from the original +predictions. We run this optimization as a post-processing +step to refine the prediction results of the network. +4. Experiments +4.1. Experimental Setting +4.1.1 +Datasets and Evaluation Metrics +We conduct experiments on three popular 3D lane detection +benchmarks, including ApolloSim [8], OpenLane [5], and +ONCE-3DLanes [39]. +ApolloSim is a photo-realistic synthetic dataset created +with Unity 3D engine which contains 10.5K images from +various virtual scenes, including highway, urban, residen- +tial, downtown, etc. In addition, the data is also diverse in +daytime, weather conditions, traffic/obstacles, and road sur- +face qualities. +OpenLane is a large-scale real-world 3D lane detection +dataset constructed upon the Waymo Open dataset [30]. +It contains 200K frames and over 880K lanes are anno- +tated. Camera intrinsics and extrinsics are provided for each +frame. All lanes are annotated including lanes in the oppo- +site direction if no curbside in the middle. Categories and +scene tags (e.g., weather and locations) are also provided. +ONCE-3DLanes is a real-world 3D lane detection +dataset with 1 million scenes. It consists of 211K images +with labeled 3D lane points. It covers different time peri- +ods (sunny, cloudy, rainy) and various regions (downtown, +suburbs, highway, bridges, and tunnels). Only camera in- +trinsics are provided in ONCE-3DLanes. +During the evaluation, the predictions and ground truth +lanes are matched via minimum-cost flow where the pair- +wise cost is defined as the square root of the sum of point- +wise Euclidean distance. A prediction is considered as true + +Scene +Method +AP(%)↑ +F1(%)↑ +x err/C(m) ↓ +x err/F(m) ↓ +z err/C(m) ↓ +z err/F(m) ↓ +Balanced Scene +3DLaneNet [7] +89.3 +86.4 +0.068 +0.477 +0.015 +0.202 +Gen-LaneNet [8] +90.1 +88.1 +0.061 +0.496 +0.012 +0.214 +CLGo [20] +94.2 +91.9 +0.061 +0.361 +0.029 +0.250 +PersFormer [5] +- +92.9 +0.054 +0.356 +0.010 +0.234 +GP [16] +93.8 +91.9 +0.049 +0.387 +0.008 +0.213 +Anchor3DLane (Ours) +97.2 +95.6 +0.052 +0.306 +0.015 +0.223 +Anchor3DLane†(Ours) +97.1 +95.4 +0.048 +0.299 +0.013 +0.220 +Rare Subset +3DLaneNet [7] +74.6 +72.0 +0.166 +0.855 +0.039 +0.521 +Gen-LaneNet [8] +79.0 +78.0 +0.139 +0.903 +0.030 +0.539 +CLGo [20] +88.3 +86.1 +0.147 +0.735 +0.071 +0.609 +PersFormer [5] +- +87.5 +0.107 +0.782 +0.024 +0.602 +GP [16] +85.2 +83.7 +0.126 +0.903 +0.023 +0.625 +Anchor3DLane (Ours) +96.9 +94.4 +0.094 +0.693 +0.027 +0.579 +Anchor3DLane† (Ours) +96.1 +93.9 +0.086 +0.678 +0.025 +0.562 +Visual Variations +3D-LaneNet [7] +74.9 +72.5 +0.115 +0.601 +0.032 +0.230 +Gen-LaneNet [8] +87.2 +85.3 +0.074 +0.538 +0.015 +0.232 +CLGo [20] +89.2 +87.3 +0.084 +0.464 +0.045 +0.312 +PersFormer [5] +- +89.6 +0.074 +0.430 +0.015 +0.266 +GP [16] +92.1 +89.9 +0.060 +0.446 +0.011 +0.235 +Anchor3DLane (Ours) +93.6 +91.4 +0.068 +0.367 +0.020 +0.232 +Anchor3DLane† (Ours) +92.6 +92.3 +0.049 +0.363 +0.019 +0.242 +Table 1. Comparison with state-of-the-art methods on ApolloSim dataset with three different split settings. “C” and “F” are short for close +and far respectively. † denotes iterative regression. +positive if over 75% of its points’ distances to ground-truth +points are less than a threshold, i.e., 1.5m. With the defi- +nition above, Average Precision (AP) and the maximum F1 +score are further calculated, and x/z errors are counted sepa- +rately at close (0-40m) and far (40-100m) ranges. We report +the results of F1 score, AP, and x/z-errors on ApolloSim +dataset. On OpenLane dataset, except for F1 score and x/z +errors, we further report category accuracy which calculates +the proportion of predictions whose categories are correctly +predicted to all true positive predictions. ONCE-3DLanes +adopts a different way to match predictions and ground truth +lanes. The matching degree is firstly decided by IoU on the +top-view plane and pairs above the threshold are further cal- +culated with their unilateral Chamfer Distance (CD) as the +matching error. A true positive is counted when CD is un- +der the threshold. We report F1 score, precision, recall, and +CD error for results on ONCE-3DLanes. +4.1.2 +Implementation Details +We choose ResNet-18 [9] as the backbone of our An- +chor3DLane. +To maintain feature resolution, we set the +downsampling stride of its last two stages to 1 and replace +the 3 × 3 convolutions with dilated convolutions. The start- +ing positions xs of 3D anchors are evenly placed along the +x-axis with an interval of 1.3m. For each xs, different yaws +φ ∈ {0◦, ±1◦, ±3◦, ±5◦, ±7◦, ±10◦, ±15◦, ±20◦, ±30◦} +and pitches θ ∈ {0◦, ±1◦, ±2◦, ±5◦} are set respectively. +The number of points N for each anchor is set to 10 for ex- +periments on ApolloSim and ONCE and 20 for OpenLane. +We resize the image to 360 × 480 before feeding it to the +backbone and the shape of F is 45 × 60 × 64. During train- +ing, λcls and λreg are both set to 1 and the number of posi- +tive proposals is set as 3. The distance threshold for NMS is +2 during inference. For multi-frame Anchor3DLane, each +time we randomly select 1 frame from the previous 5 frames +to interact with current frame during training, and select +the first frame of the previous 5 frames during inference. +Since car poses are only available in OpenLane dataset, we +only conduct temporal experiments on this dataset. We use +Adam optimizer [14] with weight decay set as 1e−4, and +set the initial learning rate to 1e−4. Step learning rate de- +cay is used during training. αl is set to 0.5 and γ is set to +2 for focal loss. More details about our Anchor3DLane are +included in supplementary materials. +4.2. Quantitative Results +4.2.1 +Results on ApolloSim +Table 1 shows the experimental results under three differ- +ent split settings of the ApolloSim dataset, including bal- +anced scene, rare subset and visual variations. +We re- +port the results of both our original Anchor3DLane and +Anchor3DLane with iterative regression optimized with +equal-width constraint. It is shown that our original An- + +Method +F1(%)↑ +Cate Acc(%)↑ +x err/C(m) ↓ +x err/F(m) ↓ +z err/C(m) ↓ +z err/F(m) ↓ +3D-LaneNet [7] +44.1 +- +0.479 +0.572 +0.367 +0.443 +GenLaneNet [8] +32.3 +- +0.591 +0.684 +0.411 +0.521 +PersFormer [5] +50.5 +92.3 +0.485 +0.553 +0.364 +0.431 +Anchor3DLane (Ours) +53.1 +90.0 +0.300 +0.311 +0.103 +0.139 +Anchor3DLane† (Ours) +53.7 +90.9 +0.276 +0.311 +0.107 +0.138 +Anchor3DLane-T† (Ours) +54.3 +90.7 +0.275 +0.310 +0.105 +0.135 +Table 2. Comparison with state-of-the-art methods on OpenLane validation set. † denotes iterative regression. Anchor3DLane-T denotes +incorporating multi-frame information. “Cate Acc” means category accuracy. +Method +All +Up & Down +Curve +Extreme Weather +Night +Intersection +Merge & Split +3D-LaneNet [7] +44.1 +40.8 +46.5 +47.5 +41.5 +32.1 +41.7 +GenLaneNet [8] +32.3 +25.4 +33.5 +28.1 +18.7 +21.4 +31.0 +PersFormer [5] +50.5 +42.4 +55.6 +48.6 +46.6 +40.0 +50.7 +Anchor3DLane (Ours) +53.1 +45.5 +56.2 +51.9 +47.2 +44.2 +50.5 +Anchor3DLane† (Ours) +53.7 +46.7 +57.2 +52.5 +47.8 +45.4 +51.2 +Anchor3DLane-T† (Ours) +54.3 +47.2 +58.0 +52.7 +48.7 +45.8 +51.7 +Table 3. Comparison with state-of-the-art methods on OpenLane validation set. F1 score is presented for each scenario. † denotes iterative +regression. Anchor3DLane-T denotes incorporating multi-frame information. +chor3DLane outperforms previous methods with large mar- +gins on AP and F1 score on all the three splits with sim- +ple design, i.e., +3.0% AP and +2.7% F1 score on bal- +anced scene, +8.6% AP and +6.9% F1 score on rare sub- +set, +2.4% F1 score and +1.5% AP on visual variations, +showing the superiority of our method. Our Anchor3DLane +also achieves comparable or lower x/z errors compared with +previous methods, especially for x error far, indicating re- +gressing over 3D anchors have greater advantages for dis- +tant predictions. Furthermore, by iteratively regressing over +the proposals predicted by Anchor3DLane, x/z errors can +be further reduced to better fit the shape of 3D lanes. +4.2.2 +Results on OpenLane +We present the experimental results of our method opti- +mized with the equal-width constraint on OpenLane dataset +in Table 2. Our original Anchor3DLane outperforms Pers- +Former by 2.6% F1 score improvement. +Moreover, our +method achieves much more precise predictions than Pers- +Former, i.e., −0.185m on x error close, −0.242m on x er- +ror far, −0.261m on z error close, and −0.292m on z error +far respectively, which are crucial for driving safety. The +gap in x/z errors indicates that under real scenarios with di- +verse conditions, directly sampling features from FV repre- +sentation could maintain more environment context infor- +mation, thus producing more precise predictions. By in- +corporating iterative regression and temporal information +in Anchor3DLane, the overall performances can be fur- +ther boosted. In Table 3, we compare with previous meth- +ods under different scenarios and report F1 score for each +scenario. Our method produces much better performance +in Up&Down scenarios, showing the advantage of 3D an- +chor regression in uneven ground. It is also worth noting +that we adopt a lightweight CNN, i.e., ResNet-18 as the +backbone of Anchor3DLane, which still outperforms Pers- +Former with a larger backbone, i.e., EfficientNet-B7 [33]. +4.2.3 +Results on ONCE-3DLanes +In Table 4, we present the experimental results on the +ONCE-3DLanes dataset. Since camera extrinsics are not +available in ONCE-3DLanes, we define the 3D anchors in +the camera coordinate system and make predictions in the +same space. Our method also achieves state-of-the-art per- +formances on this dataset. Compared with PersFormer, our +Anchor3DLane still produces a higher F1 score and reduces +CD error by 18.9% relatively, which indicates that 3D an- +chors are able to adapt different 3D coordinate systems. +4.2.4 +Ablation Study +In this section, we follow previous work [5] to conduct +most ablation studies on OpenLane-300, which is a sub- +set of OpenLane. As for feature sampling experiments, we +present the results on the original OpenLane to verify the ef- +fectiveness of our method. More ablation studies and qual- +itative results are included in the supplementary materials. + +Method +F1(%)↑ +P(%)↑ +R(%)↑ +CD Error(m)↓ +3D-LaneNet [7] +44.73 +61.46 +35.16 +0.127 +Gen-LaneNet [8] +45.59 +63.95 +35.42 +0.121 +SALAD [39] +64.07 +75.90 +55.42 +0.098 +PersFormer [5] +74.33 +80.30 +69.18 +0.074 +Anchor3DLane (Ours) +74.44 +80.50 +69.23 +0.064 +Anchor3DLane† (Ours) +74.87 +80.85 +69.71 +0.060 +Table 4. +Comparison with state-of-the-art methods on ONCE- +3DLanes validation set. Results under τCD = 0.3 are displayed +here. † denotes iterative regression. “P” and “R” are short for +precision and recall respectively. +Input +Feat +F1(%) +x err/C(m) +x err/F(m) +z err/C(m) +z err/F(m) +BEV +BEV +47.6 +0.466 +0.421 +0.119 +0.170 +FV +BEV +47.6 +0.443 +0.446 +0.118 +0.160 +FV +FV +53.1 +0.300 +0.31 +0.103 +0.139 +Table 5. Comparison between sampling anchor features from BEV +features and FV features. +Sampling anchor features from FV features. To illus- +trate the superiority of FV features, we compare the per- +formances of extracting anchor features from FV features +and BEV features. The results are shown in Table 5. We +explore different ways of obtaining BEV features, includ- +ing warping FV image to BEV image (line 1) and warping +FV feature to BEV feature (line 2), and keep the other set- +tings same as our original Anchor3DLane. Results show +that sampling anchor features from FV features produces +the best F1 score and x/z errors, especially for x errors, +where more than 10cm gap exists between FV anchor fea- +tures and BEV anchor features. The above performance gap +indicates that the context information contained in raw FV +features is beneficial for accurate lane predictions. +Iter +F1(%) +x err/C(m) +x err/F(m) +z err/C(m) +z err/F(m) +1 +54.8 +0.318 +0.349 +0.101 +0.147 +2 +56.3 +0.287 +0.335 +0.103 +0.152 +3 +57.0 +0.287 +0.327 +0.104 +0.148 +Table 6. Ablation study on the steps of iterative regression. +Steps of iterative regression. +Table 6 presents the +results of different steps of iterative regression for An- +chor3DLane. Compared with no iterative regression, 2 iter- +ations produces relatively large performance improvements. +More steps of iterative regression can further reduce lateral +errors as well as elevate F1 score by refining the shape of +proposals progressively. +Temporal integration methods. In this section, we ex- +plore different methods to integrate anchor features of mul- +tiple frames. Besides the cross-frame attention that we men- +tioned in Section 3.3, we also try linear fusion which con- +Method +F1(%) +x err/C(m) +x err/F(m) +z err/C(m) +z err/F(m) +w/o Temporal +54.8 +0.318 +0.349 +0.101 +0.147 +Linear Fusion +54.9 +0.322 +0.343 +0.102 +0.148 +Weighted Sum +55.8 +0.320 +0.346 +0.101 +0.150 +Attention +55.2 +0.308 +0.330 +0.099 +0.145 +Table 7. Ablation study on temporal integration methods. +catenates features of the same anchor along their channels +and fuses them with a linear layer, and weighted sum which +learns to predict a group of weights for each y-coordinate +to fuse features of the same anchor elementwisely, As +shown in Table 9, comparing with the baseline, incorporat- +ing temporal information into Anchor3DLane can improve +the overall performance significantly due to the richer con- +text information obtained from previous frames. Weighted +sum produces better results than linear fusion, indicating +that dynamic weights are necessary for different points at +different distances. Although weighted sum achieves a bet- +ter F1 score compared with single frame setting, x/z errors +increase at the same time. Among the 3 integration meth- +ods, cross-frame attention, which aggregates anchor fea- +tures with more anchor points from previous frames, im- +proves both F1 score and x errors and achieves the best per- +formance balance. +Method +F1(%) +x err/C(m) +x err/F(m) +w/o EWC +54.8 +0.318 +0.349 +w/ EWC +55.0 +0.318 +0.337 +Table 8. Ablation study on Equal-Width Constraint (EWC). +Effect of equal-width constraint. We also illustrate the +comparison between predictions without and with equal- +width constraint optimization. +As shown in Table 8, by +applying the equal-width constraint to the lane predictions, +errors of the distant parts of the lane lines can be further +reduced by restricting them to have the same width as the +close parts. More visualization results of this constraint can +be found in the supplementary materials. +5. Conclusion and Limitations +In this work, we propose a novel Anchor3DLane frame- +work for 3D lane detection which bypasses the transforma- +tion to BEV space and predicts 3D lanes from FV directly. +By defining anchors in the 3D space and projecting them to +the FV features, accurate anchor features are sampled for +lane prediction. 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We resample 10 points for the ONCE- +3DLanes dataset at y-coordinates of {2, 5, 8, 10, 15, 20, 25, +30, 40, 50}. Since experiments are conducted in the camera +coordinate system where the origin is above the ground, the +starting positions of 3D anchors are set at (xs, 0, −1.5m). +Other training settings are the same as those on the Open- +Lane dataset as mentioned above. +Appendix B. Quantitative Results +B.1. The Range of Training Frames +Frame Range +F1(%) +x err/C(m) +x err/F(m) +z err/C(m) +z err/F(m) +3 frames +55.0 +0.306 +0.326 +0.099 +0.148 +5 frames +55.2 +0.308 +0.330 +0.099 +0.145 +7 frames +56.1 +0.312 +0.335 +0.101 +0.150 +Table 9. Ablation study on the range of training frames. +For temporal context modeling, we sample one frame +from different ranges of previous frames to aggregate its +feature to the current frame during training. The first frame +of the previous 5 frames is sampled during inference. As +shown in Table 9, the F1 score increases as the frame +range becomes larger, indicating that aggregating informa- +tion from farther frames yields a better estimation for the +current frame. +B.2. Computational Cost Analysis +Method +F1 Score(%) +FLOPs +Param +FPS +PersFormer [5] +50.5 +572.4G +54.9M +5.58 +Anchor3DLane (ours) +53.1 +38.1G +12.2M +87.29 +Anchor3DLane† (ours) +53.7 +42.4G +13.2M +73.73 +Anchor3DLane-T† (ours) +54.3 +82.3G +13.3M +30.22 +Table 10. +Comparison of computational cost and F1 score +on OpenLane validation set. +† denotes iterative regression. +Anchor3DLane-T denotes incorporating multi-frame information. +We report the computational cost comparison in Ta- +ble 10. Our Anchor3DLane achieves a higher F1 score on +the OpenLane dataset with much fewer FLOPs and param- +eters compared with PersFormer [5]. The inference speeds +(FPS) of these methods are measured using the code re- +leased by PersFormer on a single 2080 Ti GPU. Our orig- +inal Anchor3DLane achieves nearly 16 times faster infer- +ence speed than PersFormer. By adopting iterative regres- +sion and temporal context modeling, the F1 score is further +improved, while the inference speed decreases but is still +much faster than PersFormer. These results demonstrate our +Anchor3DLane is both effective and efficient. +B.3. Experimental Results with EfficientNet +To verify the adaptability and performance potential +of our method, we further conduct experiments with +EfficientNet-B3 [33] to compare with PersFormer which +adopts EfficientNet-B7 as the backbone. Results are shown +in Table 11, Table 12 and Table 13. On OpenLane dataset, +utilizing EfficientNet-B3 as the backbone could boost the +performance of our Anchor3DLane from 53.1% F1 score to +56.0% and reduce the x/z errors at the same time, indicating +that our method adapts well to stronger backbones. +Appendix C. Qualitative Results +ApolloSim. +We compare our Anchor3DLane with +CLGo [20] on the ApolloSim dataset and the results are +included in Figure 4. Our method has better lateral predic- +tions in the distant parts when lanes turn in the distance (row +2 and row 3). In addition, when encountering uphill (row 6) +or downhill (row 4 and row 5), our method can better cap- +ture the height changes than CLGo, which demonstrates the +superiority of directly regressing 3D anchors for 3D lanes. +OpenLane. We also compare with PersFormer [5] on +the OpenLane dataset in Figure 5. +Our Anchor3DLane +can better recover the whole lanes occluded by vehicles as +shown in column 2 of Figure 5 (a) and (b). +ONCE-3DLanes. +In Figure 6, we show the qualita- +tive results of our Anchor3DLane on the ONCE-3DLanes +dataset. Our method performs well in different scenes, such +as bad weather like rainy days (column 1 of row 1 and row +2). Since the 2D annotations of ONCE-3DLanes are gener- +ated by the lane detection model, annotations of some cases +are inaccurate or incomplete but our method still produces +fine predictions as shown in column 3. +Equal-Width Constraint. We show the visualization +results of equal-width constraint (EWC) optimization in +Figure 7. After adjusting the x coordinates of lanes with +EWC, lane predictions are parallel to each other and errors +in the distant parts are reduced as well. It is also worth +noting that the ground-truth lanes do not satisfy the equal- +width hypothesis in the close parts of some cases, which is +possibly due to annotation defects (column 3). Therefore, + +Method +Backbone +F1(%)↑ +Cate Acc(%)↑ +x err/C(m) ↓ +x err/F(m) ↓ +z err/C(m) ↓ +z err/F(m) ↓ +3D-LaneNet [7] +VGG-16 [29] +44.1 +- +0.479 +0.572 +0.367 +0.443 +GenLaneNet [8] +ERFNet [28] +32.3 +- +0.591 +0.684 +0.411 +0.521 +PersFormer [5] +EfficientNet-B7 [33] +50.5 +92.3 +0.485 +0.553 +0.364 +0.431 +Anchor3DLane (Ours) +ResNet-18 [9] +53.1 +90.0 +0.300 +0.311 +0.103 +0.139 +Anchor3DLane (Ours) +EfficientNet-B3 +56.0 +89.9 +0.293 +0.317 +0.103 +0.130 +Table 11. Comparison with state-of-the-art methods on OpenLane validation set with stronger backbone. +Method +Backbone +All +Up & Down +Curve +Extreme Weather +Night +Intersection +Merge & Split +3D-LaneNet [7] +VGG-16 +44.1 +40.8 +46.5 +47.5 +41.5 +32.1 +41.7 +GenLaneNet [8] +ERFNet +32.3 +25.4 +33.5 +28.1 +18.7 +21.4 +31.0 +PersFormer [5] +EfficientNet-B7 +50.5 +42.4 +55.6 +48.6 +46.6 +40.0 +50.7 +Anchor3DLane (Ours) +ResNet-18 +53.1 +45.5 +56.2 +51.9 +47.2 +44.2 +50.5 +Anchor3DLane (Ours) +EfficientNet-B3 +56.0 +50.3 +59.1 +53.6 +52.8 +47.4 +53.3 +Table 12. Comparison with state-of-the-art methods on OpenLane validation set with stronger backbone. F1 score is presented for each +scenario. +Method +Backbone +F1 Score(%)↑ +Precision(%)↑ +Recall(%)↑ +CD Error(m)↓ +3D-LaneNet [7] +VGG-16 +44.73 +61.46 +35.16 +0.127 +Gen-LaneNet [8] +ERFNet +45.59 +63.95 +35.42 +0.121 +SALAD [39] +SegFormer [37] +64.07 +75.90 +55.42 +0.098 +PersFormer [5] +EfficientNet-B7 +74.33 +80.30 +69.18 +0.074 +Anchor3DLane (Ours) +ResNet-18 +74.44 +80.50 +69.23 +0.064 +Anchor3DLane (Ours) +EfficientNet-B3 +75.02 +83.22 +68.29 +0.064 +Table 13. Comparison with state-of-the-art methods on ONCE-3DLanes validation set with stronger backbone. +adjusting with EWC may not be beneficial to reducing x +error. + +(a) CLGo +(b) Anchor3DLane +Figure 4. Comparison between CLGo [20] and our Anchor3DLane on the ApolloSim dataset. (a): Qualitative results of CLGo. (b): +Qualitative results of our Anchor3DLane. Blue:Ground-truth. Red: Prediction. + +0.10 +0.100.10 +0.25 +0.30 +500.10 +0.05 +0.00 +0.05 +0.100.10 +0.05 +0.00 +0.05 +0.10 +300.10 +0.100.10 +0.05 +0.00 +-0.05 +-0.100.10 +-0.10 +500.10 +0.100.100.70.7 +500.10 +50gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +gt +pred +(a) +PersFormer +(c) +PersFormer +(b) +Anchor3DLane +(d) +Anchor3DLane +Figure 5. Comparison between PersFormer [5] and our Anchor3DLane on the OpenLane dataset. (a)(c): Qualitative results of PersFormer. +(b)(d): Qualitative results of our Anchor3DLane. + +gt +0.3 +0.2 +0.1 +-axis +0.0 +-0.1 +-0.2 +N +-0.3 +-0.4 +80 +70 +60 +50 +40 +y-axis +30 +5 +20 +10pred +gt +gt +gt +0.6 +0.4 +-axis +0.2 +N +0.0 +0.2 +80 +70 +-5 +60 +50 +0 +40 +30 +5 +20 +10 +10pred +gt +gt I +gt +0.7 +z-axis +0.0 +-0.1 +-0.2 +80 +70 +-5 +60 +50 ++ +40 +30 +5 +20 +10 +10pred +pred +gt +gt +gt +0.10 +0.05 +0.00 +.S +-0.05 +!xe- +-0.10 +-0.15 +N +-0.20 +-0.25 +-0.30 +80 +70 +60 +50 +40 +y-axis +30 +20 +5 +10pred +pred +gt +gt +gt +0.10 +0.05 +0.00 +.S +-0.05 +axi +-0.10 +-0.15 +N +-0.20 +-0.25 +-0.30 +-10 +80 +70 +09 +-5 +50 +40 +y-axis +30 +20 +5 +10gt +gt +gt +gt +0.0 +0.2 + SIxe-z +-0.4 +-0.6 +-0.8 +-10 +80 +70 +-5 +60 +50 +X-axis +40 +y-axis +0 +30 +20 +10 +5gt +gt +gt +gt +0.0 +-0.2 +-axis +-0.4 +-0.6 +N +-0.8 +-10 +80 +70 +60 +-5 +50 ++ +40 +y-axis +0 +30 +20 +10 +5pred +gt +0.1 +0.0 +-axis +-0.1 +-0.2 +N +-0.3 +-0.4 +80 +70 +60 +-2.5 +50 +40 +y-axis +30 +-2.0 +20 +10pred +gt +0.10 +0.05 +0.00 +.S +-0.05 +-axi +-0.10 +2 +-0.15 +-0.20 +80 +70 +60 +2.5 +X-axis +50 +40 +y-axis +-2.0 +20 +10pred +gt +gt +4.0 +3.5 +3.0 +z-axis +1.5 +1.0 +0.5 +0.0 +80 +70 +60 +-5 +50 +40 +y-axis +30 +20 +10pred +gt +gt +2.5 +0.0 +-2.5 +z-axis +-5.0 +-7.5 +-10.0 +-12.5 +-15.0 +17.5 +80 +70 +60 +-5 +50 +40 +y-axis +30 +0 +20 +10gt +0.3 +0.2 +0.1 +-axis +0.0 +-0.1 +N +-0.2 +0.3 +80 +70 +60 +50 +40 +y-axis +5 +30 +20 +10Ground-Truth +Prediction +Figure 6. Qualitative results on the ONCE-3DLanes dataset. + +0.00 +0.25 +-0.50 +-0.75 +-1.00 +-1.25 +N +-1.50 +1.75 +2.00 +5.0 +50 +2.5 +% +2.5 +20 +10 +5.00.00 +-0.25 +-0.50 +-0.75 +z-axis +-1.00 +1.25 +1.50 +1.75 +-2.00 +50 +40 +axis +20z-axis +50 +20 +100.00 +-0.75 +-1.00 +xe-z +1.25 +2.00 +10 +50 +40 +30 +y-axis +20 +100.00 +Z-axis +2.00 +2.25 +50 +40 +X-axis +20 +100.0 +Z-axis +50 +40 +30 +y-axis +20 +10Z-axis +50 +40 +30 +y-axis +200.0 +Z-axis +1.6 +50 +0.0 +40 +2.5 +30 +y-axis +20 +7.5Z-axis +40 +y-axis +20(a) +(b) +Ground-Truth +Original Prediction +Prediction Adjusted by EWC +Figure 7. Visualization of equal-width constraint (EWC). (a) Results on 2D images after EWC adjustment. (b) Results on the x-y plane. + +0.00 +si8e z +0.10 +0.15 +0.20 +x/m0.1 +axs… +-0.3 +x/msie : +-15 +Se xot- +x/m0.10 +33 +0.35 +K/m0.0 + xis +y/m +ote +-10 X axis +%a +x/m0.1 +0.0 +xe +y/m +-0.3 +axis +-10 X axis +x/m0.10 +y/m +-0.30 +axis +-10 + axis +a +x/m0.0 +sxe +y/m +x/m \ No newline at end of file diff --git a/z9E0T4oBgHgl3EQfdACa/content/tmp_files/load_file.txt b/z9E0T4oBgHgl3EQfdACa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..859e62c916b6180c2a6fd8172a441e8fdb9fd870 --- /dev/null +++ b/z9E0T4oBgHgl3EQfdACa/content/tmp_files/load_file.txt @@ -0,0 +1,1182 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf,len=1181 +page_content='Anchor3DLane: Learning to Regress 3D Anchors for Monocular 3D Lane Detection Shaofei Huang1,2 Zhenwei Shen3* Zehao Huang3 Zihan Ding4 Jiao Dai1,2 Jizhong Han1,2 Naiyan Wang3 Si Liu4 1 Institute of Information Engineering, Chinese Academy of Sciences 2 School of Cyber Security, University of Chinese Academy of Sciences 3 TuSimple 4 Institute of Artificial Intelligence, Beihang University {nowherespyfly, zehaohuang18, zihanding819, winsty}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='com shenzhenwei@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='com {hanjizhong, daijiao}@iie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='cn liusi@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='cn Abstract Monocular 3D lane detection is a challenging task due to its lack of depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A popular solution to 3D lane detection is to first transform the front-viewed (FV) im- ages or features into the bird-eye-view (BEV) space with inverse perspective mapping (IPM) and detect lanes from BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' However, the reliance of IPM on flat ground assumption and loss of context information makes it inac- curate to restore 3D information from BEV representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' An attempt has been made to get rid of BEV and predict 3D lanes from FV representations directly, while it still under- performs other BEV-based methods given its lack of struc- tured representation for 3D lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this paper, we define 3D lane anchors in the 3D space and propose a BEV-free method named Anchor3DLane to predict 3D lanes directly from FV representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D lane anchors are projected to the FV features to extract their features which contain both good structural and context information to make ac- curate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We further extend Anchor3DLane to the multi-frame setting to incorporate temporal informa- tion for performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In addition, we also develop a global optimization method that makes use of the equal-width property between lanes to reduce the lat- eral error of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Extensive experiments on three popular 3D lane detection benchmarks show that our An- chor3DLane outperforms previous BEV-based methods and achieves state-of-the-art performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Introduction Monocular 3D lane detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' which aims at estimating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='the 3D coordinates of lane lines from a frontal-viewed im- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Work done while at TuSimple ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Front-Viewed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Image/Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='IPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='BEV Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2D Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Depth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Projecting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Dense Depth Map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Segmentation Mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Front-Viewed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Image/Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Bird-Eye-Viewed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Image/Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Lane Heights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Front-Viewed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Image/Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Anchor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3D Anchors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Iterative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='BEV Lanes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3D Lanes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3D Lanes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3D Lanes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='[ "$%& ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='[ "$%& ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='[ "$%& ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3D Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (a) BEV-based methods, which perform lane detection in the warped BEV images or features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (b) Non-BEV method, which projects 2D lane predictions back to 3D space with esti- mated depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (c) Our Anchor3DLane projects 3D anchors into FV features to sample features for 3D prediction directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' age, is one of the essential modules in autonomous driv- ing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Accurate and robust perception of 3D lanes is not only critical for stable lane keeping, but also serves as an important component for downstream tasks like high- definition map construction [21, 25], and trajectory plan- ning [1, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' However, due to the lack of depth informa- tion, estimating lanes in 3D space directly from 2D image domain still remains very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='15 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='05 2:0 0 lane] [ane2 lane3 lane42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 0.' 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0 5 20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 40 25 60 80 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' GOr 25A straightforward way to tackle the above challenges is to detect lanes from the bird-eye-viewed (BEV) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As illustrated in Figure 1(a), a common practice of BEV- based methods [5, 7, 8, 20] is to warp images or features from frontal-viewed (FV) space to BEV with inverse per- spective mapping (IPM), thereby transforming the 3D lane detection task into 2D lane detection task in BEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To project the detected BEV lanes back into 3D space, coordinates of the lane points are then combined with their corresponding height values which are estimated by a height estimation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Though proven effective, their limitations are still ob- vious: (1) IPM relies on a strict assumption of flat ground, which does not hold true for uphill or downhill cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (2) Since IPM warps the images on the basis of ground, some useful height information as well as the context information above the road surface are lost inevitably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' For example, ob- jects like vehicles on the road are severely distorted after warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Therefore, information lost brought by IPM hin- ders the accurate restoration of 3D information from BEV representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given the above limitations of BEV, some works tried to predict 3D lanes from FV directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As illustrated in Fig- ure 1(b), SALAD [39] decomposes 3D lane detection task into 2D lane segmentation and dense depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The segmented 2D lanes are projected into 3D space with cam- era intrinsic parameters and the estimated depth informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Even though getting rid of the flat ground assumption, SALAD lacks structured representations of 3D lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As a result, it is unnatural to extend it to more complex 3D lane settings like multi-view or multi-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Moreover, their performance is still far behind the state-of-the-art methods due to the unstructured representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this paper, we propose a novel BEV-free method named Anchor3DLane to predict 3D lanes directly from FV concisely and effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As shown in Figure 1(c), our Anchor3DLane defines lane anchors as rays in the 3D space with given pitches and yaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Afterward, we first project them to corresponding 2D points in FV space us- ing camera parameters, and then obtain their features by bilinear sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A simple classification head and a re- gression head are adopted to generate classification proba- bilities and 3D offsets from anchors respectively to make final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Unlike the information loss in IPM, sam- pling from original FV features retains richer context infor- mation around lanes, which helps estimate 3D information more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Moreover, our 3D lane anchors can be it- eratively refined to sample more accurate features to better capture complex variations of 3D lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Furthermore, An- chor3DLane can be easily extended to the multi-frame set- ting by projecting 3D anchors to adjacent frames with the assistance of camera poses between frames, which further improves performances over single-frame prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In addition, we also utilize global constraints to refine the challenging distant parts due to low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The moti- vation is based on an intuitive insight that lanes in the same image appear to be parallel in most cases except for the fork lanes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', distances between different point pairs on each lane pair are nearly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' By applying a global equal- width optimization to non-fork lane pairs, we adjust 3D lane predictions to make the width of lane pairs consistent from close to far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The lateral error of distant parts of lane lines can be further reduced through the above adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our contributions are summarized as follows: We propose a novel Anchor3DLane framework that di- rectly defines anchors in 3D space and regresses 3D lanes directly from FV without introducing BEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' An extension to the multi-frame setting of Anchor3DLane is also proposed to leverage the well-aligned temporal information for further performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We develop a global optimization method to utilize the equal-width properties of lanes for refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Without bells and whistles, our Anchor3DLane out- performs previous BEV-based methods and achieves state-of-the-art performances on three popular 3D lane detection benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2D Lane Detection 2D lane detection [12, 22, 24, 32, 40] aims at obtaining the accurate shape and locations of 2D lanes in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Earlier works [2, 10, 13, 36, 42] mainly focus on extract- ing low-level handcrafted features, such as edge and color information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' However, these approaches often have com- plex feature extraction and post-processing designs and are less robust under changing scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' With the development of deep learning, CNN-based methods have been explored recently and achieve notable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Segmentation- based methods [11, 23, 24, 26] formulate 2D lane detection task as a per-pixel classification problem and typically focus on how to explore more effective and semantically informa- tive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To make predictions more sparse and flexible, keypoint-based methods [15,27,35,38] model lane lines as sets of ordered keypoints and associate keypoints belonged to the same lane together by postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Apart from the above methods, anchor-based methods [17, 19, 31, 41] are also popular in 2D lane detection task due to their concise- ness and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' LineCNN [17] first defines straight rays emitted from the image boundary to fit the shape of 2D lane lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Non-Maximum Suppression (NMS) is then applied to the 2D lanes to keep only lanes with higher confi- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' LaneATT [31] develops anchor-based feature pool- ing to extract features for the 2D anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' CLRNet [41] learns to refine the initial anchors iteratively through the feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D Lane Detection Since projecting 2D lanes back into 3D space suffers from inaccuracy as well as less robustness, 3D lane detec- tion task is proposed to predict lanes in 3D space end to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Some works utilize multiple sensors, such as stereo cameras [4] and Lidar-camera [3] to restore 3D informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' However, the collection and annotation cost of multi- sensor data is expensive, restricting the practical applica- tion of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Therefore, monocular camera image based 3D lane detection [6–8,20,39] attracts more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Due to the good geometric properties of lanes in the perspective of BEV, 3DLaneNet [7] utilizes IPM to trans- form features from FV into BEV and then regresses the an- chor offsets of lanes in the BEV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' CLGo [20] trans- forms raw images into BEV images with the estimated camera pitches and heights and fits the lane lines by pre- dicting polynomial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Since IPM relies heavily on the flat ground assumption, lanes represented in BEV space may be misaligned with 3D space in rough ground cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To deal with the above issue, Gen-LaneNet [8] fur- ther makes a distinction between the virtual top view gen- erated by IPM and the true top view in 3D space for better space alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Persformer [5] utilizes deformable atten- tion to generate BEV features more adaptively and robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' SALAD [39] tries to get rid of BEV by decomposing 3D lane detection into 2D lane segmentation and dense depth estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Different from the above methods, our Anchor3DLane defines anchors in the 3D space to explicitly model 3D lanes and bridge the gap between FV space and 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The projection and sampling operations ensure the accuracy of anchor feature extraction, enabling effec- tively predicting 3D lanes directly from FV representations without introducing BEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method The overall architecture of our Anchor3DLane is il- lustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given a front-viewed image I ∈ RH×W ×3 as input, where H and W denote the height and width of the input image, a CNN backbone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', ResNet- 18 [9]) is adopted to extract 2D visual features represented in FV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To enlarge the receptive field of the net- work, we further insert a single Transformer layer [34] af- ter the backbone to obtain the enhanced 2D feature map F ∈ RHf ×Wf ×C, where Hf, Wf, and C represent the height, width and channel number of feature map respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D anchors are then projected to this feature map F with the assistance of camera parameters, and the cor- responding anchor features are sampled using bilinear in- terpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Afterward, we apply a classification head and a regression head to the sampled anchor features to make pre- dictions, with each head composed of several lightweight fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=" Furthermore, the predictions can be 𝑿𝒈 𝒀𝒈 𝒁𝒈 𝑶𝒈 𝜽 𝝓 𝑶𝒄 𝒁𝒄 𝒀𝒄 𝑿𝒄 𝒒𝟒 𝒑𝟑 𝒒𝟐 𝒒𝟏 𝒒'𝟒 𝒒'𝟑 𝒒'𝟐 𝒒'𝟏 𝒙𝒔 Road 3D Anchor Front-viewed Image/Feature Anchor Projection Camera 𝒑𝟏 𝒒𝟑 𝒑𝟐 𝒑𝟒 3D Lane Figure 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Illustration of 3D anchor and 3D lane in the ground coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' regarded as new 3D anchors for iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D Lane Representation We first revisit the representation of 3D lanes in this sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As shown in Figure 2, two different coordinate sys- tems are involved in our paper, including the camera coordi- nate system and the ground coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The camera coordinate directly corresponds with the FV image and is a right-handed coordinate system defined by origin Oc and Xc, Yc, Zc axes, with Oc located at the center of the cam- era and Zc pointing forward vertical to the camera plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D lanes are commonly annotated in the ground coordinate system, of which the origin Og is set right below Oc, x- axis Xg points positive to the right, y-axis Yg points posi- tive forwards and z-axis Zg points positive upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A 3D lane is described by 3D points with N uniformly sampled y-coordinates y = {yk}N k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Thus, we denote the i-th 3D lane as Gi = {pk i }N k=1 and its k-th point is represented as pk i = (xk i , yk, zk i , visk i ), where the first 3 elements denote the location of pk i in the ground coordinate system and the last one denotes the visibility of pk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It is worth noting that we elaborate our method based on the ground coordinate system following the common practices adopted in previ- ous works [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' However, our Anchor3DLane is able to work in an arbitrary 3D coordinate system as long as cam- era calibration is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Anchor3DLane 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 Representation of 3D Lane Anchors Our 3D lane anchors are defined in the same coordinate sys- tem as 3D lanes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', ground coordinate, for ease of posi- tion regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As illustrated in Figure 2, a 3D anchor is a ray starting from (xs, 0, 0) with pitch θ and yaw φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=" Sim- ilar to 3D lanes, we also sample N points for each anchor Transformer Layer 3D Anchors Projecting [+, 7%⟶'] … Anchor Features Iterative Regression Camera Parameters Backbone Classification Head Regression Head 4 ∈ ℝ'!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='×)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='×* 3D Proposals Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The overall architecture of Anchor3DLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given a front-viewed input image, a CNN backbone and a Transformer layer are adopted to first extract visual feature F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3D anchors are then projected to sample their features from F given camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Afterward, a classification head and a regression head are applied to make the final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The lane predictions can also serve as new 3D anchors for iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' by the same y-coordinates and represent the j-th 3D an- chor by Aj = {qk j }N k=1, and its k-th point is denoted by qk j = (xk j , yk, zk j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Different from previous works [5, 7] that define anchors in the BEV plane, our 3D anchors have pitches to the ground and could fit the lane shape better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 Anchor Projection and Feature Sampling To obtain features of 3D anchors, we first project them into the plane of FV feature F using camera parameters as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given an anchor Aj, we take its k-th point qk j as an example to explain the projection operation and omit the subscript j for simplicity as follows: � � ˜uk ˜vk dk � � = KTg→c � ��� xk yk zk 1 � ��� , (1) uk = Wf/W · ˜uk dk , (2) vk = Hf/H · ˜vk dk , (3) where K ∈ R3×3 denotes camera intrinsic parameters, Tg→c ∈ R3×4 denotes the transform matrix from ground coordinate to camera coordinate, and dk denotes the depth of qk to the camera plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Through the above formula- tions, qk is projected to position (uk, vk) in FV feature F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Finally, the feature of anchor Aj is obtained through bilin- ear interpolation within the neighborhood of the projected points and is represented as {F(uk,vk)}N k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 3D Lane Prediction We concatenate features of points belonging to the same anchor as its feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Then we apply a clas- sification head and a regression head to the anchor fea- tures for predicting classification probabilities cj ∈ RL, anchor points offsets (∆xj ∈ RN, ∆zj ∈ RN) = {(∆xk j , ∆zk j )}N k=1 and visibility of each point visj ∈ RN respectively, with j ∈ [1, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' L and M denote the num- bers of lane types and 3D anchors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this way, 3D lane proposals are generated as {Pj = (cj, xj + ∆xj, y, zj + ∆zj, visj)}M j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Furthermore, these 3D lane proposals can also serve as new anchors in the following it- erative regression steps as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Through iterative regression, proposals can be refined progressively to better fit the lane shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' During training, we associate n nearest anchors to each ground-truth lane and the rest are defined as negative sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Distance metric between ground-truth Gi and anchor Aj is calculated as follows: D(Gi, Aj) = �N k=1 visk i · � (xk i − xk j )2 + (zk i − zk j )2 �N k=1 visk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (4) This metric is also used in Non-Maximum Suppression (NMS) during inference to keep a reasonable number of proposals except that distances are calculated between visi- ble parts of two proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We adopt focal loss [18] for training classification to bal- ance the positive and negative proposals as follows: Lcls = − M � j=1 L � l=1 αl(1 − cl j)γ log cl j, (5) where αl and γ are the hyperparamters for focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The regression loss is only calculated between the positive pro- 07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='.0 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 281 4812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 015 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0 0 5 2461 461 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 80] 20posals and their assigned ground-truth lanes following [8]: Lreg = Mp � i=1 N � k=1 (∥ ˆ vis k i · (xk i + ∆xk i − ˆxk i )∥1 + Mp � i=1 N � k=1 ∥ ˆ vis k i · (zk i + ∆zk i − ˆzk i )∥1) + Mp � i=1 N � k=1 ∥ ˆ vis k i − visk i ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (6) Mp represents the total number of positive proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Here we use ˆxk i , ˆzk i and ˆ vis k i to denote the x, z coordinates and visibility of the ground-truth lane points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The total loss function of our Anchor3DLane is a com- bination of the above two losses with corresponding coeffi- cients: L = λclsLcls + λregLreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Temporal Context Modeling Thanks to the design of 3D anchors, our Anchor3DLane can be easily extended to multi-frame 3D lane detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given a 3D point (xt, yt, zt) in the t-th frame’s ground co- ordinate system, we transform it to the t′-th frame’s ground coordinate system with the following formulation: � � xt′ yt′ zt′ � � = Tg(t)→g(t′) � ��� xt yt zt 1 � ��� , (8) where Tg(t)→g(t′) ∈ R3×4 denotes the transformation ma- trix from t-th frame to t′-th frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Together with Equa- tion 1, anchors defined in the current frame can be projected to previous frames for sampling their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' For each an- chor, we take its points from the current frame as query and points from previous frames as key and value to conduct cross-frame attention for feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' By integrat- ing the well-aligned anchor features from multiple frames, temporal context is incorporated into our Anchor3DLane to enlarge its perception range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Optimization with Equal-Width Constraint In most cases, lanes in 3D space are nearly parallel with each other, which is helpful in generating robust 3D estima- tions from monocular 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this work, we lever- age this geometry property of 3D lanes and formulate it as an equal-width constraint to adjust the x-coordinates of lane predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Given two lane predictions Pj = {pk j }N k=1 and Pj′ = {pk j′}N k=1, width between Pj and Pj′ at point pair pk j and pk j′ is calculated as: wk j,j′ = | cos θk j (xk j + ˜∆xk j − xk j′ − ˜∆xk j′)|, (9) where ˜∆xk j and ˜∆xk j′ denote the adjustment to xk j and xk j′ to be optimized respectively and θk j denotes the normal di- rection of the adjusted lane at pk j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The objective function of equal-width constraint is as follows: min { ˜∆xj}j∈[1,Q] 1 Q(Q − 1) Q � j=1 Q � j′=1,j′̸=j L(wj,j′) + α 1 Q Q � j=1 ∥ ˜∆xj∥2, (10) where L(wj,j′) = N � k=1 |wk j,j′ − 1 N N � k′=1 wk′ j,j′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (11) We use Q to denote the number of lane predictions after NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' L(wj,j′) restricts the width between Pj and Pj′ to be consistent and the second item serves as a regularization to avoid the adjusted results being too far from the original predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We run this optimization as a post-processing step to refine the prediction results of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Experimental Setting 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 Datasets and Evaluation Metrics We conduct experiments on three popular 3D lane detection benchmarks, including ApolloSim [8], OpenLane [5], and ONCE-3DLanes [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' ApolloSim is a photo-realistic synthetic dataset created with Unity 3D engine which contains 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5K images from various virtual scenes, including highway, urban, residen- tial, downtown, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In addition, the data is also diverse in daytime, weather conditions, traffic/obstacles, and road sur- face qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' OpenLane is a large-scale real-world 3D lane detection dataset constructed upon the Waymo Open dataset [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It contains 200K frames and over 880K lanes are anno- tated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Camera intrinsics and extrinsics are provided for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' All lanes are annotated including lanes in the oppo- site direction if no curbside in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Categories and scene tags (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', weather and locations) are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' ONCE-3DLanes is a real-world 3D lane detection dataset with 1 million scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It consists of 211K images with labeled 3D lane points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It covers different time peri- ods (sunny, cloudy, rainy) and various regions (downtown, suburbs, highway, bridges, and tunnels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Only camera in- trinsics are provided in ONCE-3DLanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' During the evaluation, the predictions and ground truth lanes are matched via minimum-cost flow where the pair- wise cost is defined as the square root of the sum of point- wise Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A prediction is considered as true Scene Method AP(%)↑ F1(%)↑ x err/C(m) ↓ x err/F(m) ↓ z err/C(m) ↓ z err/F(m) ↓ Balanced Scene 3DLaneNet [7] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='562 Visual Variations 3D-LaneNet [7] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='235 Anchor3DLane (Ours) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='232 Anchor3DLane† (Ours) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='242 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on ApolloSim dataset with three different split settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' “C” and “F” are short for close and far respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' † denotes iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' positive if over 75% of its points’ distances to ground-truth points are less than a threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' With the defi- nition above, Average Precision (AP) and the maximum F1 score are further calculated, and x/z errors are counted sepa- rately at close (0-40m) and far (40-100m) ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We report the results of F1 score, AP, and x/z-errors on ApolloSim dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' On OpenLane dataset, except for F1 score and x/z errors, we further report category accuracy which calculates the proportion of predictions whose categories are correctly predicted to all true positive predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' ONCE-3DLanes adopts a different way to match predictions and ground truth lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The matching degree is firstly decided by IoU on the top-view plane and pairs above the threshold are further cal- culated with their unilateral Chamfer Distance (CD) as the matching error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A true positive is counted when CD is un- der the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We report F1 score, precision, recall, and CD error for results on ONCE-3DLanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 Implementation Details We choose ResNet-18 [9] as the backbone of our An- chor3DLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To maintain feature resolution, we set the downsampling stride of its last two stages to 1 and replace the 3 × 3 convolutions with dilated convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The start- ing positions xs of 3D anchors are evenly placed along the x-axis with an interval of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' For each xs, different yaws φ ∈ {0◦, ±1◦, ±3◦, ±5◦, ±7◦, ±10◦, ±15◦, ±20◦, ±30◦} and pitches θ ∈ {0◦, ±1◦, ±2◦, ±5◦} are set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The number of points N for each anchor is set to 10 for ex- periments on ApolloSim and ONCE and 20 for OpenLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We resize the image to 360 × 480 before feeding it to the backbone and the shape of F is 45 × 60 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' During train- ing, λcls and λreg are both set to 1 and the number of posi- tive proposals is set as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The distance threshold for NMS is 2 during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' For multi-frame Anchor3DLane, each time we randomly select 1 frame from the previous 5 frames to interact with current frame during training, and select the first frame of the previous 5 frames during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Since car poses are only available in OpenLane dataset, we only conduct temporal experiments on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We use Adam optimizer [14] with weight decay set as 1e−4, and set the initial learning rate to 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Step learning rate de- cay is used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' αl is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 and γ is set to 2 for focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' More details about our Anchor3DLane are included in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Quantitative Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 Results on ApolloSim Table 1 shows the experimental results under three differ- ent split settings of the ApolloSim dataset, including bal- anced scene, rare subset and visual variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We re- port the results of both our original Anchor3DLane and Anchor3DLane with iterative regression optimized with equal-width constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It is shown that our original An- Method F1(%)↑ Cate Acc(%)↑ x err/C(m) ↓ x err/F(m) ↓ z err/C(m) ↓ z err/F(m) ↓ 3D-LaneNet [7] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='443 GenLaneNet [8] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='521 PersFormer [5] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='431 Anchor3DLane (Ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='139 Anchor3DLane† (Ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='138 Anchor3DLane-T† (Ours) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='135 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on OpenLane validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' † denotes iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Anchor3DLane-T denotes incorporating multi-frame information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' “Cate Acc” means category accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method All Up & Down Curve Extreme Weather Night Intersection Merge & Split 3D-LaneNet [7] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 GenLaneNet [8] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 PersFormer [5] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 Anchor3DLane (Ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 Anchor3DLane† (Ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 Anchor3DLane-T† (Ours) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on OpenLane validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' F1 score is presented for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' † denotes iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Anchor3DLane-T denotes incorporating multi-frame information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' chor3DLane outperforms previous methods with large mar- gins on AP and F1 score on all the three splits with sim- ple design, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0% AP and +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7% F1 score on bal- anced scene, +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6% AP and +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9% F1 score on rare sub- set, +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4% F1 score and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5% AP on visual variations, showing the superiority of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our Anchor3DLane also achieves comparable or lower x/z errors compared with previous methods, especially for x error far, indicating re- gressing over 3D anchors have greater advantages for dis- tant predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Furthermore, by iteratively regressing over the proposals predicted by Anchor3DLane, x/z errors can be further reduced to better fit the shape of 3D lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 Results on OpenLane We present the experimental results of our method opti- mized with the equal-width constraint on OpenLane dataset in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our original Anchor3DLane outperforms Pers- Former by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6% F1 score improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Moreover, our method achieves much more precise predictions than Pers- Former, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='185m on x error close, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='242m on x er- ror far, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='261m on z error close, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='292m on z error far respectively, which are crucial for driving safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The gap in x/z errors indicates that under real scenarios with di- verse conditions, directly sampling features from FV repre- sentation could maintain more environment context infor- mation, thus producing more precise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' By in- corporating iterative regression and temporal information in Anchor3DLane, the overall performances can be fur- ther boosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In Table 3, we compare with previous meth- ods under different scenarios and report F1 score for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our method produces much better performance in Up&Down scenarios, showing the advantage of 3D an- chor regression in uneven ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It is also worth noting that we adopt a lightweight CNN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', ResNet-18 as the backbone of Anchor3DLane, which still outperforms Pers- Former with a larger backbone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=', EfficientNet-B7 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 Results on ONCE-3DLanes In Table 4, we present the experimental results on the ONCE-3DLanes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Since camera extrinsics are not available in ONCE-3DLanes, we define the 3D anchors in the camera coordinate system and make predictions in the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our method also achieves state-of-the-art per- formances on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Compared with PersFormer, our Anchor3DLane still produces a higher F1 score and reduces CD error by 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9% relatively, which indicates that 3D an- chors are able to adapt different 3D coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 Ablation Study In this section, we follow previous work [5] to conduct most ablation studies on OpenLane-300, which is a sub- set of OpenLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As for feature sampling experiments, we present the results on the original OpenLane to verify the ef- fectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' More ablation studies and qual- itative results are included in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method F1(%)↑ P(%)↑ R(%)↑ CD Error(m)↓ 3D-LaneNet [7] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='73 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='46 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='127 Gen-LaneNet [8] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='59 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='95 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='121 SALAD [39] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='07 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='90 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='098 PersFormer [5] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='33 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='074 Anchor3DLane (Ours) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='44 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='50 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='064 Anchor3DLane† (Ours) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='87 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='85 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='060 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on ONCE- 3DLanes validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Results under τCD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 are displayed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' † denotes iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' “P” and “R” are short for precision and recall respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Input Feat F1(%) x err/C(m) x err/F(m) z err/C(m) z err/F(m) BEV BEV 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='170 FV BEV 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='160 FV FV 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='139 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison between sampling anchor features from BEV features and FV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Sampling anchor features from FV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' To illus- trate the superiority of FV features, we compare the per- formances of extracting anchor features from FV features and BEV features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The results are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We explore different ways of obtaining BEV features, includ- ing warping FV image to BEV image (line 1) and warping FV feature to BEV feature (line 2), and keep the other set- tings same as our original Anchor3DLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Results show that sampling anchor features from FV features produces the best F1 score and x/z errors, especially for x errors, where more than 10cm gap exists between FV anchor fea- tures and BEV anchor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The above performance gap indicates that the context information contained in raw FV features is beneficial for accurate lane predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Iter F1(%) x err/C(m) x err/F(m) z err/C(m) z err/F(m) 1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='147 2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='152 3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='148 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Ablation study on the steps of iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Steps of iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Table 6 presents the results of different steps of iterative regression for An- chor3DLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Compared with no iterative regression, 2 iter- ations produces relatively large performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' More steps of iterative regression can further reduce lateral errors as well as elevate F1 score by refining the shape of proposals progressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Temporal integration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this section, we ex- plore different methods to integrate anchor features of mul- tiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Besides the cross-frame attention that we men- tioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3, we also try linear fusion which con- Method F1(%) x err/C(m) x err/F(m) z err/C(m) z err/F(m) w/o Temporal 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='147 Linear Fusion 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='148 Weighted Sum 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='150 Attention 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='145 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Ablation study on temporal integration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' catenates features of the same anchor along their channels and fuses them with a linear layer, and weighted sum which learns to predict a group of weights for each y-coordinate to fuse features of the same anchor elementwisely, As shown in Table 9, comparing with the baseline, incorporat- ing temporal information into Anchor3DLane can improve the overall performance significantly due to the richer con- text information obtained from previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Weighted sum produces better results than linear fusion, indicating that dynamic weights are necessary for different points at different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Although weighted sum achieves a bet- ter F1 score compared with single frame setting, x/z errors increase at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Among the 3 integration meth- ods, cross-frame attention, which aggregates anchor fea- tures with more anchor points from previous frames, im- proves both F1 score and x errors and achieves the best per- formance balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method F1(%) x err/C(m) x err/F(m) w/o EWC 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='349 w/ EWC 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='337 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Ablation study on Equal-Width Constraint (EWC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Effect of equal-width constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We also illustrate the comparison between predictions without and with equal- width constraint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As shown in Table 8, by applying the equal-width constraint to the lane predictions, errors of the distant parts of the lane lines can be further reduced by restricting them to have the same width as the close parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' More visualization results of this constraint can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Conclusion and Limitations In this work, we propose a novel Anchor3DLane frame- work for 3D lane detection which bypasses the transforma- tion to BEV space and predicts 3D lanes from FV directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' By defining anchors in the 3D space and projecting them to the FV features, accurate anchor features are sampled for lane prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We further extend our Anchor3DLane to the multi-frame setting to incorporate temporal information, which improves performances due to the enriched context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In addition, a global equal-width optimization method is proposed to utilize the parallel property of lanes for refine- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Experimental results show that our Anchor3DLane outperforms previous methods on three 3D lane detection benchmarks with a simple architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In this work, we leverage the equal-width constraint in an offline manner to make adjustments over lane predictions, which is inflexible for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In the future, we plan to leverage the equal-width property to facilitate the training process.' metadata={'source': 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Segformer: Simple and ef- ficient design for semantic segmentation with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 12 [38] Shenghua Xu, Xinyue Cai, Bin Zhao, Li Zhang, Hang Xu, Yanwei Fu, and Xiangyang Xue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Rclane: Relay chain pre- diction for lane detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2 [39] Fan Yan, Ming Nie, Xinyue Cai, Jianhua Han, Hang Xu, Zhen Yang, Chaoqiang Ye, Yanwei Fu, Michael Bi Mi, and Li Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Once-3dlanes: Building monocular 3d lane detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2, 3, 5, 8, 12 [40] Jiaxing Yang, Lihe Zhang, and Huchuan Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Lane detec- tion with versatile atrousformer and local semantic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Pattern Recognition, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2 [41] Tu Zheng, Yifei Huang, Yang Liu, Wenjian Tang, Zheng Yang, Deng Cai, and Xiaofei He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Clrnet: Cross layer re- finement network for lane detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2 [42] Shengyan Zhou, Yanhua Jiang, Junqiang Xi, Jianwei Gong, Guangming Xiong, and Huiyan Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' A novel lane detection based on geometrical model and gabor filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In IV, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 2 [43] Sheng Zhu and Bilin Aksun-Guvenc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Trajectory planning of autonomous vehicles based on parameterized control op- timization in dynamic on-road environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' J INTELL ROBOT SYST, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 1 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Implementation Details ApolloSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We resample 10 points for the ApolloSim dataset at y-coordinates of {5, 10, 15, 20, 30, 40, 50, 65, 80, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The training batch size is set to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We train Anchor3DLane on this dataset with one NVIDIA RTX 2080 Ti GPU for 50, 000 iterations and decay the learning rate at the 45, 000-th iteration by 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' OpenLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We resample 20 points for the OpenLane dataset at y-coordinates of {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The train- ing batch size is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We train Anchor3DLane on this dataset with eight NVIDIA RTX 2080 Ti GPUs for 60, 000 iterations and decay the learning rate at the 50, 000-th iter- ation by 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' ONCE-3DLanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We resample 10 points for the ONCE- 3DLanes dataset at y-coordinates of {2, 5, 8, 10, 15, 20, 25, 30, 40, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Since experiments are conducted in the camera coordinate system where the origin is above the ground, the starting positions of 3D anchors are set at (xs, 0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Other training settings are the same as those on the Open- Lane dataset as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Quantitative Results B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The Range of Training Frames Frame Range F1(%) x err/C(m) x err/F(m) z err/C(m) z err/F(m) 3 frames 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='148 5 frames 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='145 7 frames 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='150 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Ablation study on the range of training frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' For temporal context modeling, we sample one frame from different ranges of previous frames to aggregate its feature to the current frame during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The first frame of the previous 5 frames is sampled during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' As shown in Table 9, the F1 score increases as the frame range becomes larger, indicating that aggregating informa- tion from farther frames yields a better estimation for the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Computational Cost Analysis Method F1 Score(%) FLOPs Param FPS PersFormer [5] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4G 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='58 Anchor3DLane (ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1G 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2M 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='29 Anchor3DLane† (ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4G 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='73 Anchor3DLane-T† (ours) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3G 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3M 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='22 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison of computational cost and F1 score on OpenLane validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' † denotes iterative regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Anchor3DLane-T denotes incorporating multi-frame information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We report the computational cost comparison in Ta- ble 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our Anchor3DLane achieves a higher F1 score on the OpenLane dataset with much fewer FLOPs and param- eters compared with PersFormer [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' The inference speeds (FPS) of these methods are measured using the code re- leased by PersFormer on a single 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our orig- inal Anchor3DLane achieves nearly 16 times faster infer- ence speed than PersFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' By adopting iterative regres- sion and temporal context modeling, the F1 score is further improved, while the inference speed decreases but is still much faster than PersFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' These results demonstrate our Anchor3DLane is both effective and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Experimental Results with EfficientNet To verify the adaptability and performance potential of our method, we further conduct experiments with EfficientNet-B3 [33] to compare with PersFormer which adopts EfficientNet-B7 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Results are shown in Table 11, Table 12 and Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' On OpenLane dataset, utilizing EfficientNet-B3 as the backbone could boost the performance of our Anchor3DLane from 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1% F1 score to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0% and reduce the x/z errors at the same time, indicating that our method adapts well to stronger backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Qualitative Results ApolloSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We compare our Anchor3DLane with CLGo [20] on the ApolloSim dataset and the results are included in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our method has better lateral predic- tions in the distant parts when lanes turn in the distance (row 2 and row 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In addition, when encountering uphill (row 6) or downhill (row 4 and row 5), our method can better cap- ture the height changes than CLGo, which demonstrates the superiority of directly regressing 3D anchors for 3D lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' OpenLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We also compare with PersFormer [5] on the OpenLane dataset in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our Anchor3DLane can better recover the whole lanes occluded by vehicles as shown in column 2 of Figure 5 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' ONCE-3DLanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' In Figure 6, we show the qualita- tive results of our Anchor3DLane on the ONCE-3DLanes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Our method performs well in different scenes, such as bad weather like rainy days (column 1 of row 1 and row 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Since the 2D annotations of ONCE-3DLanes are gener- ated by the lane detection model, annotations of some cases are inaccurate or incomplete but our method still produces fine predictions as shown in column 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Equal-Width Constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' We show the visualization results of equal-width constraint (EWC) optimization in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' After adjusting the x coordinates of lanes with EWC, lane predictions are parallel to each other and errors in the distant parts are reduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' It is also worth noting that the ground-truth lanes do not satisfy the equal- width hypothesis in the close parts of some cases, which is possibly due to annotation defects (column 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Therefore, Method Backbone F1(%)↑ Cate Acc(%)↑ x err/C(m) ↓ x err/F(m) ↓ z err/C(m) ↓ z err/F(m) ↓ 3D-LaneNet [7] VGG-16 [29] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='443 GenLaneNet [8] ERFNet [28] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='521 PersFormer [5] EfficientNet-B7 [33] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='431 Anchor3DLane (Ours) ResNet-18 [9] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='139 Anchor3DLane (Ours) EfficientNet-B3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='130 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on OpenLane validation set with stronger backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method Backbone All Up & Down Curve Extreme Weather Night Intersection Merge & Split 3D-LaneNet [7] VGG-16 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 GenLaneNet [8] ERFNet 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 PersFormer [5] EfficientNet-B7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='7 Anchor3DLane (Ours) ResNet-18 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='5 Anchor3DLane (Ours) EfficientNet-B3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='3 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on OpenLane validation set with stronger backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' F1 score is presented for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Method Backbone F1 Score(%)↑ Precision(%)↑ Recall(%)↑ CD Error(m)↓ 3D-LaneNet [7] VGG-16 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='73 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='46 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='127 Gen-LaneNet [8] ERFNet 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='59 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='95 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='121 SALAD [39] SegFormer [37] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='07 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='90 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='098 PersFormer [5] EfficientNet-B7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='33 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='074 Anchor3DLane (Ours) ResNet-18 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='44 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='50 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='064 Anchor3DLane (Ours) EfficientNet-B3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='02 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='22 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='064 Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison with state-of-the-art methods on ONCE-3DLanes validation set with stronger backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' adjusting with EWC may not be beneficial to reducing x error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (a) CLGo (b) Anchor3DLane Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Comparison between CLGo [20] and our Anchor3DLane on the ApolloSim dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (a): Qualitative results of CLGo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' (b): Qualitative results of our Anchor3DLane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Blue:Ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' Red: Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='30 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E0T4oBgHgl3EQfdACa/content/2301.02371v1.pdf'} +page_content='05 0.' metadata={'source': 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